Dunham - Data Mining PDF
Dunham - Data Mining PDF
Dunham - Data Mining PDF
nd Advanced Topics
KEY FEATURES:
..
2517
Prentice Hall
Upper Saddle River, NJ 07458
www.
prenhall.com
227
1
Hail
MARGAREf H. DUNHJ
Contents
Preface
xi
Part One
1
Introduction
1.1
1.2
1.1.1
Classification . . .
1.1.2
Regression . .
1.1.3
1.1.4
Prediction . . .
1.1.5
Clustering
....
1.1.6
Summarization . .
1.1.7
Association Rules
1.1.8
Sequence Discovery
Introduction
9
12
1.3
14
1.4
15
1.5
16
1.6
16
1.7
17
1.8
Exercises ..... .
19
1.9
Bibliographic Notes.
19
Related Concepts
21
2.1
Database/OLTP Systems
21
2.2
23
2.3
Information Retrieval . . .
26
2.4
28
2.5
2.6
Dimensional Modeling . . .
2.5.1
Multidimensional Schemas
2.5.2
Indexing
29
31
34
Data Warehousing
35
2.7
OLAP ....... .
39
2.8
41
2.9
Statistics . . . . . .
41
42
44
2.1 2 Summary . . . . .
44
2.1 3 Exercises . . . . .
45
45
v
vi
3
Contents
Data Mining Techniques
46
3.1
Introduction . . . . .
4 6
3.2
4 7
3.2.1
Point Estimation . . . . .. . . .
4 7
3.2.2
5 1
3.2.3
Bayes Theorem . . . . .. .
52
3.2.4
5 4
3.2.5
5 5
3.3
Similarity Measures .
5 7
3.4
Decision Trees . . . . . . .
5 8
3.5
Neural Networks . . . . . .
61
3.5.1
64
Activation Functions
3. 6
Genetic Algorithms .
76
3.7
Exercises . . . . . .
70
3.8
Bibiographic Notes .
Part Two
4
Contents
4.1
75
4.3
4.4
4.5
4.6
Introduction
12 5
.
5 2
12 9
5.3
Outliers . . . . . . . . . . . . . .
13 0
5.4
Hierarchical Algorithms . . . . .
13 1
5.4.1
Agglomerative Algorithms .
13 2
5.4.2
Divisive Clustering . . . .
5.6
Statistical-Based Algorithms ..
80
4.2.1
Regression . . . . . . .
80
4 .2.2
Bayesian Classification .
86
Distance-Based Algorithms ..
89
4.3.1
89
4.3.2
K Nearest Neighbors
90
92
4.4.1
97
ID3 ..... .
4 . 4 .2
C 4 .5 and C5 . 0 .... .
100
4.4.3
CART . . . . . . . . . .
102
4.4.4
Scalable D T Techniques
103
103
Propagation . . . . . . . . .
105
4..
5 2
NN Supervised Learning . .
106
4.5.3
112
4.5.4
Perceptrons . . . . . . . . . .
112
Rule-Based Algorithms . . . . . . . . . .
114
.
4 6.1
114
4.6.2
115
4.6.3
116
..
119
4.8
Summary . ... . .
12 1
4.9
Exercises . . . . . .
12 1
122
13 8
Partitional Algorithms . . . . . . .
13 8
5.5.1
13 8
.
5 .
5 2
13 9
5.5.3
14 0
5.5.4
14 2
5.5.5
PAM Algorithm . . .. . . .
14 2
.
5 .
5 6
14 5
5.5.7
14 6
5.5.8
14 7
14 9
5.6.1
BIRCH . . . . . .
15 0
5.6.2
DBSCAN . . . . . .
15 2
5.6.3
CURE Algorithm . .
15 4
15 7
.
5 8
Comparison . . . . .
15 9
5.9
Exercises
. . . . ..
16 1
16 1
Association Rules
164
5.7
77
Issues in Classification .
Combining Techniques
4.7
125
5.1
5.5
73
Core Topics
75
4 .2
Clustering
71
Classification
4.1
.1
vii
6.1
Introduction .
16 4
6.2
Large Itemsets
16 7
6.3
Basic Algorithms
16 9
6.3.1
16 9
6.4
Apriori Algorithm
6.3.2
Sampling Algorithm
17 3
6.3.3
Partitioning . . . . .
17 7
17 8
6.4.1
Data Parallelism
17 9
6. 4.2
Task Parallelism
18 0
6.5
Comparing Approaches .
18 1
6.6
Incremental Rules . .
18 4
6.7
18 4
6.7.1
18 4
6.7.2
18 5
6.7.3
18 5
6.7.4
18 6
6.7.5
Correlation Rules . . . .
18 7
6.8
18 8
6.9
Exercises . . . . . .
19 0
19 1
Contents
viii
Part Three
7
193
Advanced Topics
7.2
7.3
7 4
.
19 7
7.2.2
Harvest System .
2 01
7.2.3
2 01
.
7 2.4
Personalization
2 02
204
7.3.2
Clever ... . .
.
7 4 .2
Data Structures .
2 11
Pattern Analysis
2 18
7.6
Bibliographic Notes.
2 19
221
8.1
Introduction
8.2
22 1
222
Spatial Queries
8.2.2
8.2.3
Thematic Maps .. . . .
.
8 2.4
Image Databases . . . .
8.3
8.4
8.4 .2
Generalization ..
.
8 4.3
Nearest Neighbor .
.
8 4.4
STING
8.6
8. 7
8.6 .2
..
CLARANS Extensions .
8.7.2
SD(CLARANS)
8.7.3
DBCLASD .
8.7. 4
BANG ....
8..
7 5
WaveCluster
8.7.6
Approximation
8.8
Exercises
8.9
Bibliographic Notes ..
9.3.1
25 3
25 5
.
9 3.3
Transformation
9.3.4
Similarity .
o
9 3 o5
Prediction
252
25 5
Pattern Detection ..
25 6
25 7
String Matching
25 7
Sequences . . . . .
26 0
.
9 5.1
AprioriAll..
9.5.2
SPADE ...
262
2 62
9.5.3
Generalization
9. 5.4
26 4
Feature Extraction
26 6
Intertransaction Rules
26 6
26 7
9. 6.2
9.6 03
Trend Dependencies
9.6.4
270
271
907
Exercises
9.8
Bibliographic Notes .
..
26 7
26 8
2 72
272
22 8
APPENDICES
A
22 8
Bibliographic Notes .
22 9
23 3
ID3 Extension
22 6
22 7
231
8.5
24 8
2 52
22 6
231
8.6.1
22 3
24 5
9.3.2
9.6. 5
222
8.2.1
.
8 5.1
.
9 6
2 18
Spatial Mining
8.4 . 1
Time Series . . . . . . . .
2 09
Pattern Discovery
9.3
2 08
Preprocessing .
7 4.4
.
9.5
2 06
7.4 . 1
Exercises
9.2
9.4.1
2 05
7.5
Introduction . . . . .. . . .
9.4
2 05
7.3.1
ix
245
9.1
19 8
7.2.1
7.4
.3
19 5
Introduction
Temporal Mining
195
Web Mining
7.1
Contents
234
236
236
236
23 7
238
239
24 0
24 1
2 41
24 1
24 3
24 3
Bibliography
274
28 9
290
Index
305
315
Preface
Data doubles about every year, but useful information seems to be decreasing. The area
of data mining has arisen over the last decade to address this problem. It has become
not only an important research area, but also one with large potential in the real world.
Current business users of data mining products achieve millions of dollars a year in
savings by using data minif\g techniques to reduce the cost of day to day business
operations. Data mining techniques are proving to be extremely useful in detecting and
predicting terrorism.
The purpose of this book is to introduce the reader to various data mining con
cepts and algorithms. The book is concise yet thorough in its coverage of the many
data mining topics. Clearly written algorithms with accompanying pseudocode are used
to describe approaches. A database perspective is used throughout. This means that I
examine algorithms, data structures, data types, and complexity of algorithms and space.
The emphasis is on the use of data mining concepts in real-world applications with large
database components.
Data mining research and practice is in a state similar to that of databases in the
1960s. At that time applications programmers had to create an entire database environ
ment each time they wrote a program. With the development of the relational data model,
query processing and optimization techniques, transaction management strategies, and ad
hoc query languages (SQL) and interfaces, the current environment is drastically differ
ent. The evolution of data mining techniques may take a similar path over the next few
decades, making data mining techniques easier to use and develop. The objective of this
book is to help in this process.
The intended audience of this book is either the expeiienced database professional
who wishes to learn more about data mining or graduate level computer science students
who have completed at least an introductory database course. The book is meant to
be used as the basis of a one-semester graduate level course covering the basic data
mining concepts. It may also be used as reference book for computer professionals and
researchers.
Introduction
Chl Introduction
1-
Core Topics
rl
r-H
Ch4 Classification
1- y
11-r-
Ch5 Clustering
Appendix
xi
xii
Preface
Preface
The book is divided into four major parts: Introduction, Core Topics, Advanced
Topics, and Appendix. The introduction covers background information needed to under
stand the later material. In addition, it examines topics related to data mining such as
OLAP, data warehousing, information retrieval, and machine learning. In the first chapter
of the introduction I provide a very cursory overview of data mining and how it relates
to the complete KDD process. The second chapter surveys topics related to data min
ing. While this is not crucial to the coverage of data mining and need not be read to
understand later chapters, it provides the interested reader with an understanding and
appreciation of how data mining concepts relate to other areas. To thoroughly under
stand and appreciate the data mining algorithms presented in subsequent chapters, it is
important that the reader realize that data mining is not an isolated subject. It has its basis
in many related disciplines that are equally important on their own. The third chapter
in this part surveys some techniques used to implement data mining algorithms. These
include statistical techniques, neural networks, and decision trees. This part of the book
provides the reader with
serves as
an
mining area.
The Core Topics covered are classification, clustering, and association rules. I view
xiii
temporal databases, and I have used some of the information from his dissertation in
the temporal mining chapter. Nat Ayewah has been very patient with his explanations
of hidden Markov models and helped improve the wording of that section. Zhigang Li
has introduced me to the complex world of time series and helped write the solutions
manual. I've learned a lot, but still feel a novice in many of these areas.
The students in my CSE 8 3 3 1 class (Spring 1 9 9 9 , Fall 2000, and Spring 2002) at
SMU have had to endure a great deal. I never realized how difficult it is to clearly word
algorithm descriptions and exercises until I wrote this book. I hope they learned something
even though at times the continual revisions necessary were, I'm sure, frustrating. Torsten
Staab wins the prize for findng and correcting the most errors. Students in my CSE8 3 3 1
class during Spring 2002 helped me prepare class notes and solutions to the exercises. I
thank them for their input.
My family has been extremely supportive in this endeavor. My husband, Jim, has
been (as always) understanding and patient with my odd work hours and lack of sleep.
A more patient and supportive husband could not be found. My daughter Stephanie has
put up with my moodiness caused by lack of sleep. Sweetie, I hope I haven't been too
short-tempered with you (ILYMMTYLM). At times I have been impatient with Kristina
these as the major data mining functions. Other data mining concepts (such as prediction,
but you know how much I love you. My Mom, sister Martha, and brother Dave as always
regression, and pattern matching) may be viewed as special cases of these three. In each
Some of the research required for this book was supported by the National Science
each type. Our coverage includes pseudocode for these algorithms, an explanation of
Foundation under Grant No. IIS- 9 8 208 4 1. I would finally like to thank the reviewers
(Michael Huhns, Julia Rodger, Bob Cimikowski, Greg Speegle, Zoran Obradovic,
The advanced topics part looks at various concepts that complicate data mining
applications. I concentrate on temporal data, spatial data, and Web mining. Again, algo
rithms and pseudocode are provided.
In the appendix, production data mining systems are surveyed. I will keep a more
up to data list on the Web page for the book. I thank all the representatives of the various
companies who helped me correct and update my descriptions of their products.
All chapters include exercises covering the material in that chapter. In addition to
conventional types of exercises that either test the student's understanding of the material
or require him to apply what he has learned. I also include some exercises that require
implementation (coding) and research. A one-semester course would cover the core topics
and one or more of the advanced ones.
ACKNOWLEDG MENTS
Many people have helped with the completion of this book. Tamer Ozsu provided initial
advice and inspiration. My dear friend Bob Korfhage introduced me to much of computer
science, including pattern matching and information retrieval. Bob, I think of you often.
I particularly thank my graduate students for contributing a great deal to some of
the original wording and editing. Their assistance in reading and commenting on earlier
drafts has been invaluable. Matt McBride helped me prepare most of the original slides,
many of which are still available as a companion to the book. Yongqiao Xiao helped
write much of the material in the Web mining chapter. He also meticulously reviewed
an earlier draft of the book and corrected many mistakes. Le Gruenwald, Zahid Hossain,
Yasemin Seydim, and Al Xiao performed much of the research that provided information
found concerning association rules. Mario Nascimento introduced me to the world of
T.Y. Lin, and James Buckly) for their many constructive comments. I tried to implement
as many of these I could.
PART
ONE
INTRODUCTION
CHAPTER
Introduction
1.1
1.2
1.3
1.4
1.5
1.6
1.7
THE FUTURE
1.8
EXERCISES
1.9
BIBLIOGRAPHIC NOTES
The amount of data kept in computer files and databases is growing at a phenomenal rate.
At the same time, the users of these data are expecting mo!l'e sophisticated information
from them. A marketing manager is no longer satisfied with a simple listing of marketing
contacts, but wants detailed information about customers' past purchases as well as pre
dictions of future purchases. Simple structured/query language queries are not adequate
to support these increased demands for information. Data mining steps in to solve these
needs. Data mining is often defined as finding hidden information in a database. Alterna
tively, it has been called exploratory data analysis, data driven discovery, and deductive
learning.
Traditional database queries (Figure 1.1), access a database using a well-defined
query stated in a language such as SQL. The output of tht: query consists of the data
from the database that satisfies the query. The output is usually a subset of the database,
but it may also be an extracted view or may contain aggregations. Data mining access
of a database differs from this traditional access in several ways:
Query: The query might not be well formed or precisely stated. The data miner
might not even be exactly sure of what he wants to see.
Data: The data accessed is usually a different version from that of the original
operational database. The data have been cleansed and modified to better support
the mining process.
Output: The output of the data mining query probably is not a subset of the
database. Instead it is the output of some analysis of the contents of the database.
The current state of the art of data mining is similar to that of database query processing
in the late 1960s and early 1970s. Over the next decade there undoubtedly will be great
3
Chapter 1
Section 1.1
Introduction
SQL
Q I
DBMS
( }
Data mining
- Ds
---Predictive
Results
Descriptive
-------
A predictive model makes a prediction about values of data using known results
found from different data. Predictive modeling may be made based on the use of
other historical data. For example, a credit card use might be refused not because of
application.
the user's own credit history, but because the current purchase is similar to earlier
purchases that were subsequently found to be made with stolen cards. Example 1.1
EXAMPL1.1
uses predictive modeling to predict the credit risk. Predictive model data mining tasks
include classification, regression, time series analysis, and prediction. Prediction may
Credit card companies must determine whether to authorize credit card purchases. Sup
pose that based on past historical information about purchases, each purchase is placed
section 1.1.4.
into one of four classes: (1) authorize, (2) ask for further identification before authoriza
tion, (3) do not authorize, and (4) do not authorize but contact police. The data mining
functions here are twofold. First the historical data must be examined to determine how
model, a descriptive model serves as a way to explore the properties of the data examined,
the data fit into the four classes. Then the problem is to apply this model to each new
not to predict new properties. Clustering, summarization, association rules, and sequence
purchase. Although the second part indeed may be stated as a simple database query, the
Data mining involves many different algorithms to accomplish different tasks. All
follow the basic outline of tasks shown in Figure 1.2. This list is not intended to be
of these algorithms attempt to fit a model to the data. The algorithms examine the data
exhaustive, but rather illustrative. Of course, these individual tasks may be combined to
and determine a model that is closest to the characteristics of the data being examined.
Preference: Some criteria must be used to fit one model over another.
In Example 1.1 the data are modeled as divided into four classes. The search requires
examining past data about credit card purchases and their outcome to determine what
criteria should be used to define the class structure. The preference will be given to
criteria that seem to fit the data best. For example, we probably would want to authorize
a credit card purchase for a small amount of money with a credit card belonging to a
long-standing customer. Conversely, we would not want to authorize the use of a credit
card to purchase anything if the card has been reported as stolen. The search process
requires that the criteria needed to fit the data to the classes be properly defined.
As seen in Figure 1.2, the model that is created can be either predictive or descrip
1.1.1 i
Classification
Classification maps data into predefined groups or classes. It is often referred to as
supervised learning because the classes are determined before examining the data. Two
examples of classification applications are determining whether to make a bank loan and
identifying credit risks. Classification algorithms require that the classes be defined based
on data attribute values. They often describe these classes by looking at the character
istics of data already known to belong to the classes. Pattern recognition is a type of
classification where an input pattern is classified into one of several classes based on
its similarity to these predefined classes. Example 1.1 illustrates a general classification
problem. Example 1.2 shows a simple example of pattern recognition.
EXAMPLE 1.2
An airport security screening station is used to determine: if passengers are potential
tive in nature. In this figure, we show under each model type some of the most common
terrorists or criminals. To do this, the face of each passenger is scanned and its basic
pattern (distance between eyes, size and shape of mouth, shape of head, etc.) is identified.
Chapter 1
Introduction
Section 1.1
This pattern is compared to entries in a database to see if it matches any patterns that
are associated with known offenders.
---o-X
__
1.1.2
.,___
--z
Regression
al
ity, regression involves the learning of the function that does t is mappig. Regre si?n
assumes that the target data fit into some known type of functiOn (e.g., linear, logistic,
etc.) and then determines the best function of this type that models the given data. orne
type of error analysis is used to determine which function is "best." standard hnear
.
regression, as illustrated in Example 1.3, is a simple example of regressiOn.
EXAMPLE 1.3
A college ptofessor wishes to reach a certain level of savings before her retirement.
. urret value
Periodically, she predicts what her retirement savings will be based on Its
1.1.4
and several past values. She uses a simple linear regression foula to predict this value
.
.
by fitting past behavior to a linear function and then using this functiOn
to ?redict the
Prediction
Many real-world data mining applications can be seen as predicting future data states
based on past and current data. Prediction can be viewed as
values at points in the future. Based on these values, she then alters her mvestment
This is a data mining task that is different from the prediction model, although the pre
portfolio.
diction task is a type of prediction model.) The difference is that prediction is predicting
a future state rather than a current state. Here we are referring to a type of application
rather than to a type of data mining modeling approach, as discussed earlier. Prediction
1.1.3
applications include flooding, speech recognition, machine learning, and pattern recog
nition. Although future values may be predicted using time series analysis or regression
With time series analysis, the value of an attribute is examined as it varies over time. The
techniques, other approaches may be used as well. Example 1.5 illustrates the process.
values usually are obtained as evenly spaced time points (daily, weeki, hourly, etc.). A
1.3),
EXAMPLE 1.5
easily see that the plots for Y and Z have similar behavior, while X appears to have less
volatility. There are three basic functions performed in time series analysis In on case,
:
.
distance measures are used to determine the similarity between different tlme senes. In
the second case, the structure of the line is examined to determine (and perhaps classi y)
its behavior. A third application would be to use the historical time series plot to predict
future values. A time series example is given in Example
1.4.
EXAMPLE 1.4
Mr. Smith is trying to determine whether to purchase stock from Companies X, Y,
or z. For a period of one month he charts the daily stock price for eah copany.
Figure
1.3
shows the time series plot that Mr. Smith ha geneated . Usmg this and
similar information available from his stockbroker, Mr. Sllllth decides to purchase stock
X because it is less volatile while overall showing a slightly larger relative amount of
growth than either of the other stocks . As a matter of fact, the to cks or Y and Z have
.
a similar behavior. The behavior of Y between days 6 and 20 IS Identical to that for Z
between days
13
and 27.
1.1.5
Clustering
Clustering is similar to classification except that the groups are not predefined, but rather
defined by the data alone. Clustering is alternatively referred to as unsupervised learn
ing or segmentation. It can be thought of as partitioning or segmenting the data into
groups that might or might not be disjointed. The clustering is usually accomplished by
determining the similarity among the data on predefined attributes. The most similar data
are grouped into clusters. Example
1.6
clusters are not predefined, a domain expert is often required to interpret the meaning of
the created clusters.
Chapter 1
Section 1.2
Introduction
products are frequently purchased with bread. He finds that 60% of the time that bread is
sold so are pretzels and that 70% of the time jelly is also sold. Based on these facts, he
EXAMPLE 1 . 6
tries to capitalize on the association between bread, pretzels, and jelly by placing some
A certain national department store chain creates special catalogs targeted t o various
pretzels and jelly at the end of the aisle where the bread is placed. In addition, he decides
demographic groups based on attributes such as income, location, and physical charac
teristics of potential customers (age, height, weight, etc.). To determine the target mailings
of the various catalogs and to assist in the creation of new, more specific catalogs, the
company performs a clustering of potential customers based on the determined attribute
Users of association rules must be cautioned that these are not causal relation
values. The results of the clustering exercise are then used by management to create
ships. They do not represent any relationship inherent in the actual data (as is true with
special catalogs and distribute them to the correct target population based on the cluster
that this association will apply in the future. However, association rules can be used to
assist retail store management in effective advertising, marketing, and inventory control.
1.1.8
Sequence Discovery
associations in that data (or events) are found to be related, but the relationship is based
1.1.6
Summarization
on time. Unlike a market basket analysis, which requires the items to be purchased at
the same time, in sequence discovery the items are purchased over time in some order.
Summarization maps data into subsets with associated simple descriptions. Summariza
tion is also called characterization or generalization. It extracts or derives representative
Example 1.9 illustrates the discovery of some simple patterns. A similar type of discovery
can be seen in the sequence within which data are purchased. For example, most people
information about the database. This may be accomplished by actually retrieving portions
who purchase CD players may be found to purchase CDs within one week. As we will
of the data. Alternatively, summary type information (such as the mean of some numeric
attribute) can be derived from the data. The summarization succinctly characterizes the
contents of the database. Example 1.7 illustrates this process.
EXAMPLE 1.9
EXAMPLE 1.7
The Webmaster at the XYZ Corp. periodically analyzes the Web log data to determine
how users of the XYZ's Web pages access them. He is interested in determining what
is the average SAT or AC T score [GM99]. This is a summarization used to estimate the
sequences of pages are frequently accessed. He determines that 70 percent of the users
of page
or
1.1.7
Association Rules
1.2
data mining task of uncovering relationships among data. The best example of this
The terms
interchangeably. In fact, there have been many other names given to this process of
knowledge discovery in databases (KDD) and data mining are often used
identifies specific types of data associations. These associations are often used in the retail
sales community to identify items that are frequently purchased together. Example 1.8
illustrates the use of association rules in market basket analysis. Here the data analyzed
Over the last few years KDD has been used to refer to a process consisting of many
consist of information about what items a customer purchases. Associations are also used
steps, while data mining is only one of these steps. This is the approach taken in this
book. The following definitions are modified from those found in [FPSS96c, FPSS96a].
DEFINITION 1.1. Knowledge discovery in databases (KDD) is the process of
EXAMPLE 1.8
A grocery store retailer is trying to decide whether to put bread on sale. To help determine
the impact of this decision, the retailer generates association rules that show what other
10
Introduction
Chapter 1
Section 1.2
The KDD process is often said to be nontrivial; however, we take the larger view that
KDD is an all-encompassing concept. A traditional SQL database query can be viewed
as the data mining part of a KDD process. Indeed, this may be viewed as somwhat
simple and trivial. However, this was not the case 30 years ago. If we were to advance
30 years into the future, we might find that processes thought of today as nontrivial and
complex will be viewed as equally simple. The definition of KDD includes the keyword
useful. Although some definitions have included the term "potentially useful," we believe
that if the information found in the process is not useful, then it really is not information.
Of course, the idea of being useful is relative and depends on the individuals involved.
'
KDD is a process that involves many different steps. The input to this process is
the data, and the output is the useful information desired by the users. However, the
objective may be unclear or inexact. The process itself is interactive and may require
11
modified to facilitate use by techniques that require specific types of data distributions.
Some attribute values may be combined to provide new values, thus reducing the com
plexity of the data. For example, current date and birth date could be replaced by age.
One attribute could be substituted for another. An example would be replacing a sequence
of actual attribute values with the differences between consecutive values. Real valued
attributes may be more easily handled by partitioning the values into ranges and using
these discrete range values. Some data values may actually be removed. Outliers, extreme
values that occur infrequently, may actually be removed. The data may be transformed
by applying a function to the values. A common transformation function is to use the log
of the value rather than the value itself. These techniques make the mining task easier by
reducing the dimensionality (number of attributes) or by reducing the variability of the
data values. The removal of outliers can actually improve the quality of the results. As
much elapsed time. To ensure the usefulness and accuracy of the results of the process,
with all steps in the KDD process, however, care must be used in performing transfor
interaction throughout the process with both domain experts and technical experts might
mation. If used incorrectly, the transformation could actually change the data such that
be needed. Figure 1.4 (modified from [FPSS96c]) illustrates the overall KDD process.
Selection: The data needed for the data mining process may be obtained from
many different and heterogeneous data sources. This first step obtains the data
from various databases, files, and nonelectronic sources.
Preprocessing: The data to be used by the process may have incorrect or miss
ing data. There may be anomalous data from multiple sources involving different
Visualization refers to the visual presentation of data. The old expression "a picture
is worth a thousand words" certainly is true when examining the structure of data. For
example, a line graph that shows the distribution of a data variable is easier to understand
and perhaps more informative than the formula for the corresponding distribution. The use
of visualization techniques allows users to summarize, extra.ct, and grasp more complex
results than more mathematical or text type descriptions of the results. Visualization
techniques include:
data types and metrics. There may be many different activities performed at this
time. Erroneous data may be corrected or removed, whereas missing data must be
format for processing. Some data may be encoded or transformed into more usable
Sltion
Prepro=&og
"'"'formtioo
Dt. m
Initial
Target
Preprocessed
Transformed
data
data
data
data
hU
lot<or><ot.tion
Model
Pixel-based: With these techniques each data value is shown as a uniquely colored
Hierarchical: These techniques hierarchically divide the display area (screen) into
Transformation techniques are used to make the data easier to mine and more use
ful, and to provide more meaningful results. The actual distribution of the data may be
Icon-based: Using figures, colors, or other icons can improve the presentation of
pixel.
Interpretation/evaluation: How the data mining results are presented to the users
Various visualization and GUI strategies are used at this last step.
the results.
Data mining: Based on the data mining task being performed, this step applies
Geometric:
techniques.
formats. Data reduction may be used to reduce the number of possible data values
being considered.
Graphical: Traditional graph structures including bar charts, pie charts, histograms,
Knowledge
tools can be used to summarize data as a data mining technique itself. In addition,
visualization can be used to show the complex results of data mining tasks.
The data mining process itself is complex. As we will see in later chapters, there
are many different data mining applications and algorithms. These algorithms must be
carefully applied to be effective. Discovered patterns must be correctly interpreted and
properly evaluated to ensure that the resulting information is meaningful and accurate.
Section 1 .2
12
Chapter 1
Information
retrieval
Databases
Statistics
Time
Area
Contribution
Reference
Late 1 700s
Stat
[Bay63]
Early 1 900s
Stat
Regression analysis
Early 1 920s
Stat
Early 1 940s
AI
Neural networks
[MP43]
Nearest neighbor
[FJ5 1 ]
Early 1 950s
Algorithms
Machine
learning
Single link
[FLP+ 5 1 ]
Perceptron
[Ros58]
Late 1 950s
Stat
Early 1 960s
AI
ML started
Early 1 960s
DB
Batch reports
result of years of
g functions and products is the
The current evolution of data minin
val, statistics,
retrie
n
matio
infor
ases,
including datab
influence from many disciplines,
area that has
ce
scien
uter
comp
er
Anoth
.5).
(Figure 1
algorithms, and machine learning
goal of KDD
r
ss is multimedia and graph ics. A majo
had a majo r impact on the KDD proce
er. Because
the KDD process in a meaningful mann
is to be able to describe the results of
lization
Visua
em.
probl
ced, this is a nontrivial
many different results are often produ
on,
additi
In
ns.
ntatio
prese
ics
ed multimedi a and graph
techniques often involve sophisticat
s.
ation
applic
d to multimedia
data mining techniques can be applie
datab ase
disparate areas , a majo r trend in the
Unlike previous research in these
one
into
lines
from these seemingly different discip
community is to combine results
this
of
goal
ate
ultim
ach. Although in its infancy, the
unifying data or algorithmi c appro
of
ration
integ
tate
facili
will
re" view of the area that
evolution is to develop a "big pictu
ins.
doma
real-world user
the various types of applications into
information
the areas of artificial intelligence (AI),
in
nts
opme
devel
s
show
1
.
1
Table
view of data
nt
curre
the
to
ng
leadi
(Stat)
statistics
retrieval (IR), databases (DB) , and
development of the
influences, which have led to the
ical
histor
ent
differ
These
g.
minin
mining functions
rise to different views of what data
total data mining area, have given
tics
mining is to describe some characteris
Because the primary objective of data
com
of
type
a
as
d
viewe
be
this approach can
of a set of data by a general model,
ressed
the database are abstracted and comp
within
data
ed
detail
the
Here
.
press ion
l.
mode
the
in
found
are
that
tics
characteris
to a smaller description of the data
ing
ss itself can be viewed as a type of query
As stated earlier, the data mining proce
g research is
minin
data
of
ion
direct
ng
ongoi
an
the underlying datab ase. Indeed,
Decision trees
[HMS66]
[Nil65]
IR
Similarity measures
IR
Clustering
Stat
Late 1 960s
DB
[Cod70]
Early 1 970s
IR
SMART IR systems
[Sal7 1 ]
Mid 1 970s
AI
Genetic algorithms
[Hol75]
Late 1 970s
Stat
[DLR77]
Late 1 970s
Stat
K-means clustering
Early 1 980s
AI
(Koh82]
Mid 1 980s
AI
[Qui86]
Early 1 990s
DB
1 990s
DB
Data warehousing
1 990s
DB
how to define a data mining query and whether a query language (like SQL) can
be developed to capture the many different types of data mining queries.
[FF63]
Stat
Mid 1 960s
Mid 1 960s
[Fis2 1 ]
AI
Early 1 950s
Late 1 950s
1 .2.1
13
I ntroduction
When dealing with large databases, the impact of size and efficiency of developing
an abstract model can be thought of as a type of search problem.
It is intresting to t nk about the various data mining problems and how each may be
.
VIewed m several different perspectives based on the viewpoint and background of the
"W_
14
Chapter 1
Introduction
Section 1 .4
apprehension among the different players. You can see statisticians voice cocern over
15
6. Large datasets : The massive datasets associated with data mining create problems
the compounded use of estimates (approximation) with results being generalized when
when applying algorithms designed for small datasets. Many modeling applica
they should not be. Database researchers often voice concern about the inefficiency of
tions grow exponentially on the dataset size and thus are too inefficient for larger
many proposed AI algorithms, particularly when used on very large databases. IR and
datasets. Sampling and parallelization are effective tools to attack this scalability
those interested in data mining of textual databases might be concerned about the fact
problem.
that many algorithms are targeted only to numeric data. The approach taken in this book
is to examine data mining contributions from all these different disciplines together.
different attributes. The problem here is that not all attributes may be needed to
There are at least two issues that characterize a database perspective of examining
solve a given data mining problem. In fact, the use of some attributes may interfere
data mining concepts: efficiency and scalability . Any solutions to data mining problems
with the correct completion of a data mining task. The use of other attributes may
must be able to perform well against real-world databases. As part of the efficiency, we
simply increase the overall complexity and decrease the efficiency of an algorithm.
are concerned about both the algorithms and the data structures used. Parallelization may
be used to improve efficiency. In addition, how the proposed algorithms behave as the
associated database is updated is also important. Many proposed data mining algorithms
which ones should be used. One solution to this high dimensionality problem is
may work well against a static database, but they may be extremely inefficient as changes
are made to the database. As database practitioners, we are interested in how algorithms
However, determining which attributes not needed is not always easy to do.
perform against very large databases, not "toy" problems. We also usually assume that
8. Multimedia data: Most previous data mining algorithms are targeted to traditional
data types (numeric, character, text, etc.). The use of multimedia data such as is
found in GIS databases complicates or invalidates many proposed algorithms.
1.3
9. Missing data: During the preprocessing phase of KDD, missing data may be
replaced with estimates. This and other approaches to handling missing data can
There are many important implementation issues associated with data mining :
10. Irrelevant data: Some attributes in the database might not be of interest to the
1. Human interaction: Since data mining problems are often not precisely stated,
interfaces may be needed with both domain and technical experts. Technical experts
11. Noisy data: Some attribute values might be invalid or incorrect. These values are
are used to formulate the queries and assist in interpreting the results. Users are
12. Chan ging data: Databases cannot be assumed to be static. However, most data
mining algorithms do assume a static database. This requires that the algorithm be
state it is desirable that the model also fit future database states. Overfitting occurs
whe
the model does not fit future states. This may be caused by assumptions that
are made about the data or may simply be caused by the small size of the training
13. Integration: The KDD process is not currently integrated into normal data pro
needs. This makes them inefficient, ineffective, and not general enough to be used
is quite small, the model might erroneously indicate that a short person is anyone
under five feet eight inches because there is only one entry in the training database
under five feet eight. In this case, many future employees would be erroneously
14. Application: Determining the intended use for the information obtained from the
classified as short. Overfitting can arise under other circumstances as well, even
data mining function is a challenge. Indeed, how business executives can effectively
use the output is sometimes considered the more difficult part, not the running of
3. Outliers: There are often many data entries that do not fit nicely into the derived
the algorithms themselves. B ecause the data are of a type that has not previously
model. This becomes even more of an issue with very large databases. If a model
is developed that includes these outliers , then the model may not behave well for
5. Visualization of results : To easily view and understand the output of data mining
algorithms, visualization of the results is helpful.
1 .4
16
Chapter 1
Introduction
Section 1.7
also based on the interest level. From an overall business or usefulness perspective, a
measure such as
of data mining applications and techniques. However, we are interested primarily in those
between what the data mining technique costs and what the savings or benefits from
that are of practical interest. While our interest is not limited to any particular type of
its use are. Of course, this would be difficult to measure because the return is hard to
percentage of catalog recipients and the amount of urchase per rectptent would provtde
.
one means to measure the effectiveness of the mmhngs.
measure various data mining approaches. We assume that the business management
has determined that a particular data mining application be made. They subsequently
Real-world data: Real-world data are noisy and have many missing attribute
values. Algorithms should be able to work even in the presence of these problems.
will determine the overall effectiveness of the approach using some ROI (or related)
data mining task. T he metrics used include the traditional met cs of s?ace and ttme
.
based on complexity analysis. In some cases, such as accuracy m classtficatwn, more
Update: Many data mining algorithms work with static datasets. This is not a
realistic assumption.
Ease of use: Although some algorithms may work well, they may not be well
received by users if they are difficult to use or understand.
These issues are crucial if applications are to be accepted a:nd used in the workplace.
T hroughout the text we will mention how techniques perforn1 in these and other imple
The integration of data mining techniques into normal day-to - ay activities has become
.
.
commonplace. We are confronted daily with targeted adverttsmg, and busmesses have
mentation categories.
Data mining today is in a similar state as that of databases in the early 1960s. At
become more efficient through the use of data mining activities to reduce costs. Data
mining adversaries, however, are concerned that this informati n is being obtained
that time, each database application was implemented independently even though there
the cost of reduced privacy. Data mining applications can denve mch d mographtc
information concerning customers that was previously not known or
were many similarities between different applications. In the mid 1960s, an abundance
of database management systems (DBMS) like tools (such as bill of material systems
including DBOMP and CFMS) emerged. While these made the development of applica
The unauthorized use of such data could result in the disclosure of mformat10n that ts
deemed to be confidential.
tions easier, there were still different tools for different applications. The rise of DBMS
occurred in the early 1970s. Their success has been due partly to the abstraction of data
definition and access primitives to a small core of needed requirements. This abstraction
process has yet to be performed for data mining tasks. Each task is treated separately.
Most data mining work (to date) has focused on specific algorithms to realize each indi
involved. Indeed, many classification techniques work by tdenttfymg the attnbute values
.
that commonly occur for the target class. Subsequent records will be then cl ssified
.
based on these attribute values. Keep in mind that these approaches to classificatiOn are
vidual data mining task. T here is no accepted abstraction to a small set of primitives.
akes
card purchases that are similar to those often made when a card
IS
One crucial part of the database abstraction is query processing support. One reason
relational databases are so popular today is the development of SQL. It is easy to use
a series of credit
(at least when compared with earlier query languages such as the DBTG or IMS DML)
and has become a standard query language implemented by all major DBMS vendors.
SQL also has well-defined optimization strategies. Although there currently is no corre
Users of data mining techniques must be sensitive to these issues and must not
sponding data mining language, there is ongoing work in the area of extending SQL to
1 .6
Scalability: Algorithms that do not scale up to perform well with massive real
world datasets are of limited application. Related to this is the fact that techniques
1 .5
17
The study of data mining from a database perspective involves looking at all types
The Future
might look primarily at the historical techniques, includi g time eries an lsis,
ypoth
esis testing, and applications of Bayes theorem; a machme learrung sp cialist might be
interested primarily in data mining algorithms that learn; and an algonthms researc er
would be interested in studying and comparing algorithms based on type and complexity.
1 .7
THE FUTURE
The advent of the relational data model and SQL were milestones
in the evolution of
database systems. Currently, data mining is little more than a set of tools that
can be used
to uncover previously hidden information in a database. While there are
many tools to aid
in this process, there is no all-encompassing model or approach.
Over the next few years,
not only will there be more efficient algorithms with better interface techniques,
but also
steps will be taken to develop an all-encompassing model for data mining
. While it may
18
Chapter 1
Introduction
Section 1 . 9
lorithms,
rung tools
re uire much human interction not only to define the rquest, bu also to merpret he
re;ults. As the tools become better and more integrated, his extensive burna mteractwn
is likely to decrease. The various data mining applictwns are of man diverse types,
.
so the development of a complete data mining model ts desrrable. A maJ or e:'elopment
will be the creation of a sophisticated "query anguage" that include tradittonat s:;-L
B i bl iographic N otes
19
not look like the relational model, it probably will include similar items:
all steps in the KDD process, including the maintenance of the results of the data mining
data model, and metrics for goodness (like normal forms). Current data
step. The CRISP-DM life cycle contains the following steps: business understanding,
data understanding, data preparation, modeling, and evaluation deployment. The steps
involved in the CRISP-DM model can be summarized as the "the 5As:" assess, access,
analyze, act, and automate.
1 .8
on ne
the retrieved data need not be a subset or aggregate of data from relatiOns . Thus, a MQL
.
statement must indicate the type of knowledge to be mined. Another dtfference IS that
.
a DMQL statement can indicate the necessary importance dt threshol that any mmed
tf
EXERCISES
{DMQL) : :
USE DATABASE {database _name)
{USE HIERARCHY {hi erarchy_name) FOR {attribute ) }
{ru l e_spec)
RELATED TO {attr_or_agg_l i s t )
FROM {relation ( s ) )
[WHERE (condit ion)]
[ORDER B Y {order -l i st)]
{threshold_value)
{WITH [{kinds_o f ) ] THRESHOLD
[ FOR {attr ibute ( s ) )] }
=
The heart of a DMQL statement is the rule specification portion. This is where the
1 .9
true data mining request is made. The data mining request can be one of the following [HFW+96] :
..
The term knowledge and data discovery management system (KDDMS) has been
_
coined to describe the future generation of data mining systems that mclu e not o ly data
Its constst ncy
ensur
,
mining tools but also techniques to manage the underlying dat
d
.M_S will provide access vta a
hoc data mining queries that have been optimized for effictent access.
A
Mining)
BIBLIOGRAPHIC NOTES
Although many excellent books have been
published that examine data mining and
knowledge discovery in databases, most
are high-level books that target users of
data
mining techniques and business professional
s. There have been, however, some other
technical books that examine data minin
g approaches and a1gorithms. An excelle
nt text
that is written by one of the foremost expert
s in the area is Data Mining Concepts and
Techniques by Jiawei Han and Micheline
Katnber [HKOl ] . 1bis book not only exami
nes
data mining algorithms, but also includ
es a thorough coverage of data wareh
ousing,
OLAP, preprocessing, and data mining langua
ge developments. Other books that provid
e
a technical survey of portions of data minin
g algorithms include [AdaOO] and [IiMS
Ol].
There have been several recent surveys
and overviews of data mining, including
special issues of the Communications of
the ACM in November 1 996 and Novem
ber
1 999 , IEEE Transactions on Knowledge
and Data Engineering in December 1 996,
and
Computer in August 1 999. Other survey
articles can be found: [FPSS 96c], [FPSS
96b],
[GGR99a], [Man9 6], [Man9 7], and [RG99
] . A popular tutorial booklet has been produ
ced
by Two Crows Corporation [Cor99] . A
complete discussion of the KDD process
is found
in [BA96 ]. Articles that examine the
intersection between databases and data
mining
include [Cha97], [Cha9 8], [CHY96], [Fay9
8], and [HKMT95] . There have also been
20
Chapter 1
Introduction
several tutorials surveying data mining concepts: [Agr94], [Agr95], [Han96] , and [RS99].
C H A P T E R
The aspect of parallel and distributed data mining has become an important research
topic. A workshop on Large-Scale Parallel KDD Systems was held in 1999 [ZHOO].
The idea of developing an approach to unifying all data mining activities has
been proposed in [FPSS96b], [Man96], and [Man97]. The term KDDMS was first pro
posed in [IM96]. A recent unified model and algebra that supports all maj or data mining
2.1
tasks has been proposed [JLNOO]. The 3W model views data as being divided into three
2.2
dimensions. An algebra, called the dimension algebra, has been proposed to access this
three-dimensional world.
There are several KDD and data mining resources. The ACM (Association for
2.3
IN FORMATION RETRIEVAL
2.4
2.5
2.6
Computing Machinery) has a special interest group, SIGKDD, devoted to the promotion
and dissemination of KDD information. SIGKDD Explorations is a free newsletter pro
duced by, ACM SIGKDD. The ACM SIGKDD home page contains a wealth of resources
DATA WAREHOUSING
2.7
OLAP
2.8
2.9
DATABASE/OLTP SYSTEMS
STATISTICS
2. 1 1 PATTERN MATCHING
data mining standards. Information about DMG can be found at www.dmg. org. The
ISO/IEC standards group has created a final committee draft for an SQL standard includ
2 . 1 3 EXERCISES
ing data mining extensions [ComO l ] . In addition, a proj ect begun by a consortium of
data mining vendors and users resulted in the development of the data mining process
model, CRISP-DM (see: www.crisp-dm.org).
There currently are several research journals related to data mining. These include
IEEE Transactions on Knowledge and Data Engineering published l:>y IEEE Computer
Society and Data Mining and Knowledge Discovery from Kluwer Academic Publish
ers. International KDD conferences include the ACM SIGKDD International Conference
on Knowledge Discovery and Data Mining (KDD), the Conference on Information and
Data mining applications have existed for thousands of years. For example, the classifi
cation of plants as edible or nonedible is a data mining task. The development of the data
(ICDM), the European Conference on Principles of Data Mining and Knowledge Dis
mining discipline has its roots in many other areas. In this chapter we examine many
covery (PKDD), and the Pacific-Asia Conference on Knowledge Discovery and Data
concepts related to data mining. We briefly introduce each concept and indicate how it
It contains a wealth of KDD and data mining information for practitioners, users, and
researchers. Subscriptions are free at www.kdnuggets.com. Additional KDD resources
can be found at Knowledge Discovery Central (www.kdcentral.com).
2.1
DATABASE/OLTP SYSTEMS
A database i s a collection of data usually associated with some organization or enterprise.
Unlike a simple set, data in a database are usually viewed to have a particular structure
or schema with which it is associated. For example, (/D, Name, Address, Salary, JobNo)
may be the schema for a personnel database. Here the schema indicates that each record
(or tuple) in the database has a value for each of these five attributes. Unlike a file, a
database is independent of the physical method used to store it on disk (or other media).
It also is independent of the applications that access it. A database management system
(DBMS) is the software used to access a database.
Data stored in a database are often viewed in a more abstract manner or data
model. This data model is used to describe the data, attributes, and relationships among
them. A data model is independent of the particular DBMS used to implement and
access the database. In effect, it can be viewed as a documentation and communication
tool to convey the type and structure of the actual data. A common data model is the
21
Section 2.2
Re lated Concepts
Chapter 2
22
23
SELECT Name
FROM R
WHERE Salary >
100,000
Figure 2.2 shows a sample SQL statement issued against the relation
joins, aggregates, and views. Traditional database queries usually involve retrieving data
from a database based on a well-defi ned query. As shown in Figure 2.2, a user may
ask to find all employees wh o earn over $ 100, 000. This could be viewed as a type of
classifi cation application as we segment the database into two classes: those who have
salaries satisfying the predicate and those who do not. A simple database application
is not thought of as a data mining task, however, because the queries are well defined
the r latioship_s.
with precise results. Data mining applicati ons, conversely, are often vaguely defined with
entity is ass ociated with a real-world object and has a key that umquely tdenttfies tt.
relationship is used to describe as ass ociation that exists between entities.
imprecise results. Users might not even be able to precisely defi ne what th ey want, let
alone be able to tell if the results of thei r request are accurate. A database user usually
can tell if the results of his query are not correct. Thus, it is usually assumed that a
EXAMPLE 2 . 1
DBMS returns the correct results for a query. Metrics (instead of quality) often include
An employee database consists of emp loyees and information concerning the j_ob that
they perform. An entity would be an Employee and the key could be the ID
. Stmt larly,
different jobs can be associated with a job number so that we can think of the Job as an
Data mining problems are often ill-posed with many diffe rent s olutions. Judging the
When viewed as a query sys tem, data mining queries extend database concepts.
entity with key JobNo. In Figure 2.1, there is a rectangle for each entity. The diamond is
used to represent the relationship between the two entities. Here the relationship HasJob
between data mining queries and th ose of database s ystems is the output. Basic databas e
indicates that a specific employee with key ID has a particular job with key JobNo. The
queries always output either a subset of the database or aggregates of the data. A data
attributes associated with Employee are {ID, Name, Address, Salary} and the attributes
cluster. These objects do not exis t before executing the query, and they are not part of the
database being queried. Aggregation operators have existed in SQL for years. They do
not return objects existing in the database, but return a model of the data. For example,
The ER model is often used to abstractly view the data independent of DBMS.
an average operator returns the average of a set of attribute values rather than the values
DBMS sys tems often view the data in a structure more like a table. This gives ris e to
relational model, where data are viewed as being composed of relations. Taking a
mathematical pers pective, a relation is a subset of a Cartesian product
. I magine looking
at the domain, or set of values, associated with each atttibute in the Employee example.
A relation R could then be viewed as a subset of the product of the domains:
the
R ; dom(ID )
dom(Name)
dom(Address)
dom(Salary)
dom(JobNo)
(2.1)
Access to a relation can be performed based on operations in the traditional set algebra
such as union and intersection. This extended group of set operations is referred to as
2.2
the set
F = { l , 2, 3 , 4 , 5 }
(2.2)
E z+ and
:::: 5}
(2. 3 )
query language may be based on relational algebra or calculus. Although many query
A fuzzy
languages have been proposed, the standard language used by most DBMSs is SQL.
set is a set, F, in which the set membership function, f, is a real valued (as
24
Chapter 2
Sectio n 2.2
Related Concepts
25
Fuzzy logic is reasoning with uncertainty. That is, instead of a two valued logic
1 - f (x) . In actuality, this is not a true probability, but rather the degree of truth associated
(true and false), there are multiple values (true, false, maybe). Fuzzy logic has been
with the statement that x is in the set. To show the difference, let us look at a fuzzy set
used in database systems to retrieve data with imprecise or missing values. In this case,
operation. Suppose the membership value for Mary being tall is 0.7 and the value for
the membership of records in the query result set is fuzzy. As with traditional boolean
her being thin is 0.4. The membership value for her being both is 0.4, the minimum of
the two values. If these were really probabilities, we would look at the product of the
two values.
mem(-.x)
classification problem, all records in a database are assigned to one of the predefined
classification areas. A common approach to solving the classification problem is to assign
mem(x 1\ y)
a set membership function to each record for each class. The record is then assigned to
mem(x
used to describe other data mining functions. Association rules are generated given a
confidence value that indicates the degree to which it holds in the entire database. This
can be thought of as a membership function.
Qqeries can be thought of as defining a set. With traditional database queries,
however, the set membership function is boolean. The set of tuples in relation R that
satisfy the SQL statement in Figure 2.2 can be defined as
{x I x E R and x .Salary
>
and
v.
Fuzzy sets have been used in many computer science and database areas. In the
the class that has the highest membership function value. Similarly, fuzzy sets may be
-., 1\,
logic statements and that mem(x) defines the membership value, the following values are
y)
1 - mem(x)
(2. 6)
min(mem(x), mern(y))
(2.7)
max(mem(x) , mem(y))
(2. 8 )
Fuzzy logic uses rules and membership functions to estimate a continuous function. Fuzzy
logic is a valuable tool to develop control systems for such things as elevators, trains,
and heating systems. In these cases, instead of providing a crisp on-off environment, the
fuzzy controller provides a more continuous adjustment.
Most real-world classification problems are fuzzy. This is illustrated by Figure 2.4.
In this figure we graphically show the threshold for approving a loan based on the
income of the individual and the loan amount requested. A loan officer may make the
(2.4)
100, 000}
loan decision by simply approving any loan requests on or above the line and rejecting
any requests that fall below the line [Figure 2.4(a)]. This type of decision would not
be fuzzy. However, this type of decision could lead to elToneous and perhaps costly
not have a membership function that is boolean. For example, suppose that we wished
decisions by the loan officer. From a data mining perspective, this application is a clas
X.
sification problem; that is, classify a loan application into the approval or reject class.
R and x is tall}
(2.5)
This membership function is not boolean, and thus the results of this query are fuzzy. A
good example of queries of this type are searches performed on the Web.
Figure 2.3 shows the real difference between traditional and fuzzy set membership.
Suppose there are three sets (short, medium, and tall) to which a person can be classified
There are many other factors (other than income) that should be used to predict the
classification problem (such as net worth and credit rating). Even if all the associated
predictors could be identified, the classification problem is not a black-and-white issue.
It is possible that two individuals with exactly the same predictor values should be
placed in two different classes. This is due to the fuzzy nature of this classification. This
is shown by the shading around the line in Figure 2.4(b). We could perhaps classify
based on his height. In Figure 2.3(a) the traditional (or crisp) set membership values are
shown. Part (b) shows the triangular view of set membership values. Notice that there
is a gradual decrease in the set membership value for short; there is a gradual increase
and decrease for set membership in the medium set; there is a gradual increase in the set
membership value for tall.
Short
Medium
Height
Height
Tall
Loan
Loan
amount
amount
Income
(a) Simplistic loan approval
26
Chapter 2
Section 2.3
the individuals into multiple classes: approve, reject, unknown, probably approve, and
probably reject. This approach attacks the fuzzy miture of the classification by flagging
the applications requiring more analysis into the three new fuzzy classes. Procedural
policies could . then e used to determine the ultimate classification of these cases into
the final approve or reject classes. This is one of many possible approaches to defuzzify
the classification problem.
2.3
Relevant
Not relevant
Information retriewil (IR) (and more recently digit! libraries and Internet searching)
involves retrieving desired information from textual data. The historical development of
IR was based on effective use of libraries. So a typical 1R request would be to find all
library documents related to a particular subject, for example "data mining." This is,
in fact, a classification task because the set of documents in the library is divided into
classes based on the keywords involved. In IR systems, documents are represented by
document surrogates consisting of data, such as identifiers, title, authors, dates, abstracts,
extracts, reviewg, and keywords. As can be seen, the data consist of both formatted and
unformatted (text) data. The retrieval of documents is based on calculation of a similarity
measure showing how close each document is to the desired results (i .e., the stated query).
Similarity measures are also used in classification and clustering problems.
{ D1 , . . . , Dn } . The input is a
An IR system consists of a set of documents D
query, q , often stated as a list of keywords. The similarity between the query and eah
document is then calculated: sim(q , D; ) . This similarity measure is a set membership
function describing the likelihood that the document is of interest (relevant) to the user
based on the user's interest as stated by the query. The effectiveness of the system in
processing the query is often measured by looking at precision and recall:
=
Precision
Recall
I Retrieved I
I Relevant I
(2_9)
(2. 1 0)
Precision is used to answer the question: "Are all documents retrieved ones that I am
interested in?" Recall answers: "Have all relevant documents been retrieved?" Here a
document is relevant if it should have been retrieved by the query. Figure 2.5 illustrates
the four possible query results available with IR queries. Of these four quadrants, two
represent desirable outcomes: relevant and retrieved or not relevant and not retrieved.
The other two quadrants represent error situations. Documents that are relevant and not
retrieved should have been retrieved but were not. Documents that are not relevant and
retrieved should not have been retrieved but were. Figure 2.6 illustrates the basic structure
of a conventional information retrieval query .
Many similarity measures have been proposed for use in information retrieval. As
stated earlier, sim (q , D; ) 1 s i s n is used to determine the results of a query q applied
{D1 , . . . , Dn }. Similarity measures may also be used to clus
to a set of documents D
ter or classify documents by calculating sim (D; , D J) for all documents in the database.
Thus, similarity can be used for document-document, query-query, and query-document
measurements. The inverse document frequency (IDF) is often used by similarity mea
sures. IDF assumes that the importance of a keyword in calculating similarity measures is
=
Relevant
Not retrieved
Retrieved
IN FORMATION RETRIEVAL
Retrieved
FIGURE 2 . 5 : IR
Not relevant
Not retrieved
Keywords
Q::::: G Documents
FIGURE 2.6:
Information Retrieval
Documents
Feline -- Cat
'
Lion
Domestic
Cheetah
Tiger
FIG U R E 2.7:
Concept hierarchy.
inversely proportional to the total number of documents that contain it. Given a keyword,
IDFk
lg
I documents containing k 1
+ 1
(2. 1 1)
Concept hierarchies are often used in information retrieval systems to show the
relationships between various keywords (concepts) as related to documents. Suppose
you wish to find all documents about cats. Figure 2.7 illustrates a concept hierarchy that
could be used to answer this query. This figure shows that feline and cat are similar
terms. In addition, a cat may be domestic or tiger or lion or cheetah. In turn, a tiger
may be a Siberian, White, Indochinese, Sumatran, or South Chinese. Nodes lower in the
tree represent more specific groups of tigers. When a user requests a book on tigers, this
query could be modified by replacing the keyword "tiger" with a keyword at a higher
27
28
Chapter 2
Related Concepts
Section 2.5
D i mensional Mod e l i ng
29
Tall
Classified tall
20 10
45 25
Not tall
Classified tall
Tall
Classified
not tall
ing tool need not be contained in a DSS system. A decision support system could be
enterprise-wide, thus allowing upper-level managers the data needed to make intelligent
Not tall
Classified
not tall
business decisions that impact the entire company. A DSS typically operates using data
warehouse data. Alternatively, a DSS could be built around a single user and a PC. The
bottom line is that the DSS gives managers the tools needed to make intelligent decisions.
2.5
F I G U R E 2.8: Precision and recall applied to classific
ation.
DSS is much more broad than the term data mining. While
a DSS usually contains data mining tools, this need not be so. Likewise, a data min
. has
EXAMP LE 2.2
The accuracy of a predictive modeling techniqu
e can be described based on precision
and recall. Suppose 1 00 college students are to be
classified based on height. In actuality,
there are 30 tall students and 70 who are not tall. A
classification technique classifies 65
student as tall and 35 as not tall. The precision and
recall applied to this problem are
.
shown m Figure 2.8. The precision is 20/65, while
the recall is 20/30. The precision is
low because so many students who are not tall are classified
as such.
required, for efficiency purposes the data may be stored using different data structures
as well. Decision support applications often require that information be obtained along
many dimensions. For example, a sales manager may want to obtain information about
the amount of sales in a geographic region, particular time frame, and by-product type.
This query requires three dimensions. A
attributes and is viewed as an axis for modeling the data. The time dimension could
be divided into many different granularities: millennium, century, decade, year, month,
day, hour, minute, or second. Within each dimension, these entities form levels on which
various DSS questions may be asked. The specific data stored are called the
facts and
usually are numeric data. Facts consist of measures and context data. The measures are
the numeric attributes about the facts that are queried. DSS queries may access the facts
from many different dimensions and levels. The levels in each dimension facilitate the
retrieval of facts at different levels. For example, the sales information could be obtained
for the year 1999, for the month of February in the year 2000, or between the times
of 1 0 and 1 1
A.M.
general level,
Table 2 . 1 shows a relation with three dimensions: Products, Location, and Date.
Determining a key for this relation could be difficult because it is possible for the same
product to be sold multiple times on the same day. In this case, product 1 50 was sold at
two different times in Dallas on the same day. A finer granularity on the time (perhaps
down to the minute rather than date as is here) could make a key. However, this illustrates
that choice of key may be difficult. The same multidimensional data may also be viewed
as a cube. Figure 2.9 shows a view of the data from Table 2. 1 as a cube. Each dimension
is seen as an axis for the cube. This cube has one fact for each unique combination of
2.4
at deve1opmg
the business structure and comput
er techmques
to
the relation showed only 10 tuples). Obviously, this sparse amount of data would need
to be stored efficiently to reduce the amount of space required.
The levels of a dimension may support a partial order or a total order and can be
viewed via a directed path, a hierarchy, or a lattice. To be consistent with earlier uses of
the term, we use
relationship. X
<
<
30
Chapter 2
Section 2.5
Related Concepts
Di mensiona l M o d e l i n g
31
Year
TABLE 2. 1 : Relational View of Multidimensional Data
Quantity
Date
LociD
ProdiD
1\
UnitPrice
123
Dallas
022900
25
1 23
Houston
020100
10
20
1 50
Dallas
03 1 500
100
95
150
Dallas
03 1 500
150
Fort Worth
021000
80
150
Chicago
0 1 2000
20
75
200
Seattle
030100
50
300
Rochester
021500
200
500
Bradenton
022000
15
20
500
Chicago
0 1 2000
10
25
Planet
Month Season
1\
Country Continent
Day
Company
Product type
.1 \
1\
Hour AM/PM
Minute
State
1\
City
Second
Product
(a) Product dimension
Region
F I G U RE 2 . 1 0: Aggregation hierarchies.
such aggregate operations as average, maximum, and minimum prices. Figure 2 . 1 0(b)
shows a hierarchical relationship among levels in the time dimension, and Figure 2. 1 0(c)
shows a lattice for the location dimension. Here Day
<
aggregation can be applied only to levels that can be found in the same path as defined
by the
Seattle
--1--+--+-+---V
Rochester e.--+---+---+-f---V
Houston
Fort Worth
Dallas
<
relationship. When levels for a dimension satisfy tlJ.is structure, the facts along
a day, we get the sales data for that day. This is not always the case. Looking at the
location dimension, if we were to sum up the sales data for all zip codes in a given
1----l--+--+-1---V
county, however, we would not get the sales data for the county. Thus, these dimensions
e.--+---+---+-f---V
are not additive. This is due to the fact that zip codes may span different counties. The
e.--+---+---+-f---1"
Chicago
1----l--+--+-1--V
Bradenton L-...L__L---L_L-__v
use of nonadditive dimensions complicate the roll up and drill down applications.
2.5.1
Multidimensional Schemas
Specialized schemas have been developed to portray multidimensional data. These in
clude star schema, snowflake schema, and fact constellation schema.
Products
A
F I G U R E 2.9: Cube.
star schema shows data as a collection of two types: facts and dimensions.
Unlike a relational schema, which is flat, a star schema is a graphical view of the data.
fact tables
major tables). On the outside of the facts, each dimension is shown
separately in dimension tables (sometimes called minor tables). The simplest star schema
At the center of the star, the data being examined, the facts, are shown in
in Y. Figure 2.1 0(a) shows a total order relationship among levels in the Product dimen
sion from Figure 2.9. Here Product
<
Type
<
(sometimes called
using in this example are Quantity and UnitPrice. When this order relationship is satis
has one fact table with multiple dimension tables. In tlJ.is case each fact points to one
fied between two levels, there is an aggregate type of relationship among the facts. Here
tuple in each of the dimensions. The actual data being accessed are stored in the fact
the Quantity of products sold of a particular type is the sum of the quantities for all
tables and thus tend to be quite large. Descriptive information about the dimensions
products within that type. Similarly, the quantity of products sold for a company is the
is stored in the dimensions tables, which tend to be smaller. Figure 2. 1 1 shows a star
sum of all products sold across all product types. The aggregate operation is not always
schema based on the data in Figure 2.9. Here one extra dimension, division, is shown.
the sum, however. When looking at the unit price, it would be reasonable to look at
The facts include the quantity and price, while the dimensions are the product, time,
32
Chapter 2
Type Description
Product
Salesman ID
1\
I
Day iD
Product ID
Day iD
Salesman ID
Location ID
Quantity
Unit Price
Dept
Sales
Dept Desc
1/
Day
Month
Quarter
Year
Day
Location iD
Zip Code
City
Div Desc
Location
Division
F I G U R E 2 . 1 1 : Star schema.
33
to the storage of data in dimension tables [PB99]. Each dimension table can be stored in
_
one of these four manners. Figure 2 . 1 2 illustrates these four approaches w1 h the sals
_
data The first technique, the flattened technique, stores the data for each d1menson m
exa tly one table. There is one row in the table for each row in the lwest le: el
dimensional model. The key to the data are the attributes for all levels
m. the
tht d1mens1?n.
State
Div
Dimensional Modeling
Price
Unit Price)
(a) Flattened
Quant ity,
Unit Price)
Sales ( Product ID , Day ID, salesman ID, Location ID, Quant ity ,
Product ( Product I D , Description , Type , Type Description)
Types ( Type Description)
Day ( Day ID, Month , Quarte r , Year)
Months (Month, Quart er , Year)
Quarters ( Quarter, Year)
Years (Year)
sale sman ( Salesman ID , Dept , Dept Desc , Div, Div Desc)
Dept s ( Dept , Dept Desc , Div, Div Desc)
Divs ( Div, Div D e s c )
Locat ion ( Locat ion ID , Z i p Code , State , C i ty )
Zip ( Z ip Code, State , City)
Citie s ( S tate , City)
stat e s ( )
Unit Price)
(b) Normalized
Locat ion
( Fa c t s . Locat ioniD
Locat i on . Locat i on i D )
and
( Lo c a t i on . C i t y
(c) Expanded
(d) Levelized
34
Chapter 2
Data Warehousing
Section 2.6
number of attributes grows with the number of levels, it does facilitate the simple
Product ID
Description
Type
Type Description
implementation of many DSS applications via the use of traditional SQL aggregation
operations.
The second technique to store a dimension table is called the normalized technique,
1\
Product
where a table exists for each level in each dimension. Each table has one tuple for every
occurrence at that level. As with traditional normalization, duplication is removed at the
expense of creating more tables and potentially more expensive access to factual data
due to the requirement of more joins. Each lower level dimension table has a foreign
key pointing to the next higher level table.
Using expanded dimension tables achieves the operational advantages of both the
flattened and the normalized views, while actually increasing the space requirements
Dept
Dept Desc
Div
Div Desc
I Salesman ID r
Dept
Department
Salesman
beyond that of the flattened approach. The number of dimension tables is identical to
Product ID
Day iD
Salesman ID
Location ID
Quantity
Unit Price
Sales
35
DayiD
Day
Month
Quarter
Year
Day
Location IDI
Zip Code f--- Zip Code
State
Location
City
Zip Codes
that in the normalized approach, and the structure of the lowest level dimension table
is identical to that in the flattened technique. Each higher level dimension table has, in
F I G U RE 2 . 1 3 : Snowflake schema.
addition to the attributes existing for the normalized structure, attributes from all higher
level dimensiol}S.
The levelized approach has one dimension table as does the flattened technique.
Traditional B-tree indices may be constructed to access each entry in the fact table.
However, the aggregations have already been performed. There is one tuple for each
Here the key would be the combination of the foreign keys to the dimension tables.
instance of each level in the dimension, the same number existing in all normalized
tables combined. In addition, attributes are added to show the level number.
An extension of the star schema, the snowflake schema facilitates more complex
2.6
DATA WAREHOUSING
data views. In this cas. the aggregation hierarchy is shown explicitly in the schema itself.
Decision support systems (DSS) are subject-oriented, integ:rated, time-variant, and non
volatile. The term data warehouse was first used by William Inmon in the early 1 980s. He
defined data warehouse to be a set of data that supports DSS and is "subject-oriented,
ing star schema. The division and location dimension tables have been normalized in
ths figure.
data (current and )listorical) are merged into a single repository. Traditional databases
contain operational data that represent the day-to7day needs of a company. Traditional
2.5.2
Indexing
With multidimensional data, indices help to reduce the overhead of scanning the extrem
ely large tables. Although the indices used are not defined specifically to support multi
dimensional data, they do have inherent advantages in their use for these types of data.
With bitmap indices each tuple in the table (fact table or dimension table) is repre
sented by a predefined bit so that a table with n tuples would be represented by a vector
of n bits. The first tuple in the table is associated with the first bit, the second with the
second bit, and so on. There is a unique bit vector for each value in the domain. This
vector indicates which associated tuples in the table have that domain value. To find the
precise tuples, an address or pointer to each tuple would also have to be associated with
each bit position, not each vector. Bitmap indices facilitate easy functions such as join
and aggregation through the use of bit arithmetic operations. Bitmap indices also save
space over more traditional indices where pointers to records are maintained.
business data proces&!ng (such as billing, inventory control, payroll, and manufactur
ing support) support online transaction processing and. batch reporting applications. A
data warehouse, however, COJ1tains informational data, which are used to support other
functions suet> as planning and forecasting. Although much of the content is similar
between the operational and informational data, much is different. As a matter of fact,
the operational data are transformed into the informational data. Example 2.3 illustrates
the difference between the two.
EXAMPLE 2.3
The ACME Manufacturing Company maintains several operational databases: sales,
billing, employee, manufacturing, and warehousing. These are used to support the day
to-day functions such as writing paychecks, placing orders for supplies needed in the
manufacturing process, billing customers, and so on. The president of ACME, Stephanie
Join indices support joins by precomputing tuples from tables that join together and
pointing to the tuples in those tables. When used for multidimensional data, a common
products. To perform this task; she asks several "what if' questions, does a projection of
approach s to 'cteate a join index between a dimension table and a fact table. This
current sales into the future, and examines data at different geographic and time dimen
facilitates the efficient identification of facts for a specific dimension level and/or value.
sions. All the data that she needs to perform this task can be found in one or more of
Join indices can be created for multiway joins across multiple dimension tables. Join
indices can be constructed using bitmaps as opposed to pointers.
the existing databases. However, it is not easily retrieved in the exact format that she
desires. A data warehouse is created with exactly the sales information she needs by
36
Chapter 2
Related Concepts
Section 2.6
location and time. OLAP retrieval tools are provided to facilitate quick response
to her
.
questions at any and all dimension granularities.
The data wehouse market supports such diverse industries as manufacturing
.
,
.
retat, telecmmumcatw
ns, and health care. Think of a personnel database for a company
.
that ts cotmally mo?tfi
ed as personnel are added and deleted. A personnel databas
that contams tformatwn about the current set of employees is sufficient. Howeve e
r, if
management wtshes t analyze trends with respect to employment history, more
data are
nedd. They may wtsh to determine if there is a problem with too many
employees
qmttmg. To analyze this problem, they would need to know which employe
es have
.
left, when they left, why
they left, and other information about their employment. For
mangement to make these types of high-level business analyses, more historica
l data
(not JUst the curret s apshot that is typically stored) and data from other sources (perhaps
employment applicatiOns and results of exit interviews) are required. In addition, some of
the data in the pesonnel database, such as address, may not be needed. A data warehouse
provides just thist information. In a nutshell, a data warehouse is a data repository used
to support decision support systems .
.The basic motivation for this shift to the strategic use of data is to increase business
protbili. Tradii nal data processing applications support the day-to-day clerical and
Opet>tionl '"'
/
Transformation
Que<y tooffi
/l
OLAP tools
TABLE 2.2:
Data watehouse.
OLTP
Precise queries
Snapshot
Dynamic
Application
Operational values
Gigabits
Detailed
Often
Few seconds
Relational
Data Warehouse
OLAP
Ad hoc
Historical
Static
Business
Integrated
Terabits
Summarized
Less often
Minutes
Star/snowflake
Table 2.2 summarizes the differences between operational data stored in traditional
databases and the data stored in a data warehouse. The traditional database applications
are related to OLTP where the users' requests are stated in a high-level query language
(such as SQL) and the results are subsets of the relationships. Data warehouse applica
tions are directly related to business decisions and analysis of the data, OLAP. While
operational data usually represent facts concerning a snapshot in time (the current time),
a warehouse has historical data as well. Data in the warehouse are not modified as fre
quently as data in a conventional database. Updates are hatched and merged into the
warehouse at weekly or monthly intervals. Although this means that the warehouse data
are not completely up-to-date, this usually is not a problem with the normal decision
support applications. A conventional database usually is related to one application type.
This is a fallout of the normalization design process used. The warehouse is associated
with the business enterprise, not with an application. Traditional databases may be on
the order of megabytes or gigabytes, whereas a warehouse may be terabytes in size. The
fact that conventional data are stored in many diverse fo:rmats and locations makes it
inefficient to support decision support applications. OLTP users expect to get a response
in a few seconds. As a result of the complexity of OLAF' application, their users may
have to wait minutes for a response to their query.
The data transformation process required to convert operational data to informa
tional involves many functions including:
Converting heterogeneous sources into one common schema. This problem is the
same as that found when accessing data from multiple heterogeneous sources. Each
operational database may contain the same data with different attribute names. For
example, one system may use "Employee ID," while another uses "EID" for the
same attribute. In addition, there may be multiple data types for the same attribute.
As the operational data is probably a snapshot of the data, multiple snapshots may
need to be merged to create the historical view.
Dt> W><<ho
FIGURE 2. 1 4:
37
Application
Use
Temporal
Modification
Orientation
Data
Size
Level
Access
Response
Data schema
Data Ware h o u s i n g
38
Chapter 2
Section 2.7
Related Concepts
OLAP
39
Summarizing data is performed to provide a higher level view of the data. This
Since data warehouses are not usually updated a s frequently a s operational data
are, the negatives associated with update operations are not an issue.
New derived data (e.g., using age rather than birth date) may be added to better
Handling missing and erroneous data must be performed. This could entail replac
Partitioning: Dividing the data warehouse into smaller fragments may reduce
processing time by allowing queries to access small data sets.
The relationship between data mining and data warehousing can be viewed as
ing them with predicted or default values or simply removing these entries.
symbiotic [Inm96] . Data used in data mining applications are often slightly modified
The portion of the transformation that deals with ensuring valid and consistent data is
in a data warehouse. When data are placed in a warehouse, they are extracted from the
sometimes referred to as
from that in the databases where the data permanently reside. The same is true for data
database, cleansed, and reformatted. The fact that the data are derived from multiple
There are many benefits to the use of a data warehouse. Because it provides an
sources with heterogeneous formats complicates the proble:m. In addition, the fact that
integration of data from multiple sources, its use can provide more efficient access of
the source databases are updated requires that the warehouse be updated periodically or
the data. The data that are stored often provide different levels of summarization. For
work with stale data. These issues are identical to many of those associated with data
example, sales data may be found at a low level (purchase order), at a city level (total of
mining and KDD (see Figure 1 .4). While data mining and data warehousing are actually
sales for a city), or at higher levels (county, state, country, world). The summary can be
orthogonal issues, they are complementary. Due to the types of applications and massive
provided for different types of granularity. The sales data could be summarized by both
amount of data in
salesman and department. These summarizations are provided by the conversion process
meaningful information needed for decision support systems. For example, management
instead of being calculated when the data are accessed. Thus, this also speeds up the
may use the results of classification or association rule applications to help determine
the target population for an advertising campaign. In addition, data mining activities can
The data warehouse may appear to increase the complexity of database manage
benefit from the use of data in a data warehouse. However, its use is not required. Data
ment because it is a replica of the operational data. But keep in mind that much of the
warehousing and data mining are sometimes thought of as the same thing. Even though
data in the warehouse are not simply a replication but an extension to or aggregation
they are related, they are different and each can be used without the other.
of the data. In addition, because the data warehouse contains historical data, data stored
there probably will have a longer life span than the snapshot data found in the oper
ational databases. The fact that the data in the warehouse need not be kept consistent
with the current operational data also simplifies its maintenance. The benefits obtained
by the capabilities (e.g., DSS support) provided usually are deemed to outweigh any
disadvantages.
A subset of the complete data warehouse,
data. While some of this view may actually be materialized for efficiency, it need not
all be.
There are several ways to improve the performance of data warehouse applications.
are achieved at the cost of increased processing time due to joins. With a data
warehouse, improved performance can be achieved by storing denormalized data.
2.7
OLAP
Online analytic processing (OIAP) systems are targeted to provide more complex query
results than traditional OLTP or database systems. Unlike database queries, however,
OLAP applications usually involve analysis of the actual dtata. They can be thought of
as an extension of some of the basic aggregation functions available in SQL. This extra
analysis of the data as well as the more imprecise nature of the OLAP queries is what
really differentiates OLAP applications from traditional database and OLTP applications.
OLAP tools may also be used in DSS systems.
OLAP is performed on data warehouses or data marts . The primary goal of OLAP
is to support ad hoc querying needed to support DSS. The multidimensional view of data
is fundamental to OLAP applications. OLAP is an application view, not a data structure
or schema. The complex nature of OLAP applications requires a multidimensional view
of the data. The type of data accessed is often (although not a requirement) a data
warehouse.
OLAP tools can be classified as ROLAP or MOLAP. With MOIAP (multidimen
sional OIAP), data are modeled, viewed, and physically stored in a multidimensional
database (MDD). MOLAP tools are implemented by specialized DBMS and software
systems capable of supporting the multidimensional data directly. With MOLAP, data
are stored as an n-dimensional array (assuming there are
is stored directly. Although MOLAP has extremely high storage requirements, indices
are used to speed up processing. With
data are
stored in a relational database, and a ROLAP server (middleware) creates the multi
dimensional view for the user. As one would think, the ROLAP tools tend to be less
40
Chapter 2
Related Concepts
L-
o
c
a
t
i
0
n
,,
,
:.T/
'm
Products
(a) Single cell
Section 2.8
2. 8
Roll up
41
to access the data and can be viewed as query systems much like IR systems. As with
IR queries, search engine queries can be stated as keyword, boolean, weighted, and so
(c) Slice
on. The difference is primarily in the data being searched, pages with heterogeneous data
(d) Dice
F I G U R E 2 . 1 5: OLAP operations.
Abundance: Most of the data on the Web are of no interest to most people. In
other words, although there is a lot of data on the Web, an individual query will
complex, but also less efficient. MDD systems may presummarize along all dimensions.
A third approach,
MOLAP. Queries are stated in multidimensional terms. Data that are not updated fre
quently will be stored as MDD, whereas data that are updated frequently will be stored
as RDB.
Web any time a query is requested. Instead, mos search engines create indices
that are updated periodically. When a query is requested, often only the index is
directly accessed.
OLAP tools:
A simple query may look at a single cell within the cube [Figure 2. 1 5 (a)] .
of the cube.
Limited customization: Query results are often determined only by the query
itself. However, as with traditional IR systems, the desired results are often depen
dent on the background and knowledge of the user as well. Some more advanced
performed by a slice on one dimension and then rotating the cube to select on a
search engines add the ability to do customization using user profiles or historical
second dimension. In Figure 2. 1 5(d), a dice is made because the view in (c) is
information.
rotated from all cells for one product to all cells for one location.
Limited query: Most search engines provide access based only on simple key
word-based searching. More advanced search engines may retrieve or order pages
Limited coverage: Search engines often provide results from a subset of the Web
pages. Because of the extreme size of the Web, it is impossible to search the entire
Roll up (dimension reduction, aggregation): Roll up allows the user to ask ques
from (a). Instead of looking at one single fact, we look at all the facts. Thus, we
usage mining. Web search engines are very simplistic examjJles of Web content mining.
could, for example, look at the overall total sales for the company.
Drill down: Figure 2.1 5(a) represents a drill down from (b). These functions allow
a user to get more detailed fact information by navigating lower in the aggregation
2.9
STATISTICS
Such simple statistical concepts as determining a data distribution and calculating a mean
hierarchy. We could perhaps look at quantities sold within a specific area of each
and a variance can be viewed as data mining techniques. Each of these is, in its own
of the cities.
model. A current database state may be thought of as a sample (albeit large) of the real
To assist with roll up and drill down operations, frequently used aggregations can be
precomputed and stored in the warehouse. There have been several different definitions
for a dice. In fact, the term
Part of the data mining modeling process requires searching the actual data. An
equally important part requires inferencing from the results of the search to a general
data that may not be stored electronically. When a model is generated, the goal is to
fit it to the entire data, not just that which was searched. Any model derived should
be statistically significant, meaningful, and valid. This problem may be compounded
by the preprocessing step of the KDD process, which may actually remove some of
the data. Outliers compound this problem. The fact that most database practitioners
42
Chapter 2
Related Concepts
and users are not probability experts also complicates the issues. The tools needed
t
make the computationally difficlt problems tractable may actually invalidate the results
.
Assum?twns often made about mdependence of data may be incorrect, thus leading
to
.
errors m the resultmg model.
An often-used tool i daa minig and machine learning is that of sampling. Here
a
subset f e total populatiOn IS exanuned, and a generalization (model) about the
entire
opulatwn IS made from this subset. In statistics this approach is referred to as statistical
mference . Of course, this generalization may lead to errors in the final model caused
by
.
the samplmg process.
The term exploratmy data analysis was actually coined by statisticians to describe
the fc that te data can actually drive the creation of the model and any statistical
char
actenstics. This seems contradictory to the traditional statistical view that one should
not
be corruptd or influenced by looking at the data before creating the model. This
would
.
unecessanly bias any resulting hypothesis. Of course, database practitioners would
never
.
of creating a model of the data without looking at the data in detail first and then
creatmg a schema tq describe it.
Some data mining applications determine correlations among data. These relation
.
ships, however, are not causal in nature. A discovered association rule may show
that
60 percent of the time when customers purchased hot dogs they also bought beer.
Care
must ?e taken when assigning any significance to this relationship. We do not know
why
these Items were purchased together. erhaps the beer was on sale or it was an extremely
.
hot day. There may be no relatwnshi
p between these two data items except that they
were often purchased together. There need not be any probabilistic inference that
can be
deduced.
Statisics esearch has produced many of the proposed data mining algorithms . The
.
difference lies m the goals, the fact that statisticians may deal with smaller
and more
formtted data sets, and the emphasis of data mining on the use of machine
learning
techniques . Howeve, t is often the case that the term data mining is used in a deroga
.
.
tory mann by statisticians as data mming
is "analysis without any clearly formulated
.
hypothess [Man96] . Indeed, this
may be the case because data itself, not a predefined
.
hypothesis, Is the guide.
Pr?bability distributions can be used to describe the domains found for different
data atnbutes. Statistical inference techniques can be viewed as special estimators
and
.
predctw eth?ds. Use of these approaches may not always be applicable because
the
pecse Istnbutwn of real data vlues may not actually follow any specific probability
.
distbtwn, assumptiOns on the mdependence of attributes may be invalid, and
some
heunstic-based estimator techniques may never actually converge.
I as ?een stated that "the main difference between data mining and statistics is that
data numng I eat to ?e used by the business user-not the statistician" [BS97, 292].
As
such, data numng (particularly from a database perspective) involves not only modelin
g
but also the development of effective and efficient algorithms (and data structures)
t0
perform the modeling on large data sets.
thini<:
2.10
MACH I N E LEARNING
Section 2 . 1 0
Mach i n e Learn i n g
43
numng. AI applications also may not be concerned with scalability as data sets may
be small.
Machine learning is the area of AI that examines how to write programs that can
leani. In data mining, machine learning is often used for prediction or classification.
With machine learning, the computer makes a prediction and then, based on feedback
as to whether it is correct, "learns" from this feedback. It learns through examples,
domain knowledge, and feedback. When a similar situation arises in the future, this
feedback is used to make the same prediction or to make a completely different pre
diction. Statistics are very important in machine learning programs because the results
of the predictions must be statistically significant and must perform better than a naive
prediction. Applications that typically use machine learning techniques include speech
recognition, training moving robots, classification of astronomical structures, and game
playing.
When machine learning is applied to data mining tasks, a model is used to represent
the data (such as a graphical structure like a neural network or a decision tree). During
the learning process, a sample of the database is used to train the system to properly
perform the desired task. Then the system is applied to the general database to actually
perform the task. This predictive modeling approach is divided into two phases. During
the training phase, historical or sampled data are used to create a model that represents
those data. It is assumed that this model is representative not only for this sample data,
but also for the database as a whole and for future data as well. The testing phase then
applies this model to the remaining and future data.
A basic machine learning application includes several major aspects. First, an
appropriate training set must be chosen. The quality of the training data determines
how well the program learns. In addition, the type of feedback available is important.
Direct feedback entails specific information about the results and impact of each possible
move or database state. Indirect feedback is at a higher level, with no specific informa
tion about individual moves or predictions. An important aspect is whether the learning
program can actually propose new moves or database states. Another major feature that
impacts the quality of learning is how representative the training set is of the overall final
system to be examined. If a program is to be designed to perform speech recognition, it
is hoped that the system is allowed to learn with a large sample of the speech patterns it
will encounter during its actually processing.
There are two different types of machine learning: supervised learning and unsu
pervised learning. A supervised approach learns by example. Given a training set of
data plus correct answers, the computational model successively applies each entry in
the training set. Based on its ability to correctly handle each of these entries, the model
is changed to ensure that it works better with this entry if it were applied again. Given
enough input values, the model will learn the correct behavior for any potential entry.
With unsupervised data, data exist but there is no knowledge of the correct answer of
applying the model to the data.
Although machine learning is the basis for many of the core data mining research
topics, there is a major difference between the approaches taken by the AI and database
disciplines. Much of the machine learning research has focused on the learning portion
rather than on the creation of useful information (prediction) for the user. Also, machine
learning looks at things that may be difficult for humans to do or concentrates on how
to develop learning techniques that can mimic human behavior. The objective for data
44
Chapter 2
Related Concepts
TAB LE 2.3:
Section 2. 1 4
Database Management
Machine Learning
Database is static
Databases are complete and noise-free
2.12
S U M MARY
.
not part f the database bemg que ed. Table 2.4 compares the
.
AggregatiOn operators have extsted m SQL for years. They do
not return objects existing
Query
Data
Results
Output
DB/OLTP
Precise
Precise
Analysis
Vague
Database
Documents
Multidimensional
Preprocessed
Precise
Vague
Precise
Vague
DB objects or aggregation
Documents
DB objects or aggregation
KDD objects
OLAP
DM
in the database, but return a model of the data. For example., an average operator returns
the average of a set of attribute values rather than the values themselves. This is a simple
type of data mining operator.
2. 13
EXERCISES
1. Compare and contrast database, information retrieval, and data mining queries.
What metrics are used to measure the performance of each type of query?
2. What is the relationship between a fuzzy set membership function and classifica
tion? lllustrate this relationship using the problem of assigning grades to students
in classes where outliers (extremely high and low grades) exist.
3. (Research) Data warehouses are often viewed to contain relatively static data.
Investigate techniques that have been proposed to provide updates to this data
from the operational data. How often should these updates occur?
PATTERN MATCHING
45
Area
IR
TABLE 2.4:
2.14
BI BLIOGRAPHIC NOTES
There are many excellent database books, including [DatOO], [ENOO], [GMUW02], and
[000 1]. There are several books that provide introductions to database query processing
and SQL, including [DD97] and [YM98].
Fuzzy sets were first examined by Lotti Zadeh in [Zad65]. They continue to be an
important research topic in all areas of computer science, with many introductory texts
available such as [KY95], [NW99], and [San98]. Some texts explore the relationship
between fuzzy sets and various data mining applications. Pedrycz and Gomide examine
fuzzy neural networks and fuzzy genetic algorithms. [PG98]
There are several excellent information retrieval texts, including [BYRN99] and
[SM83]. The ER (entity-relationship) data model was first proposed by Chen in 1976
[Che76].
An examination of the relationship between statistics and KDD can be found in
[IP96] . This article provides an historical perspective of the development of statistical
techniques that have influenced data mining. Much of the research concerning outliers
has been performed in the statistics community [BL94] and [Haw80] .
There is an abundance of books covering of DSS, OLAP, dimensional modeling,
multidimensional schemas, and data warehousing, including [BE97], [PKB98], [Sin98],
and [Sin99].
An excellent textbook on machine learning is [Mit97] . The relationships between
machine learning and data mining are investigated in [MBK99].
Section 3 . 2
C H A P T E R
Data M i n i n g Tech n i q u es
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
INTRODUCTION
NEURAL NETWORKS
GENETIC ALGORITHMS
EXERCISES
BIBLIOGRAPHIC NOTES
There have been many statistical concepts that are the basis for data mining techniques.
We briefly review some of these concepts.
3.2.1
Point Estimation
3.1
I NTRODUCTION
There are many different methods used to perform data mining tasks. These techniques
not only require specific types of data structures, but also imply certain types of algorith
mic approaches. In this chapter we briefly introduce some of the common data mining
techniques. These will be examined in more detail in later chapters of the book as they
are used to perform specific data mining tasks.
Parametric models describe the relationship between input and output through the
use of algebraic equations where some parameters are not specified. These unspecified
parameters are determined by providing input examples. Even though parametric model
ing is a nice theoretical topic and can sometimes be used, often it is either too simplistic or
requires more knowledge about the data involved than is available. Thus, for real-world
problems, these parametric models may not be useful.
Nonparametric techniques are more appropriate for data mining applications. A
nonparametric model is one that is data-driven. No explicit equations are used to deter
mine the model. This means that the modeling process adapts to the data at hand. Unlike
parametric modeling, where a specific model is assumed ahead of time, the nonpara
metric techniques create a model based on the input. While the parametric methods
require more knowledge about the data before the modeling process, the nonparametric
technique requires a large amount of data as input to the modeling process itself. The
modeling process then creates the model by sifting through the data. Recent nonparamet
ric methods have employed machine learning techniques to be able to learn dynamically
as data are added to the input. Thus, the more data, the better the model created. Also, .
this dynamic learning process allows the model to be created continuously as the data
is input. These features make nonparametric techniques particularly suitable to database
46
47
(3. 1 )
An unbiased estimator i s one whose bias i s 0. While point estimators for small data sets
may actually be unbiased, for larger database applications we would expect that most
estimators are biased.
One measure of the effectiveness of an estimate is the mean squared error (MSE),
which is defined as the expected value of the squared difference between the estimate
and the actual value:
(3.2)
MSE( G ) = E (G - 8) 2
The squared error is often examined for a specific prediction to measure accuracy rather
than to look at the average difference. For example, if the ttue value for an attribute was
10 and the prediction was 5, the squared error would be (5 10) 2 = 25 . The squaring is
performed to ensure that the measure is always positive and to give a higher weighting to
the estimates that are grossly inaccurate. As we will see, the MSE is commonly used in
evaluating the effectiveness of data mining prediction techniques. It is also important in
machine learning. At times, instead of predicting a simple point estimate for a parameter,
one may determine a range of values within which the true parameter value should fall.
This range is called a confidence interval.
The root mean square (RMS) may also be used to estimate error or as another
statistic to describe a distribution. Calculating the mean does not indicate the magnitude
of the values. The RMS can be used for this purpose. Given a set of n values X
{x1 , . . . , xn }, the RMS is defined by
-
RMS =
(3.3)
An alternative use is to estimate the magnitude of the error. The root mean square error
(RMSE) is found by taking the square root of the MSE.
A popular estimating technique is the jackknife estimate. With this approach, the
estimate of a parameter, e, is obtairled by omitting one value from the set of observed
48
Chapter 3
J=l
J =i+l
{x1 , . , Xn } .
.
Here the subscript (i) indicates that this estimate is obtained by omitting the i1h value.
Given a set of jackknife estimates, ecil , these can in tum be used to obtain an overall
estimate
z= e(j)
A
Bc.l =
j =l
(3 .5)
EXAMPLE 3 . 1
Suppose that a coin is tosse9 in the air five times with the following results ( 1 indicates
a head and 0 indicates a tail): { 1 , 1 , 1 , 1 , 0}. If we assume that the coin toss follows the
Bernoulli distribution, we know that
(3 .6)
1/2,
(3.7 )
However, if the coin is not perfect but has a bias toward heads such that the probability
of getting a head is 0.8, the likelihood is
(3 .8)
L( p 1 1 , 1 , 1 , 1, 0) = 0.8 0.8 0.8 0.8 0 .2 = 0 .08
Here it is more likely that the coin is biased toward getting a head than that it is not
biased. The general formula for likelihood is
5
(3 .9)
L(p 1 X J , x5) = Jl px; ( l - p) ! -x; = p LT=1 x; ( l - p) 5 - I:T=1 x;
X
. ,
i=l
't;
x , log (p)
(s-'t; ) log(! - p)
x.
5 xi
-L ---a;p
i= I
i=l
-p
Another technique for point estimation is called the maximum likelihood estimate
can be defined as a value proportional to the actual probability that
with a specific distribution the given sample exists. So the sample gives us an estimate
for a parameter from the distribution. The higher the likelihood value, the more likely
the underlying distribution will produce the results observed. Given a sample set of
values X = {x J , . . . , x11 } from a known distribution function f (xi I 8), the MLE can
estimate parameters for the population from which the sample is drawn. The approach
obtains parameter estimates that maximize the probability that the sample data occur for
the specific model. It looks at the joint probability for observing the sample data by
multiplying the individual probabilities. The likelihood function, L , is thus defined as
(MLE) . Likelihood
L(E>
XJ ,
. . , Xn )
= D t(xi
I 8)
(3. 1 3)
i=l
The value of 8 that maximizes L is the estimate chosen. This can be found by taking
the derivative (perhaps after finding the log of each side to simplify the formula) with
respect to 8. Example 3 . 1 illustrates the use of MLE.
ALGORITHM 3.1
Input :
{th , . , Op }
= {xl , . . . , Xk}
Xmi s s = {Xk+l, . . . , Xn}
e
. .
Xobs
/ / Parameters t o be e s t imated
/ / Input databa se values observed
/ / Input database values mi s s ing
Output :
EM
/ / E s t imates for 8
algori t hm :
0Ji ;
repeat
(3 . 1 0)
xiffil. SS ;
5
S - L Xi
(3 . 1 2)
i := 0 ;
Obt a i n initial parameter MLE e s t imate ,
i=l
--
For this example, the estimate for p is then p = = 0 . 8 . 'Thus, 0.8 is the value for p
that maximizes the likelihood that the given sequence of heads and tails would occur.
the likelihood
i =l
p=
(3 .4)
- I
49
I >j + L Xj
/L(i) =
Section 3 . 2
(3 . 1 1)
Oi
to
50
Chapter 3
as a mean) using a two-step process: estimation and maximization. The basic EM algo
51
and then
rithm is shown in Algorithm 3 . 1 . An initial set of estimates for the parameters is obtained.
L: x;
Given these estimates and the training data as input, the algorithm then calculates a value
for the missing data. For example, it might use the estimated mean to predict a missing
fl4
value. These data (with the new value added) are then used to determine an estimate for
the mean that maximizes the likelihood. These steps are applied iteratively until succes
i= l
n
i=k+l
+
n
x;
+ 4.92 + 4.92
= 3 .33
.9
=4 7
(3 . 1 9)
We decide to stop here because the last two estimates are only 0.05 apart. Thus, our
sive parameter estimates converge. Any approach can be used to find the initial parameter
estimate is
estimates. In Algorithm 3 . 1 it is assumed that the input database has actual observed val
ues X obs
Section 3.2
fl
4.97.
One of the basic guidelines in estimating is Ockham 's Razor, 1 which basically
L(e 1 X ) =
O J (x;
1 8)
(3 . 14)
i=l
We are looking for the 8 that maximizes L. The MLE of 8 are the estimates that satisfy
,
3.2.2
(3. 15)
as a whole. The basic well-known statistical concepts such as mean, variance, standard
deviation, median, and mode are simple models of the underlying population. Fitting a
population to a specific frequency distribution provides an even better model of the data.
The expectation part of the algorithm estimates the missing values using the current
and/or multimedia attributes, and are constantly changing is not practical (let alone always
a ln L(8 I X)
ae;
=0
estimates of e. This can initially be done by finding a weighted average of the observed
Of course, doing this with large databases that have multiple attributes, have complex
possible).
data. The maximization step then finds the new estimates for the e parameters that
There are also many well-known techniques to display the structure of the data
maximize the likelihood by using those estimates of the missing data. An illustrative
graphically. For example, a histogram shows the distribution of the data. A box plot is a
more sophisticated technique that illustrates several different features of the population at
EXAMPLE 3.2
into four equal parts called quartiles. The box in the center of the figure shows the range
once. Figure 3 . 1 shows a sample box plot. The total range of the data values is divided
We wish to find the mean, f.L, for data that follow the normal distribution where the known
{l0
3. We then use this value for the two missing values. Using this,
f.L
A
L X;
-
i=i
n
--
L X;
i=k+i
---
between the first, second, and third quartiles. The line in the box shows the median. The
lines extending from either end of the box are the values that are a distance of 1 .5 of the
interquartile range from the first and third quartiles, respectively. Outliers are shown as
points beyond these values.
Smallest value within 1.5
interquartile range from 1st quartile
= 3 . 33 +
--
= 4 . 33
(3. 1 6)
We now repeat using this as the new value for the missing items, then estimate the
mean as
A2
f.L
l: x;
x;
4.33 4.33
i=i
i=k+i
= -- + -- = 3 . 33 +
= 4.77
n
6
n
\
(3 . 1 7 )
Outliers
A3
f.L =
i=i
L Xi
i=k+ l
n
--
1 ---+-1--'-----:r:;-J-h---<:
3rd quartile
L Xi
--
'r"
1st quartile
Repeating we obtain
= 3 . 33 +
4 , 77
+ 4,77 =
6
1 Sometimes this is spelled Occum or Occam. It is named after William Ockham, who was a monk in the
4.92
(3 . 1 8)
late thirteenth and early fourteenth centuries. However, it was first used by Durand de Saint-Pourcain an earlier
French theologian.
52
Chapter 3
Section 3.2
53
..
ID
Income
Credit
Class
Xi
1
2
3
4
5
6
4
3
2
3
4
2
3
2
3
Excellent
Good
Excellent
Good
Good
Excellent
Bad
Bad
Bad
Bad
h1
h1
X4
X?
xz
X?
7
8
. .. .
.
10
hi
h1
ht
ht
hz
hz
h3
h4
xs
xz
xu
XlQ
xu
Xg
Bayes Theorem
With statistical inference, information about a data distribution are inferred by examining
data that follow that distribution. Given a set of data X
{x 1 , . . . , x11 }, a data mining
problem is to uncover properties of the distribution from which the set comes. B ayes rule,
defined in Definition 3 . 1 , is a technique to estimate the likelihood of a property given
the set of data as evidence or input. Suppose that either hypothesis h 1 or hypothesis h2
must occur, but not both. Also suppose that Xi is an observable event.
=
P(xi )
L P (xi
Thus, we have
(3 .22)
Bayes rule allows us to assign probabilities of hypotheses given a data value, P (h j I Xi ) .
Here we discuss tuples when in actuality each X i may be a n attribute value o r other data
label. Each h1 may be an attribute value, set of attribute values (such as a range), or even
a combination of attribute values.
Example 3.3 uses the training data in Table 3 . 1 to illustrate the use o ayes rule.
Example 3.3 also illustrates that we may take advantage of other probabthty laws to
determine combinations of probabilities. For example, we may find P (h t )
P (l <
$ 10, 000 1\ Good) by instead finding P (l < $10,000) P (Good I I < $10,000) .
EXAMPLE 3.3
Suppose that a credit loan authorization problem can be associated with for hypo
theses: H
{h1 , h2, h3, h4} where h 1
authorize purchase, hz
authonze after
further identification, h3
do not authorize, and h4
do not authorize but contact
police. The training data for this example are shown in Table 3 . 1 . From training data,
we find that P (h 1 ) = 60%, P (hz )
20%, P (h3) = 1 0%, and P (h4) = 10%. To
make our predictions, a domain expert has determined that the attributes we shoul be
looking at are income and credit category. Assume that income, I, has been categonzed
by ranges [0, $10,000), [$ 10,000, $50, 000) , [$50,000, $ 1 00,000), and [$ 1 00,000 ' oo).
.
These ranges are encoded and are shown in Table 3 . 1 as 1 , 2, 3, and 4 , respectively.
Suppose that credit is categorized as excellent, good, or bad. By com?inin these, we
then have 12 values in the data space: D
{xi , . . . , XJ2}. The relationship between
=
or Bayes Theorem is
(3 .20)
Here P (h1 I Xi ) is called the posterior probability, while P (h 1 ) is the prior prob
ability associated with hypothesis h 1 . P (xi) is the probability of the occurrence of
data value Xi and P (xi I h1) is the conditional probability that, given a hypothesis,
the tuple satisfies it.
(3.21)
I hj ) P (hj)
j=l
54
Chapter 3
TABLE 3.2:
Section 3.2
Excellent
Good
Bad
XJ
xs
Xg
55
and E is the expected values based on the hypothesis, the chi-squared statistic, x 2 , is
defined as:
(3.23)
X4
Xg
XJ 2
these x; values and the two attributes is shown in Table 3 .2. Using these values, the last
column in Table 3 . 1 shows the x; group into which that tuple falls. Given these, we can
then calculate P (x; I h J ) and P (x; ) . We illustrate how this is done with h I There are
six tuples from the training set that are in h 1 ; we use the distribution of these across the
x; to obtain: P (x7 I h J )
y6, P (x4 I h 1 ) = 1/6, P (x2 I h i ) = 2/6, P (xg I h i ) = 1 /6,
and P (x; I h 1 ) = 0 for all ather values of i . Suppose we wanted to predict the class for
X4. We thus need to find P (h J I x4) for each h J . We would then classify x4 to the class
EXAMPLE 3 .4
(P(x411<X(h1)) = O/i06)
Suppose that there are five schools being compared based on students' results on a set
of standardized achievement tests. The school district expects that the results will be
the same for each school. They know that the total score for the schools is 375, so the
expected result would be that each school has an average score of 75. The actual average
scores from the schools are: 50, 93, 67, 78, and 87. The district administrators want to
determine if this is statistically significant. Or in simpler te1ms, should they be worried
about the distribution of scores. The chi-squared measure here is
= 1 . We
This example illustrates some issues associated with sampling. First note that
Table 3 . 1 has no entries for XJ , x3 , xs , X6, or XJ2 This makes it impossible to use
this training sample to determine how to make predictions for this combination of input
data. If indeed these combinations never occur, this would not be a problem. However,
in this case we certainly do not know this to be true. Another issue with this sample is its
size. Of course, a sample of this size is too small. But what constitutes a good sample?
Size certainly is not the only criterion. This is a crucial issue that impacts the quality
of any data mining technique that uses sampling. There is much work in the statistics
community on good sampling strategies, so we do not cover that topic in this text.
3.2.4
Hypothesis Testing
Hypothesis testing attempts to find a model that explains the observed data by first
creating a hypothesis and then testing the hypothesis against the data. This is in contrast
to most data mining approaches, which create the model from the actual data without
guessing what it is first. The actual data itself drive the model creation. The hypothesis
usually is verified by examining a data sample. If the hypothesis holds for the sample, it
is assumed to hold for the population in general. Given a population, the initial (assumed)
hypothesis to be tested, Ho, is called the null hypothesis. Rejection of the null hypothesis
causes another hypothesis, Ht , called the alternative hypothesis, to be made.
One technique to perform hypothesis testing is based on the use of the chi-squared
statistic. Actually, there is a set of procedures referred to as chi squared. These proce
dures can be used to test the association between two observed variable values and to
determine if a set of observed variable values is statistically significant (i.e., if it dif
fers from the expected case). A hypothesis is first made, and then the observed values
are compared based on this hypothesis. Assuming that 0 represents the observed data
2
2
(93 - 75) 2
(67 - 75) 2
(78 - 75)
(87 - 75) 2 2 - (50 - 75)
+
+
+
+
- 15 . 55
75
75
75
75
75
(3.24)
Examining a chi-squared significance table, it is found that tltis value is significant. With
a degree of freedom of 4 and a significance level of 95%, the critical value is 9.488.
Thus, the administrators observe that the variance between the schools' scores and the
expected values cannot be associated with pure chance.
X
3.2.5
Both bivariate regression and correlation can be used to evaluate the strength of a
relationship between two variables. Regression is generally used to predict future values
based on past values by fitting a set of points to a curve. Correlation, however, is used
to examine the degree to which the values for two variables behave similarly.
Linear regression assumes that a linear relationship exists between the input data
and the output data. The common formula for a linear relationship is used in this model:
y = CQ + C) X[ + + CnXn
(3.25)
Here there are n input variables, which are called predictors or regressors; one output
variable (the variable being predicted), which is called the response; and n + 1 constants,
which are chosen during the modeling process to match the input examples (or sample).
This is sometimes called multiple linear regression because there is more than one predictor.
Example 3.5 is an example of the use of linear regression.
56
Chapter 3
Section 3.3
57
EXAMPLE 3.5
(3.26)
It is known that a state has a fixed sales tax, but it is not known what the amount happens
to be. The problem is to derive the equation for the amount of sales tax given an input
purchase amount. We can state the desired linear equation to be y
= co + q x 1 . So we
co and
X and Y are the means for X and Y, respectively. Suppose that X = (2, 4, 6, 8,
5, 3, 1 ) ,
10) . If Y = X, then r = 1 . When Y = ( 1 , 3, 5, 7, 9), r = 1 . If Y = (9,
r = -1.
where
really only need to have two samples of actual data to determine the values of
q . Suppose that we know ( 10, 0.5) and (25 , 1 .25) are actual purchase amount and tax
amount pairs. Using these data points, we easily determine that co = 0 and c1 = 0.05 .
Thus, the general formula is y = 0.05 x; . This would be used to predict a value o f y for
any known x; value.
When two data variables have a strong correlation, they are similar. Thus, the
correlation coefficient can be used to define similarity for clustering or classification.
3.3
Admittedly, Example
the problem of determining how much alike the two variables actually are. One standard
formula to measure linear correlation is the
SIMILARITY M EASURES
The use of similarity measures is well known to anyone who has performed Internet
searches using a search engine. In such a search, the set of all Web pages represents the
whole database, and these are divided into two classes: those that answer your query and
those that do not. Those that answer your query should be more like each other than
those that do not answer your query. The similarity in this case is defined by the query
you state, usually based on a keyword list. Thus, the retrieved pages are similar because
they all contain (to some degree) the keyword list you have specified.
The idea of similarity measures can be abstracted and applied to more general
classification problems. The difficulty lies in how the similarity measures are defined
and applied to the items in the database. Since most similarity measures assume numeric
(and often discrete) values, they may be difficult to use for more general data types.
Here negative correlation indicates that one variable increases while the other decreases
tionship exists with a value of
When looking at a scatter plot of the two variables, the closer the values ar to a straight
mapping from the attribute domain to a subset of the integers may be used.
7,
D to the range
The objective i s t o define the similarity mapping such that documents that are more
alike have a higher similarity value. Thus, the following are desirable characteristics of
a good similarity measure:
= 1
Vt; , t1 E D , sirn(t; , fJ )
Vt; , fJ , tk E D , sim(t; , fJ )
So how does one define such a similarity mapping? This, of course, is the difficult part.
Often the concept of "alikeness" is itself not well defined. When the idea of similar
ity measure is used in classification where the classes are: predefined, this problem is
somewhat simpler than when it is used for clustering where the classes are not known
in advance. Again, think of the IR example. Each IR query provides the class definition
in the form of the IR query itself. So the classification problem then becomes one of
determining similarity not among all tuples in the database but between each tuple and
the query. This makes the problem an
FIG U RE 3.3: Simple linear regression.
Here are some of the more common similarity measures used in traditional IR
Chapter 3
58
Section 3.4
Decision Trees
59
Alive?
f\\
l\
Ever alive?
N
Person?
es
Friend?
Overlap:
sim(t; , t1 )
es
...
FINISHED
mm
( "k
2 "k
2
L... h = l ti h ' L...h = l tJ h )
In these formulas it is assumed that similarity is being evaluated between two vectors
(ti J , . . . , t;k) and fJ = {fJ J , . . . , fJ k ) , and vector entries usually are assumed to be
t;
nonnegative numeric valwts. They could, for example, be a count of the number of times
an associated keyword appears in the document. If there is no overlap (i.e., one of the two
vectors always has a 0 in one of the two terms), the resulting value is 0. If the two are
identical, the resulting measure is 1 . The overlap measure, however, does not satisfy this
restriction. These formulas have their origin in measuring similarities between sets based
on the intersection between the two sets. Dice ' s coefficient relates the overlap to the
average size of the two sets together. Jaccard' s coefficient is used to measure the overlap
of two sets as related to the whole set caused by their union. The cosine coefficient relates
the overlap to the geometric average of the two sets. The overlap metric determines the
degree to which the two sets overlap.
Distance or dissimilarity measures are often used instead of similarity measures.
As implied, these measure how "unlike" items are. Traditional distance measures may
be used in a two-dimensional space. These include
=
dis(ti , lJ ) =
Euclidean:
Manhattan: dis(t; , fJ )
L = l I (t; h - fJ h ) I
To compensate for the different scales between different attribute values, the attribute
values may be normalized to be in the range [0, 1 ] . If nominal values rather than numeric
values are used, some approach to determining the difference is needed. One method is
to assign a difference of 0 if the values are identical and a difference of 1 if they
are different.
3.4
DECISION TR EES
F I G U R E 3 . 4:
at the third level in the game. Leaf nodes represent a successful guess as to the object
being predicted. This represents a correct prediction. Each question successively divides
the search space much as a binary search does. As with a binary search, questions
should be posed so that the remaining space is divided into two equal parts. Often young
children tend to ask poor questions by being too specific, such as initially asking "Is it
my Mother?" This is a poor approach because the search space is not divided into two
equal parts.
EXAMPLE 3.6
Stephanie and Shannon are playing a game of "1\venty Questions." Shannon has in mind
some object that Stephanie tries to guess with no more than 20 questions. Stephanie' s
first question is "Is this object alive?" Based on Shannon's answer, Stephanie then asks
a second question. Her second question is based on the answer that Shannon provides
to the first question. Suppose that Shannon says "yes" as her first answer. Stephanie' s
second question is "Is this a person?" When Shannon responds "yes," Stephanie asks
"Is it a friend?" When Shannon says "no," Stephanie then asks "Is it someone in my
family?" When Shannon responds "yes," Stephanie then begins asking the names of
family members and can immediately narrow down the search space to identify the
target individual. This game is illustrated in Figure 3.4.
DEFINITION 3.3. A decision tree (DT) is a tree where the root and each internal
node is labeled with a question. The arcs emanating from each node represent each
possible answer to the associated question. Each leaf node represents a prediction
of a solution to the problem under consideration.
60
Cha pter 3
Section :1 . 5
Neural Networks
61
Gender
=F
=M
Height
The building of the tree may be accomplished via an algorithm that examines
data from a training sample or could be created by a domain expert. Most decision tree
Height
<1.3 m
techniques differ in how the tree is created. We examine several decision tree technique
in later chapters of the book. Algorithm 3.2 shows the basic steps in applying a tuple to
the DT, step three in Definition 3 .4. We assume here that the problem to be performed
is one of prediction, so the last step is to make the prediction as dictated by the final
leaf node in the tree. The complexity of the algorithm is straightforward to analyze. For
>2 m
<= 2 m
Short Medium
Tall
each tuple in the databas!, we search the tree from the root down to a particular leaf.
At each level, the maximum number of comparisons to make depends on the branching
factor at that level. So the complexity depends on the product of the number of levels
and the maximum branching factor.
ALGORITHM 3.2
Inpu t :
/ /D e c i s ion t ree
T
D
/ / I nput database
Output :
/ / Model predi c t i on
M
DTProc algorithm:
/ / S i mp l i s t i c algori thm to i l lu s t r a t e predi c t ion
techn i que us ing DT
3.5
for each t E D do
n = root
whi l e
node of
not
T;
l e a f node
do
t;
f rom
at end o f t h i s arc ;
n;
EXAMPLE 3.7
Suppose that students in a particular university are to be classified as short, tall, or medium
based on their height. Assume that the database schema is {name, address, gender, height,
age, year, major}. To construct a decision tree, we must identify the attributes that are
important to the classification problem at hand. Suppose that height, age, and gender are
2 Note that we have two separate definitions: one for the tree itself and one
for the model. Although we
differentiate between the two here, the more common approach is to use the term
NE URAL N ETWORKS
The first proposal to use an artificial neuron appeared in 1943, but computer usage of
neural networks did not actually begin until the 1980s. Neural networks (NN), often
referred to as artificial neural networks (ANN) to distinguish them from biological neural
networks, are modeled after the workings of the human brain. The NN is actually an
information processing system that consists of a graph representing the processing system
as well as various algorithms that access that graph. As with the human brain, the
NN consists of many connected processing elements. The NN, then, is structured as
a directed graph with many nodes (processing elements) and arcs (interconnections)
between them. The nodes in the graph are like individual neurons, while the arcs are
their interconnections. Each of these processing elements functions independently from
the others and uses only local data (input and output to the node) to direct its processing.
This feature facilitates the use of NNs in a distributed and/or parallel environment.
The NN approach, like decision trees, requires that a graphical structure be built
to represent the model and then that the structure be applied to the data. The NN can be
viewed as a directed graph with source (input), sink (output), and internal (hidden) nodes.
The input nodes exist in a input layer, while the output nodes exist in an output layer.
The hidden nodes exist over one or more hidden layers. To perform the data mining
62
Chapter 3
Section 3 .5
task, a tuple is input through the input nodes and the output node determines what the
Neural Networks
F
(V, A) with
= { 1 , 2, . . . , n} and arcs A = { (i, j } 1 1 ::::: i, j s n } , with the following
prediction is. Unlike decision trees, which have only one input node (the root of the
vertices V
tree), the NN has one input node for each attribute value to be examined to solve the
63
restrictions:
data mining function. Unlike decision trees, after a tuple is processed, the NN may be .
1.
changed to improve future performance. Although the structure of the graph does not
nodes,
Vo .
VI ,
That is, if a poor solution to the problem is made, the network is modified to produce a
in layer
better solution to this problem the next time. A major drawback to the use of NNs is the
fact that they are difficult to explain to end users (unlike decision trees, which are easy
3. Any arc
to understand). Also, unlike decision trees, NNs usually work only with numeric data.
5. Node
for this problem in Figure 3.6. We first must determine the basic structure of the graph.
Since there are two important attributes, we assume that there are two input nodes. Since
we are to classify into three classes, we use three output nodes. The number of hidden
Definition 3 . 5 is a very simplistic view of NNs. Although there are many more compli
layers in the NN is not, easy to determine. In most cases, one or two is enough. In
cated types that do not fit this definition (as we will see later in the text) , this defines
this example, we assum that there is one hidden layer and thus a total of three layers
the most common type of NN, and the one that is used most often throughout this text.
in the general structure. We arbitrarily assume that there are two nodes in this hidden
More general NNs include some with arcs between any two nodes at any layers. Any
layer. Each node is labeled with a function that indicates its effect on the data coming
into that node. At the input layer, functions
approaches that use a more generalized view of a graph for NNs will be adequately
defined before usage.
attribute value in and replicate it as output on each of the arcs coming out of the node.
The functions at the hidden layer, h and
j4,
fs, !6 . and
h, perform more complicated functions, which are investigated later in this section. The
arcs are all labeled with weights, where
1. Neural network graph that defines the data structure of the neural network.
processing, the functions at each node are applied to the input data to produce the output.
3. Recall techniques that determine how information is obtained from the net
work. We discuss propagation in this text.
(3.27)
where h and g are the input height and gender values. Note that to determine the output
of a node we must know: the values input on each arc, the weights on all input arcs,
the technique used to combine the input values (a weighted sum is used here), and the
function h definition.
As with decision trees, we define a neural network in two parts: one for the data
structure and one for the general approach used, including the data structure and the
algorithms needed to use it.
NNs have been used in pattern recognition, speech recognition and synthesis, medi
cal applications (diagnosis, drug design), fault detection, problem diagnosis, robot control,
and computer vision. In business, NNs have been used to "advise" booking of airline
seats to increase profitability. As a matter of fact, NNs can be used to compute any func
tion. Although NNs can solve problems that seem more elusive to other AI techniques,
they have a long training time (time during which the learning takes place) and thus are
not appropriate for real-time applications. NNs may contain many processing elements
and thus can be used in massively parallel systems.
Artificial NNs can be classified based on the type of connectivity and learning. The
basic type of connectivity discussed in this text is
only to layers later in the structure. Alternatively, a NN may be feedback where some
links are back to earlier layers. Learning can be either supervised or unsupervised, as is
discussed in section 4.5.2.
. ,
, Xki . The values that flow on these arcs are shown on dashed arcs because they do
not really exist as part of the graph itself. There is one output value Yi produced. During
xu ,
. . .
propagation this value is output on all output arcs of the node. The activation function,
64
Chapter 3
F I G U R E 3.7:
Section 3 . 5
o ;..._
._ __... ........................ .
(a) Threshold
F I G U R E 3.8:
f; , is applied to the inputs, which are scaled by applying the corresponding weights.
The weight in the NN may be determined in two ways. In simple cases where much is
known about the problem, the weights may be predetermined by a domain expert. The
more common approach is to have them determined via a learning process.
The structure of ttle NN may also be viewed from the perspective of matrices.
Input and weight on the arcs into node i are
(3.28)
There is one output value from node i, y; , which is propagated to all output arcs during
the propagation process. Using summation to combine the inputs, then, the output of a
node is
Yi
f;
(t ) (
]=1
Wji Xj i
f;
[w r; . . Wk; ]
[ ])
(3 .29)
Xb
Here f; is the activation function. Because NNs are complicated, domain experts and
data mining experts are often advised to assist in their use. This in tum complicates the
process.
Overfitting occurs when the NN is trained to fit one set of data almost exactly. The
error that occurs with the given training data is quite small; however, when new data
are examined, the error is very large. In effect, the NN has "memorized" the training set
and cannot generalize to more data. Larger and more complicated NNs can be trained
to represent more complex functions. To avoid overfitting, smaller NNs are advisable.
However, this is difficult to determine beforehand. Another approach that can be used
to avoid overfitting is to stop the learning process early. Using a larger training set
also helps.
3.5.1
Activation Functions
The output of each node i in the NN is based on the definition of a function f; , activation
function, associated with it. An activation function is sometimes called a processing
element function or a squashing function. The function is applied to the set of inputs
coming in on the input arcs. Figure 3.7 illustrates the process. There have been many
proposals for activation functions, including threshold, sigmoid, symmetric sigmoid, and
Gaussian.
(b) Sigmoid
65
Neural N etworks
(c) Gaussian
..
CI:=l
+(I:=l
j; (S)
cS
(3.30)
Here c is a constant positive value. With the linear function, the output value has
no limits in terms of maximum or minimum values.
Threshold or step: The output value is either a 1 or 0, depending on the sum of the
products of the input values and their associated weights. As seen in Figure 3 .8(a),
values above a threshold, T, will be 1 or 0:
f; (S)
1
0
if S > _T
otherwise
(3 .31)
The binary output values may also be 1 or -1. Alternatively, the 1 value may
be replaced by any constant. A variation of this "hard limit" threshold function
66
Chapter 3
Section 3.6
f; (S) =
Is -
if s
>
r2
if r1 :::: s :::: r2
rl
if s
r2 - r1
0
<
r1
(3.32)
Sigmoid: As seen in Figure 3. 8(b), this is an "S"-shapd cun:e with output valm:s
.
t o g
between - 1 and 1 (or 0 and 1), and which is monotomcally tcreasmg.
there are several types of sigmoid functions, they all have thts charactenstlc S
shape. A commn one is the logistic function
?,
f; (S) =
(1 + e -cS )
(3 .33)
Here c is a constant positive value that changes the slope of the function. This
function possesses a simple derivative:
/; (1 - f; ) .
Gaussian: The Gaussian function, Figure 3.8(c), is a bell-shaped curve with output
values in the range [0, 1]. A typical Gaussian function is
(3.35)
Here s is the mean and v is the predefined positive variance of the function.
These are only a representative subset of the possible set of activation functions that
could be and have been used.
. .
Nodes in NNs often have an extra input called a bias. This bias value of 1 ts mput
on an arc with a weight of 8 The summation with bias input thus becomes
s;
L
j=l
w ji
xji
-e
67
GENETIC ALGORITH M S
Genetic algorithms are examples of evolutionary computing methods and are optimiza
tion-type algorithms. Given a population of potential problem solutions (individuals),
evolutionary computing expands this population with new and potentially better solu
tions. The basis for evolutionary computing algorithms is biological evolution, where
over time evolution produces the best or "fittest" individuals. Chromosomes, which are
DNA strings, provide the abstract model for a living organism. Subsections of the chro
mosomes, which are called genes, are used to define different traits of the individual.
During reproduction, genes from the parents are combined to produce the genes for
the child.
In data mining, genetic algorithms may be used for clustering, prediction, and even
association rules. You can think of these techniques as finding the "fittest" models from
a set of models to represent the data. In this approach a starting model is assumed and
through many iterations, models are combined to create new models. The best of these,
as defined by a fitness function, are then input into the next iteration. Algorithms differ
in how the model is represented, how different individuals in the model are combined,
and how the fitness function is used.
When using genetic algorithms to solve a problem, the first thing, and perhaps
the most difficult task, that must be determined is how to model the problem as a set
of individuals. In the real world, individuals may be identified by a complete encoding
of the DNA structure. An individual typically is viewed as an array or tuple of values.
Based on the recombination (crossover) algorithms; the values are usually numeric and
may be binary strings. These individuals are like a DNA encoding in that the structure
for each individual represents an encoding of the major features needed to model the
problem. Each individual in the population is represented as a string of characters from
the given alphabet.
DEFINITION 3.7.
This function has an output centered at zero, which may help with learning.
3.6
G e netic Algorithms
(3.3 6)
The effect of the bias input is to move the activation function on the X axis by a value
of e. Thus, a weight of -8 becomes a threshold of 8 .
The values that each character can have are called the :alleles. A population, P, is
a set of individuals.
Although individuals are often represented as bit strings, any encoding is possible.
An array with nonbinary characters could be used, as could more complicated data
structures including trees and arrays. The only real restriction is that the genetic operators
(mutation, crossover) must be defined.
In genetic algorithms, reproduction is defined by precise algorithms that indicate
how to combine the given set of individuals to produce new ones. These are called
crossover algorithms. Given two individuals (parents from the population, the crossover
technique generates new individuals (offspring or children) by switching subsequences
of the strings. Figure 3.9 illustrates the process of crossover. The locations indicating the
crossover points are shown in the figure with the vertical lines. In Figure 3 .9(a) crossover
is achieved by interchanging the last three bits of the two strings. In part (b) the center
three bits are interchanged. Figure 3.9 shows single and multiple crossover points. There
are many variations of the crossover approach, including determining crossover points
randomly. A crossover probability is used to determine how many new offspring are cre
ated via crossover. In addition, the actual crossover point may vary within one algorithm.
68
Chapter 3
l
111 1 111
ooo ooo
Data M i n i n g Techniques
l l
1 1 1 1 111 1 11
1
111 1 000
ooo ooo oo
000 1 1 1
Section 3.6
l l
1 1 1 l ooo l 11
ooo 1 1 1 oo
F I G U R E 3.9:
using the fitness function to determine the best individuals in P to keep. The
algorithm replaces a predefined number of individuals from the population
with each iteration and terminates when some threshold is met.
Crossover.
As in nature, however, mutations sometimes appear, and these also may be present
in genetic algorithms. The mutation operation randomly changes characters in the off
spring. A very small probability of mutation is set to determine whether a character
should change.
Since genetic algorithms attempt to model nature, only the strong survive. When
new individuals are created, a choice must be made about which individuals will survive.
This may be the new individuals, the old ones, or more likely a combination of the two.
The third maj or compnent of genetic algorithms, then, is the part that determines the
best (or fittest) individuals to survive.
One of the most important components of a genetic algorithm is determining how
to select individuals. A fitness function, f, is used to determine the best individuals
in a population. This is then used in the selection process to choose ? arets Given an
:
objective by which the population can be measured, the fitness functwn md1cates how
well the goodness objective is being met by an individual.
DEFINITION 3.8. Given a population, P, a fitness function, f, is a mapping
f : P --+ R.
The simplest selection process is to select individuals based on their fitness:
PI; =
f ( I; )
L f ( IJ )
(3.37 )
lj E P
Here p 1-' is the probability of selecting individual /; . This type of selection is called
roulette wheel selection. One problem with this approach is that it is still possible to
select individuals with a very low fitness value. In addition, when the distribution is
quite skewed with a small number of extremely fit individuals, these individuals may
be chosen repeatedly. In addition, as the search continues, the population becomes less
diverse so that the selection process has little effect.
DEFINITION 3.9. A genetic algorithm (GA) is a computational model consisting
of five parts:
1. Starting set of individuals, P .
2 . Crossover technique.
3. Mutation algorithm.
4. Fitness function.
69
Children
Parents
(b) Multiple crossover
Children
Parents
(a) Single crossover
Genetic A l gorithms
Suppose that each solution to the problem to be solved is represented as one of these
individuals. A complete search of all possible individuals would yield the best individual
or solution to the problem using the predefined fitness function. Since the search space
is quite large (perhaps infinite), what a genetic algorithm does is to prune from the
search space individuals who will not solve the problem. In addition, it only creates new
individuals who probably will be much different from those previously examined. Since
genetic algorithms do not search the entire space, they may not yield the best result.
However, they can provide approximate solutions to difficult problems .
ALGORITHM 3.3
Input :
// I n i t i a l populat ion
Output :
P
/ / Improved populat ion
Genetic algor ithm:
/ /Algori thm t o i l lustrate genetic algori thm
repeat
N =l P I ;
P = 0;
repeat
i 1 , i 2 = select( P) ;
o1 , 02 = cros s ( i 1 , i 2 ) ;
01 = mutate(o1 ) ;
o2 = mut ate(o2 ) ;
p
p u { 01 ' 02 } ;
unt i l I P I= N ;
=
P= P;
Chapter 3
70
Data M i n i n g Tech n i q u es
Sectio n 3.8
The major advantage to the use of genetic algorithms is that they are easily parallelized. There are, however, many disadvantages to their use:
3.7
The abstraction of the problem and method to represent individuals is quite difficult.
x; .
4. Use linear regression with one predictor to determine the formula for the output
given the following samples: ( 1 , 3} and (2, 5 } . Then predict the output value with
an input of 5.
5. Calculate the correlation coefficient
(a) X values: 1, 2, 3, 4, 5, 6, 7, 8, 9, 1 0
Y values: 5, 7, 8, 9, 10, 1 2, 1 3 , 15, 1 8, 20
(b)
X values: 1 , 2, 3, 4, 5, 6, 7, 8, 9, 10
(c) X values: 3, 5, 2, 1 , 1 0
Y values: 10, 5, 8, 7, 2
6. Find the similarity between (0, 1 , 0.5, 0.3, 1) and ( 1 , 0, 0.5, 0, 0} using the Dice,
7. Given the decision tree in Figure 3 .5, classify each of the following students:
8. Using the NN shown in Figure 3 .6, classify the same students as those used in
exercise 7. Assume that the input nodes use an identity function, the hidden nodes
use a hyperbolic tangent activation function, the output layer uses a sigmoidal
function, and a weighted sum is used to calculate input to each node in the hidden
and output layers. You may assume any value for the constants in the activation
functions. Assume that the trained weights are defined by the difference between
the node numbers at either end of the arc. For example, the weight w 1 3
0 . 2 and
W47
0.3.
=
9. Given an initial population { ( 1 0 1 0 1 0} , (001 100} , (01010 1 } , (0000 10} } , apply the
genetic algorithm to find a better population. Suppose the fitness function is defined
as the sum of the bit values for each individual and that mutation always occurs by
negating the second bit. The termination condition is that the average fitness value
for the entire population must be greater than 4. Also, an individual is chosen
71
J?ata mining owes much of its development to previous work in machine learning, statis
1. Given the following set of values { 1 , 3, 9, 15, 20}, determine the jackknife estimate
3.8
EXERCISES
B i b l i o g raphic Notes
P A R T
T W O
CO R E TO PICS
C H A P T E R
Classifi cation
4. 1
INTRODUCTION
4.2
STATISTICAL-BASED ALGORITH M S
4.3
DISTANCE-BASED ALGORITHMS
4.4
4.5
4.6
RULE-BASED ALGORITHMS
4.7
4.8
SUM MARY
4.9
EXERCISES
4. 1
INTRODUCTION
Classification is perhaps the most familiar and most popular data mining technique.
Examples of classification applications include image and pattern recognition, medical
diagnosis, loan approval, detecting faults in industry applications, and classifying financial
market trends. Estimation and prediction may be viewed as types of classification. Wheri
someone estimates your age or guesses the number of marbles in a jar, these are actually
classification problems. Prediction can be thought of as classifying an attribute value into
one of a set of possible classes. It is often viewed as forecasting a continuous value, while
classification forecasts a discrete value. Example 1 . 1 in Chapter 1 illustrates the use of
classification for credit card purchases. The use of decision trees and neural networks
(NNs) to classify people according to their height was illustrated in Chapter 3 . Before
the use of current data mining techniques, classification was frequently performed by
simply applying knowledge of the data. This is illustrated in Example 4. 1 .
EXAMPLE 4.1
Teachers classify students as A, B, C, D, or
90 ::5 grade
80 ::5 grade < 90
70 ::5 grade < 80
60 ::5 grade < 70
grade < 60
A
B
C
D
F
75
76
Chapter 4
Section 4. 1
Classification
9
8765
43-
21
0 0- 11 2I
C
C
2. Apply the model developed in step 1 by classifying tuples from the target database.
Although the second step actually does the classification (according to the definition in
Definition 4. 1 ) , most research has been applied to step 1. Step 2 is often straightforward.
As discussed in [KLR+ 98], there are three basic methods used to solve the classi
fication problem:
Specifying bou:"daries. Here classification is performed by dividing the input
space of potential database tuples into regions where each region is associated
with one class.
CJ)
P (CJ
The naive divisios used in Example 4. 1 as well as decision tree techniques are examples
of the first modehng approach. Neural networks fall into the third category.
1 In this discussion each tuple in the database is assumed to consist of a single value rather than a set of
values.
3 4 5 6 7 8
9 f- X
8 f7 ,_x
X
6
X
5 f-X
4 fX
X
3 I-
2 f- X
1
0 of- 1I 2I
lx
X
X
X
X
lx I I
3 4 5 6 7 8
Classification problem.
Tall
Medium
Short
The Output2 results require a much more complicated set of divisions using both height
and gender attributes.
In this chapter we examine classification algorithms based on the categorization
as seen in Figure 4.2. Statistical algorithms are based directly on the use of statistical
information. Distance-based algorithms use similarity or distance measures to perform
the classification. Decision tree and NN approaches use these structures to perform the
classification. Rule-based classification algorithms generate if-then rules to perform the
classification.
P(Cj)P(ti C .)
CJ.
2
1
00 1 2
2 m s Height
1 . 7 m < Height < 2 m
Height S 1. 7 m
(ti 1
Using probability distributions. For any given class,
is the PDF for
the class evaluated at one point, ti . 1 If a probability of occurrence for each class
1 1
is known (perhaps determined by a domain expert), then
.
IS used to estimate the probability that t; is in class C
J.
10
the taining data (including defined classification for each tuple) and as output a
.
defirutwn of the model developed. The model created classifies the training data
as accurately as possible.
! (CJ)
X
X
77
(x, y }
Suppose we are given that a database consists of tuples of the form t
where 0 S x S 8 and 0 S y S 10. Figure 4. 1 illustrates the classification problem.
Figure 4. l (a) shows the predefined classes by dividing the reference space, Figure 4. l (b)
provides sample input data, and Figure 4. l (c) shows the classification of the data based
on the defined classes.
A major issue associated with classification is that of overfitting. If the classification
strategy fits the training data exactly, it may not be applicable to a broader population of
data. For example, suppose that the training data has erroneous or noisy data. Certainly
in this case, fitting the data exactly is not desired.
In the following sections, various approaches to perfonning classification are exam
ined. Table 4.1 contains data to be used throughout this chapter to illustrate the various
techniques. This example assumes that the problem is to classify adults as short, medium,
or tall. Table 4. 1 lists height in meters. The last two columns of this table show two clas
sifications that could be made, labeled Output l and Output2, respectively. The Outputl
classification uses the simple divisions shown below:
1. Create a specific model by evaluating the training data. This step has as input
CJ, P
9
8
7
6
5
F I G U R E 4. 1 :
tp
Class A
3 4 5 6 7 8
Class C
Class B
(a) Definition of classes
Our definition views classification as a mapping from the database to the set of classes.
Note that the classes are predefined, are nonoverlapping, and partition the entire database.
Each tuple in the database is assigned to exactly one class. The classes that exist for a
classification problem are indeed equivalence classes. In actuality, the problem usually
is implemented in two hases :
10
10 -
I ntroduction
4. 1 . 1
Issues in Classification
Missing Data. Missing data values cause problems during both the training phase and
the classification process itself. Missing values in the training data must be handled
78
Chapter 4
Section 4.1
Classification
Introduction
79
often a fuzzy problem, the correct answer may depend on the user. Traditional algorithm
evaluation approaches such as determining the space and time overhead can be used, but
these approaches are usually secondary .
Name
Gender
Height
Output2
Outputl
placed in the correct class. This ignores the fact that there also may be a cost asso
Kristina
1 .6 m
Short
Medium
Jim
2 m
Tall
Medium
Maggie
Martha
1 .9 m
1 .88 m
1 .7 m
1 .85 m
Short
Medium
Medium
the classification, as is shown in Figure 4 . 3 . The upper left and lower right quad
Medium
1 .6 m
1 .7 m
rants [for both Figure 4.3 (a) and (b)] represent correct actions. The remaining two
Short
Medium
Short
Medium
mined by associating costs with each of the quadrants. However, this would be dif
Stephanie
Bob
Kathy
Dave
ciated with an incorrect assignment to the wrong class. This perhaps should also be
determined.
Medium
Tall
Medium
Tall
tion retrieval systems. With only two classes, there are four possible outcomes with
Worth
2.2 m
Tall
Tall
Steven
2. 1 m
Tall
Tall
Debbie
Medium
Medium
Todd
Medium
Medium
Medium
Tall
Medium
again gives us the four quadrants shown in Figure 4.3(c), which can be described in the
Medium
following ways:
Medium
Medium
Kim
Amy
1.8 m
1 .95 m
1 .9 m
1.8 m
Wynette
1 .75 m
'
l D!stance
.
Statistlca
DT
NN
Rules
classes.
Given a specific class,
assigned to that class while its actual membership may or may not be in that class. This
and may produce an inaccurate result. Missing data in a tuple to be classified must be
area to examine false alarm rates. It has also been used in information retrieval to examine
able t be andled by the resulting classification scheme. There are many approaches to
.
handlmg ffilssmg data:
fallout (percentage of retrieved that are not relevant) versus recall (percentage of retrieved
false positives and true positives. An OC curve was originally used in the communications
that are relevant) . In the OC curve the horizontal axis has the percentage of false positives
and the vertical axis has the percentage of true positives for a database sample. At the
Assu
data
IS
NOTRET
REL
RET
NOTREL
NOTRET
NOTREL
RET
REL
otice t e similarity between missing data in the classification problem and that of nulls
.
m traditional databases.
80
Chapter 4
Classification
Section 4.2
it can also be used for other applications such as forecasting. Although not explicitly
described in this text, regression can be performed using many different types of tech
niques, including NNs. In actuality, regression takes a set of data and fits the data to
a formula.
Looking at Figure 3.3 in Chapter 3, we see that a simple linear regression problem
can be thought of as estimating the formula for a straight line (in a two-dimensional
space). This can be equated to partitioning the data into two classes. With the banking
example, these would be to approve or reject a loan application. The straight line is the
break-even point or the division between the two classes.
In Chapter 2, we briefly introduced linear regression using the formula
75%
0
0..
<1)
=
50%
F:
25%
25%
50%
75%
Y = CO + Cl X l +
100%
False positives
+ CnXn
TABLE 4.2:
Confusion Matrix
Actual
Membership
Short
Medium
Tall
Assignment
Short
Medium
Tall
0
0
0
4
5
0
3
2
beginning of evaluating a sample, there are none of either category, while at the end
there are 100 percent of each. When evaluating the results for a specific sample, the
curve looks like a j agged stair-step, as seen in Figure 4.4, as each new tuple is either
a false positive or a true positive. A more smoothed version of the OC curve can also
be obtained.
A confusion matrix illustrates the accuracy of the solution to a classification prob
lem. Given m classes, a confusion matrix is an m x m matrix where entry ci,J indicates the
number of tuples from D that were assigned to class C1 but where the correct class is Ci .
Obviously, the best solutions will have only zero values outside the diagonal. Table 4.2
shows a confusion matrix for the height example in Table 4. 1 where the Outputl assign
ment is assumed to be correct and the Output2 assignment is what is actually made.
4.2
4.2.1
Regression problems deal with estimation of an output value based on input values. When
used for classification, the input values are values from the database D and the output
values represent the classes. Regression can be used to solve classification problems, but
(4. 1 )
F I G U R E 4.4:
81
FIGURE 4.5:
82
Chapter 4
There are many reasons why the linear regression model may not be used to
estimate output data. One is that the data do not fit a linear model. It is possible, however,
that the data generally do actually represent a linear model, but the linear model generated
is poor because noise or outliers exist in the data. Noise is erroneous data. Outliers are
data values that are exceptions to the usual and expected data. Example 4.2 illustrates
o?tliers: Iri these cases the observable data may actually be described by the following:
y = CO + C 1 X 1 +
+ CnXn + E
(4.2)
Here E is a random error with a mean of 0. As with point estimation, we can estimate the
accuracy of the fit of a linear regression model to the actual data using a mean squared
error function.
EXAMPLE 4.2
Suppose that a graduate level abstract algebra class has 100 students. Kristina consistently
outperforms the other 4;tudents on exams. On the final exam, Kristina gets a grade of 99.
The next highest grade is 75, with the range of grades being between 5 and 99. Kristina
clearly is resented by the other students in the class because she does not perform at
the same level they do. She "ruins the curve." If we were to try to fit a model to the
grades, this one outlier grade would cause problems because any model that attempted
to include it would then not accurately model the remaining data.
We illustrate the process using a simple linear regression formula and assuming k
points in our training sample. We thus have the following k formulas:
Yi
= co + C1X1 i + Ei , i
1, . . . , k
(xi i , Yi
L =
EXAMPLE 4.3
By looking at the data in the Qutput1 column from Table 4.1 and the basic understanding
that the class to which a person is assigned is based only on the numeric value of his or
her height, in this example we apply the linear regression concept to determine how to
distinguish between the short and medium classes. Figure 4.6(a) shows the points under
consideration. We thus have the linear regression formula of y = co + E . This implies
that we are actually going to be finding the value for co that best partitions the height
numeric values into those that are short and those that are medium. Looking at the data
in Table 4. 1, we see that only 12 of the 15 entries can be used to differentiate between
short and medium persons. We thus obtain the following values for
in our training
data: { 1 .6, 1 .9, 1 . 88, 1 .7, 1 . 85, 1 . 6, 1 . 7 , 1 . 8, 1 .95, 1 .9, 1.8, 1 .75 } . We wish to minimize
Yi
12
L =
-2
(4.3)
Taking the partial derivatives (with respect to the coefficients) and setting equal to zero,
we can obtain the least squares estimates for the coefficients, co and c1 .
Regression can be used to perfonn classification using two different approaches:
1. Division: The data are divided into regions based on class.
2. Prediction: Formulas are generated to predict the output class value.
The first case views the data as plotted in an n-dimensional space without any explicit
class values shown. Through regression, the space is divided into regions-one per
class. With the second approach, a value for each class is included in the graph. Using
regression, the formula for a line to predict class values is generated.
12
12
i=1
i= 1
L Yi + L 2co
2 i-
2-
1.9 -
1.9 i:=
:t
1.7 r-
1.6 -
1.8 I-
2.1
2.1
(4.4)
12
2
L f L (Yi - co )
i=1
i=1
E =
Taking the derivative with respect to co and setting equal to zero we get
L Ef = .L:<Yi - co - C1X 1i ) 2
i= 1
i=
83
Example 4.3 illustrates the division process, while Example 4.4 illustrates the pre
diction process using the data from Table 4. 1 . For simplicity, we assume the training data
include only data for short and medium people and that the classification is performed
using the Outputl column values. If you extend this example to all three classes, you
will see that it is a nontrivial task to use linear regression for classification. It also will
become obvious tlat the result may be quite poor.
Statistical-Based Algorithms
Section 4.2
C l assification
:=
:t
Medium
1.8 i-
1.7 r-
1.6 I-
y = 1.786
Short
1.5
1.5
'
(b) Division
84
Chapter 4
co =
We thus have the division between short and medium persons as being determined by
y = 1 .786, as seen in Figure 4.6(b).
We now look at predicting the class using the short and medium data as input and looking
at the Output! classification. The data are the same as those in Example 4.3 except that we
now look at the classes as indicated in the training data. Since regression assumes numeric
data, we assume that the value for the short class is 0 and the value for the medium class
is 1. Figure 4.7(a) sho the data for this example: { ( 1 .6, 0), ( 1 .9, 1), ( 1 .88, 1), ( 1 .7, 0),
(1 .85, 1), (1 .6, 0), ( 1 .7, 0), ( 1 .8, 1), (1 .95, 1), ( 1 .9, 1), ( 1 .8, 1), ( 1 .75, 1 ) } . In this case
we are using the regression formula with one variable:
Y = CO + C!XJ + E
We thus wish to minimize
12
12
12
- Eu o.s
6 0.5 r"'
1.8
I
2
I
2.2
I
2.4
Height
(a) Short and medium heights with classes
FIGURE 4.7:
We can now solve for co and C J . Using the data from the 1 2 points in the training data,
we have L:: xli = 21 .43, L Yi = 8, L:: <x l i y; ) = 14.83, and L:: < xr; ) = 38 .42. Thus, we
get q = 3.63 and co = -5.8 16. The prediction for the class value is thus
In Example 4.4 a line that predicts the class value is generated. This was done for
two classes, but it also could have been done for all three classes. Unlike the division
approach where class membership is obvious based on the region within which a point
occurs, with prediction the class to which a point belongs is less obvious. Here we
predict a class value. In Figure 4.7(b) the class value is predicted based on the height
value alone. Since the prediction line is continuous, howe:ver, the class membership is
not always obvious. For example, if the prediction for a value is 0.4, what would its class
be? We can determine the class by splitting the line. So a height is in the short class if
its prediction value is less than 0.5 and it is in the medium class if its value is greater
than 0.5. In Example 4.4 the value of x 1 where y
0.5 is 1 .74. Thus, this is really the
division between the short class and the medium class.
If the predictors in the linear regression function are modified by some function
(square, square root, etc.), then the model looks like
aL
- = -2 L Yi + 'L: 2co + L 2C!X1 i = 0
aco
;,1
i= 1
i=1
12
Taking the partial derivative with respect to co and setting equal to zero we get
Now taking the partial of L with respect to c 1 , substituting the value for co, and setting
equal to zero we obtain
y = - 5 . 8 1 6 + 3.63x1
12
L = L Ef = L(y; - co - C1X1i)2
i=1
i= l
1.6
L Yi - L CJ X !i
12
aL
- = 2 "'(y; - CO - C!X1i) (-X1; ) = 0
ac1
EXAMPLE 4.4
0 I-
85
To simplify the notation, in the rest of the example we drop the range values for the
summation because all are the same. Solving for co, we find that:
12
I >i
i
co = =1 = 1 .786
12
1 1-
Section 4.2
Classification
(4.5)
0
1.6
1 .8
Height
(b) Prediction
2.2
2.4
where fi is the function being used to transform the predictor. In this case the regression
is called nonlinear regression. Linear regression techniques, while easy to understand,
are not applicable to most complex data mining applications. They do not work well
with nonnumeric data. They also make the assumption that the relationship between the
input value and the output value is linear, which of course may not be the case.
86
Chapter 4
Section 4.2
Classification
Linear regression is not always appropriate because the data may not fit a straight
line, but also because the straight line values can be greater than 1 and less than 0. Thus,
they certainly cannot be used as the probability of occurrence of the target class. Another
prediction to be made. The approach is called "naive" because it assumes the indepen
commonly used regression technique is called logistic regression. Instead of fitting the
dence between the various attribute values. Given a data value Xi the probability that a
data to a straight line, logistic regression uses a logistic curve such as is illustrated in
e <co+CJxJ)
=
(4.6)
1 + e <co+cixil
of class membership. As with linear regression, it can be used when classification into
two classes is desired. To perform the regression, the logarithmic function can be applied
to obtain the logistic function
loge
p
-1 - p
The logistic curve gives a value between 0 and 1 so it can be interpreted as the probability
I ti ) .
Given a training set, the naive B ayes algorithm first estimates the prior probability
P ( C 1) for each class by counting how often each class occurs in the training data. For
each attribute, Xi , the number of occurrences of each attribute value Xi can be counted
to determine P (xi) . Similarly, the probability P (xi I Cj) can be estimated by counting
how often each value occurs in the class in the training data. Note that we are looking
at attribute values here. A tuple in the training data may have many different attributes,
each with many values. This must be done for all attributes and all values of attributes.
co + C [ XJ
(4.7)
We then use these derived probabilities when a new tuple must be classified. This is
why naive Bayes classification can be viewed as both a descriptive and a predictive
type of algorithm. The probabilities are descriptive and are then used to predict the class
Here p is the probability of being in the class and 1 - p is the probability that it is
from the training set are used to make the prediction. This is done by combining the
not. However, the process chooses values for co and q that maximize the probability of
4.2.2
87
Statistical-Based Algorith ms
When classifying a target tuple, the conditional and prior probabilities generated
effects of the different attribute values from the tuple. Suppose that tuple ti has p indepen
Bayesian Classification
dent attribute values {xi ! , Xi2 . . . , Xi p } From the descriptive phase, we know P ( Xi k
Assuming that the contribution by all attributes are independent and that each contributes
CJ ) ,
equally to the classification problem, a simple classification scheme called naive Bayes
classification has been proposed that is based on Bayes rule of conditional probability as
P (ti I Cj )
IT P (Xi k I Cj )
(4.8)
k=l
At this point in the algorithm, we then have the needed prior probabilities P ( C J) for
each class and the conditional probability P (ti I CJ ) . To calculate P (ti ) , we can estimate
the likelihood that ti is in each class. This can be done by finding the likelihood that
this tuple is in each class and then adding all these values. The probability that ti is
in a class is the product of the conditional probabilities for each attribute value. The
posterior probability P ( CJ
8
15
6b
.s
I ti)
is then found for each class. The class with the highest
probability is the one chosen for the tuple. Example 4.5 illustrates the use of naive B ayes
0.6
classification.
0.5
0.4
EXAMPLE 4.5
0.3
Using the Outputl classification results for Table 4. 1 , there are four tuples classified as
0.2
short, eight as medium, and three as tall. To facilitate classification, we divide the height
0.1
oo)
Table 4.3 shows the counts and subsequent probabilities associated with the attributes.
With these training data, we estimate the prior probabilities:
P (short)
4/ 1 5
0.267, P (medium)
8/15
3/ 1 5
0.2
88
Chapter 4
Classification
Section 4.3
Probabilities
Count
Value
Gender
M
F
Height
(0, 1 .6]
( 1 .6, 1 .7]
( 1 .7' 1 . 8]
( 1 .8, 1 . 9]
( 1 .9, 2]
(2, oo)
Short
Medium
Tall
Short
Medium
Tall
1
3
2
2
0
0
0
0
2
6
0
0
3
4
1
0
3
0
0
0
0
0
1
2
1/4
3/4
2/4
2/4
0
0
0
0
2/8
6/8
0
0
3/8
4/8
118
0
3/3
0/3
0
0
0
0
1/3
2/3
4.3
1 /4
P (t I medium)
2/8
1/8
0.03 1
P (t I tall)
3 /3
1/3
0.333
0.267
0.03 1
0.33
0.2
(4.9)
0.533
=
0.0166
0. 066
(4. 1 0)
(4. 1 1)
We estimate P (t) by summing up these individual likelihood values since t will be either
short or medium or tall:
P (t)
0 + 0.0 1 66 + 0. 066
0.0826
(4. 1 2)
0.0267
=
0.0826
0.03 1 X 0.533
0.0826
(4. 1 3)
0.2
(4. 14)
0'799
(4. 1 5 )
0.0826
0.333 X 0.2
DISTANCE-BASED ALGORITH M S
Each item that i s mapped to the same class may be thought o f as more similar to the
other items in that class than it is to the items found in other classes. Therefore, similarity
(or distance) measures may be used to identify the "alikeness" of different items in the
database. The concept of similarity measure was introduced in Chapter 2 with respect to
IR retrieval. Certainly, the concept is well known to anyone who has performed Internet
searches using a search engine. In these cases, the set of Web pages represents the whole
database and these are divided into two classes : those that answer your query and those
that do not. Those that answer your query should be more alike than those that do not
answer your query. The similarity in this case is defined by the query you state, usually
a keyword list. Thus, the retrieved pages are similar because they all contain (to some
degree) the keyword list you have specified.
The idea of similarity measures can be abstracted and applied to more general
classification problems. The difficulty lies in how the similarity measures are defined
and applied to the items in the database. Since most similarity measures assume numeric
(and often discrete) values, they might be difficult to use for more general or abstract
data types. A mapping from the attribute domain to a subset of the integers may
be used.
Using a similarity measure for classification where the classes are predefined is
somewhat simpler than using a similarity measure for clustering where the classes are
not known in advance. Again, think of the IR example. Each IR query provides the
class definition in the form of the IR query itself. So the classification problem then
becomes one of determining similarity not among all tuples in the database but between
each tuple and the query. This makes the problem an O (n) problem rather than an
2
O (n ) problem.
1
We use these values to classify a new tuple. For example, suppose we wish to classify
t
(Adam , M, 1 .95 m) . By using these values and the associated probabilities of gender
and height, we obtain the following estimates:
P (t I short)
89
The naive Bayes approach has several advantages. First, it is easy to use. Second,
unlike other classification approaches, only one scan of the training data is required.
The naive Bayes approach can easily handle missing values by simply omitting that
probability when calculating the likelihoods of membership :in each class. In cases where
there are simple relationships, the technique often does yield good results.
Although the naive Bayes approach is straightforward to use, it does not always
yield satisfactory results. First, the attributes usually are not independent. We could use a
subset of the attributes by ignoring any that are dependent on others. The technique does
not handle continuous data. Dividing the continuous values into ranges could be used to
solve this problem, but the division of the domain into ranges is not an easy task, and
how this is done can certainly impact the results.
Attribute
Distance-Based Algorithms
Therefore, based on these probabilities, w e classify the new tuple a s tall because i t has
the highest probability.
4.3 . 1
S i m ple Approach
. .
Chapter 4
90
Section 4.3
Classification
Distance-Based Algorithms
91
. .
. . .
To calculate these similarity measures, the representative vector for each class
must be determined. Referring to the three classes in Figure 4. l (a), we can determine a
representative for each class by calculating the center of each region. Thus class A is
represented by (4, 7.5), class B by (2, 2.5), and class C by (6, 2.5 ) . A simple classifica
tion technique, then, would be to place each item in the class where it is most similar
(closest) to the center of that class. The representative for the class may be found in other
ways. For example, in pattern recognition problems, a predefined pattern can be used
to represent each class. Once a similarity measure is defined, each item to be classified
will be compared to each predefined pattern. The item will be placed in the class with
the largest similarity 'value. Algorithm 4. 1 illustrates a straightforward distance-based
approach assuming that each class, Ci , is represented by its center or centroid. In the
algorithm we use Ci to be the center for its class. Since each tuple must be compared
to the center for a class and there are a fixed (usually small) number of classes, the
complexity to classify one tuple is O (n).
ALGORITHM 4.1
Input :
c1 , . . .
t
, Cm
'
I / Centers
f o r each c l a s s
/ / Input tuple t o clas s i fy
/ /C l a s s to which t i s a s s igned
'
'
'
'
10
X
9 f--
7-
3 f--
x- - - -
5 1-
2 1-
t/
6 1-
4 f--
I
X
X
X
1 1-
8
.
the process used by KNN Here the points in the training set are shown and K
3. The
three closest items in the training set are shown; t will be placed in the class to which
most of these are members.
Algorithm 4.2 outlines the use of the KNN algorithm. We use T to represent
the training data. Since each tuple to be classified must be compared to each element
in the training data, if there are q elements in the training set, this is O (q). Given n
elements to be classified, this becomes an 0 (nq) problem. Given that the training data
are of a constant size (although perhaps quite large), this can then be viewed as an O(n)
problem.
.
One common classification scheme based on the use of distance measures is that of
the K nearest neighbors (KNN). The KNN technique assumes that the entire training set
includes not only the data in the set but also the desired classification for each item. In
effect, the training data become the model. When a classification is to be made for a new
item, its distance to each item in the training set must be determined. Only the K closest
entries in the training set are considered further. The new item is then placed in the
class that contains the most items from this set of K closest items. Figure 4 . 1 0 illustrates
K Nearest Neighbors
I
I
I
', I
, I 'X
CB :l': - - x
d i s t = oo ;
for i : = 1 t o m do
i f di s ( ci , t) < di s t , then
c= i;
d i s t = dist(ci , t) ;
Figure 4.9 illustrates the use of this approach to perform classification using the
data found in Figure 4. 1 . The three large dark circles are the class representatives for the
three classes. The dashed lines show the distance from each item to the closest center.
4.3.2
'
Output :
Class A
92
Chapter 4
ALGORITHM
Classification
Section 4.4
4.2
93
The decision tree approach to classification is to divide the search space into rect
Input :
angular regions . A tuple is classified based on the region into which it falls. A definition
for a decision tree used in classification is contained in Definition 4.3. There are alter
/ /Number of
ne ighbors
/ / Input
tuple
/ /Class
to which
Output :
KNN
algori t hm :
N= 0;
/ /Algori thm t o
for each
if
native definitions; for example, in a binary DT the nodes could be labeled with the
t o c l a s s i fy
t
is
predicates themselves and each arc would be labeled with yes or no (like in the "Twenty
d E T do
I N I .:S K, then
N = NU {d} ;
uEN
begip
DEFINITION
else
if 3
Questions" game).
ass igned
s u c h t h a t s im ( t ,
u)
N,
for
_::: s im( t,
is a set of classes C
d) ,
then
to which
f o r c l a s s i f i c at ion
the mo st
uEN
{t1 , . . . , tn } where t;
{C1 ,
. .
(ti l , . . , t; h } and
, Ah } . Also given
.
Each arc is labeled with a predicate that can be applied to the attribute associated
with the parent.
end
/ / Find c l a s s
i'!= N - { u} ;
N = NU {d} ;
c = class
are c l a s s i f ie d ;
Example 4 . 6 illustrates this technique using the sample data from Table 4. 1 . The
KNN technique is extremely sensitive to the value of K. A rule of thumb is that K <
jnumber of training items [KLR+ 98] . For this example, that value is 3.46. Commerci ;;j
2. For each t;
DT represents the logic needed to perform the mapping. Thus, it implicitly defines the
sample data found in Table 4. 1 is that shown in the column labeled Output2. A different
mapping. Using the DT shown in Figure 3.5 from Chapter 3, the classification of the
Using the sample data from Table 4. 1 and the Output l classification as the training set
output value, we classify the tuple (Pat, F, 1 .6} . Only the height is used for distance calcu
lation so that both the Euclidean and Manhattan distance measures yield the same results;
that is, the distance is simply the absolute value of the difference between the values.
Suppose that K
5 is given. We then have that the K nearest neighbors to the input tuple
are
{ (Kristina, F, 1 .6} , (Kathy, F, 1 .6} , (Stephanie, F, 1 .7} , (Dave, M, 1 . 7} , (Wynette,
F, 1 .75} } . Of these five items, four are classified as short and one as medium. Thus,
=
4.4
tinuous data. These attribute domains must be divided into categories to be handled.
The approach used is that the domain space is divided into rectangular regions [such as
is seen in Figure 4. l (a)]. Not all classification problems are of this type. The division
shown by the simple loan classification problem in Figure 2.4(a) in Chapter 2 cannot be
handled by DTs. Handling missing data is difficult because correct branches in the tree
could not be taken. Since the DT is constructed from the training data, overfitting may
occur. This can be overcome via tree pruning. Finally, correlations among attributes in
the database are ignored by the DT process.
Chapter
94
Section
Classificati o n
4.4
Input :
/ / Training da t a
/ /D e c i s i o n t r e e
DTBuild algor i t hm :
Height
Short
Short
<l.
Medium Tall
Gender Tall
Height
<1.
2m
Output :
<1.3
Gender
Height
ALGORITHM 4.3
95
;/\\
Height
< = 1.8
Medium
Height
1.8 m
<1.5
Tall
Short
= 1 .5 m
Medium
for
each arc do
D = Database cre a t e d by app lying spl i t t i ng p redicate to D ;
then
1.3
else
1
T = DTBu i l d(D) ;
1.5
I
2.0
1.8
T = Add T to arc ;
There have been many decision tree algorithms . We illustrate the tree-building
phase in the simplistic DTBuild Algorithm 4.3. Attributes in the database schema that will
be used to label nodes in the tree and around which the divisions will take place are called
the splitting attributes. The predicates by which the arcs in the tree are labeled are called
the splitting predicates. In the decision trees shown in Figure 4. 1 1 , the splitting attributes
are {gender, height} . The splitting predicates for gender are { = female, = male}, while
those for height include {< 1 .3 m, > 1 . 8 m, < 1 . 5 m, >2 m} . The splitting predicates for
height differ based on whether the tuple is for a male or a female. This recursive algorithm
builds the tree in a top-down fashion by examining the training data. Using the initial
training data, the "best" splitting attribute is chosen first. Algorithms differ in how they
determine the "best attribute" and its "best predicates" to use for splitting. Once this
has been determined, the node and its arcs are created and added to the created tree.
The algorithm continues recursively by adding new subtrees to each branching arc. The
algorithm terminates when some "stopping criteria" is reached. Again, each algorithm
determines when to stop the tree differently. One simple approach would be to stop when
the tuples in the reduced training set all belong to the same class. This class is then used
to label the leaf node created.
Note that the major factors in the performance of the DT building algorithm are
the size of the training set and how the best splitting attribute is chosen. The following
issues are faced by most DT algorithms:
1.3
1 .8
1.5
2.0
Height
>=1.3 m
<1.5 m
1\
1\
Short
Medium
:,
Short
Tall
1.5
1.8
2.0
< 1 .5
Height
=2 m
Medium
H H I
1.3
Tall
M
I
1.3
I
1.5
1.8
2.0
Splits: Associated with the ordering of the attributes is the number of splits to
take. With some attributes, the domain is small, so the number of splits is obvious
based on the domain (as with the gender attribute). However, if the domain is
continuous or has a large number of values, the number of splits to use is not
easily determined.
96
Chapter 4
Section 4.4
Classification
Stopping criteria: The creation of the tree definitely stops when the training data
are perfectly classified. There may be situations when stopping earlier would be
desirable to prevent the creation of larger trees. This is a trade-off between accuracy
of classification and performance. In addition, stopping earlier may be performed
to prevent overfitting. It is even conceivable that more levels than needed would
be created in a tree if it is known that there are data distributions not represented
in the training data.
The time and space omplexity of DT algorithms depends on the size of the training
data, q ; the number of attnbutes, h ; and the shape of the resulting tree. In the worst case
the DT that is built may be quite deep and not bushy. As the tree is built, for each o
the se nodes, eah attribte will be examined to determine if it is the best. This gives
.
a time complexity to build the tree of 0 (h q log q ). The time to classify a database of
size n is based on the height of the tree. Assuming a height of 0 (log q ), this is then
O (n log q).
Training data: The structure of the DT created depends on the training data. If the
training data set is too small, then the generated tree might not be specific enough
to work properly with the more general data. If the training data set is too large,
then the created tree may overfit.
Pruning: Once a tree is constructed, some modifications to the tree might be
needed to improve the performance of the tree during the classification phase. The
pruning phase might remove redundant comparisons or remove subtrees to achieve
better performance.
To illustrate some of these design decisions, Figure 4. 1 1 shows four different deci
sion trees that can be used to classify persons according to height. The first tree is a
duplicate of that from Chapter
The first three trees of this figure all perform the same
classification. However, they all perform it differently. Underneath each tree is a table
showing the logical divisions used by the associated tree for classification. A nice feature
of Figure 4 . 1 1 (a) is that it is balanced. The tree is of the same depth for any path from
root to leaf. Figures 4 . 1 1 (b) and (c), however, are not balanced. In addition, the height of
the tree in (b) is greater than that of any of the others, implying a slightly worse behavior
when used for classification. However, all of these factors impact the time required to do
the actual classification. These may not be crucial performance issues unless the database
is extremely large. In that case a balanced shorter tree would be desirable. The tree shown
in Figure 4. l l (d) does not represent the same classification logic as the others.
The training data and the tree induction algorithm determine the tree shape. Thus,
the best-shaped tree that performs perfectly on the training set is desirable. Some algo
rithms create only binary trees. Binary trees are easily created, but they tend to be
deeper. The performance results when applying these types of trees for classification
may be worse because more comparisons usually are needed. However, since these com
parisons are simpler than those that require multiway branches, the ultimate performance
may be comparable.
The DT building algorithms may initially build the tree and then prune it for more
effective classification. With pruning techniques, portions of the tree may be removed
or combined to reduce the overall size of the tree. Portions of the tree relating to clas
sification using an unimportant attribute may be removed. This sort of change with a
node close to the root could ripple down to create major changes in the lower parts of
the tree. For example, with the data in Figure 4. 1 , if a tree were constructed by looking
3.
97
at values of the name attribute, all nodes labeled with that attribute would be removed
Lwer-level nodes wuld move up or be combined in some way. The approach to doin
this could become qmte comlicated. In the case of overfitting, lower-level subtrees may
be removed completely. Prumng may be performed while the tree is being created, thus
.
revntmg a tree from becoming too large. A second approach prunes the tree after it
Is bmlt.
Tree structure: To improve the performance of applying the tree for classification,
a balanced tree with the fewest levels is desirable. However, in this case, more
complicated comparisons with multiway branching [see Figure 4 . 1 1 (c)] may be
needed. Some algorithms build only binary trees.
103
The
echnique to building a decision tree is based on information theory and attempts
t rmmrmze the exp cted number of comparisons. The bask idea of the induction algo
nthm IS to ask questions whose answers provide the most information. This is similar to
the intuitive approach taken by adults when playing the "1\venty Questions" game. The
rst question an adult might ask could be "Is the thing alive?" while a child might ask "Is
1t II_lY Daddy?" The first question divides the search space into two large search domains,
:Vhile the second performs little division of the space. The basic strategy used by ID3
1s to choose splitting attributes with the highest information gain first. The amount of
information associated with an attribute value is related to the probability of occurrence.
ooking at the "Twenty Questions" example, the child's question divides the search space
mto two sets. One set (Daddy) has an infinitesimal probability associated with it and the
?ther set is almost . certain, while the question the adult makes divides the search space
mto two subsets with almost equal probability of occurring.
98
Chapter 4
Classification
4
3
2
0.2
0.4
0.6
(a) log(l/p)
0.8
0.5 .------,..---,---r--,
0.45
0.4
0.8
0.35
0.3
0.6
0.25
0.2
0.4
0.15
0.2
0.1
0.05
0
0
0.2 0.4 0.6 0.8
Section 4.4
Height
Short
Medium
0.2
0.4
0.6
is defined as
H( p! ,
P2 , . . . , Ps)
where
(c) H(p, 1 - p)
l:f= I Pi
L (p; log(1/p; ))
1, entropy
(4. 1 6)
H (D) -
L P (D; ) H (D;)
Medium
Tall
division into ranges is needed when the domain of an attribute is continuous or (as in this
case) consists of many possible values. While the choice of these divisions is somewhat
arbitrary, a domain expert shoulQ be able to perform the task.
EXAMPLE 4.7
The beginning state of the training data in Table 4. 1 (with the Outputl classification) is
that (4/ 1 5) are short, (8/ 1 5) are medium, and (3 / 15) are tall. Thus, the entropy of the
starting set is
4/ 1 5 log(15/4) + 8 / 1 5 log(15j8) + 3 / 1 5 log(1 5j3)
(4. 17)
i=l
Example 4.7 and associated Figure 4. 13 illustrate this process using the heig t
example. In this example, six divisions of the possible ranges of heights are used. Thts
0.4384
Choosing the gender as the splitting attribute, there are nine tuples that are F and six
that are M. The entropy of the subset that are F is
3/9 log(9/3) + 6/9 log(9/6)
Given a database state, D, H (D) finds the amount of order (or lack thereof) in that
state. When that state is split into s new states S
{DJ , D2 , . . . , Ds }, we can again look
at the entropy of those states. Each step in ID3 chooses the state that orders splitting
the most. A database state is completely ordered if all tuples in it are in the same class.
ID3 chooses the splitting attribute with the highest gain in information, where gain is
defined as the difference between how much information is needed to make a correct
classification before the split versus how much information is needed after the split.
Certainly, the split should reduce the information needed by the largest amount. This is
calculated by determining the differences between the entropies of the original dataset and
the weighted sum of the entropies from each of the subdivide datasets. he entrp e of
the split datasets are weighted by the fraction of the dataset bemg placed n that dlVlswn.
The ID3 algorithm calculates the gain of a particular split by the followmg formula:
Gain(D, S)
>1.95 m
F I G U R E 4. 1 3 : Classification problem.
i=!
Short
Tall
0.8
PI , P2, . . . , Ps
<=1 .7 m
>1.7 m
<= 1.95 m
F I G U RE 4. 1 2 : Entropy.
99
0.2764
(4. 1 8)
0.4392
(4. 1 9)
0.341 5 2
(4.20)
0.09688
(4.2 1 )
1 00
Chapter 4
Classification
0, 2 in ( 1 .6, 1 .7]
There are 2 tuples in the first division with entropy (2/2(0) + 0 + 0)
0, 4 in
with entropy (2/2(0) + 0 + 0) = 0, 3 in (1 .7, 1. 8] with entropy (0 + 3/3(0) + 0)
( 1 .8, 1 .9] with entropy (0+ 4/4(0) + 0)
0, 2 in ( 1 .9, 2.0] with entropy (0+ 1 /2(0.30 1 ) +
0. All of these
1 /2(0. 301)) = 0.30 1 , and two i n the last with entropy (0 + 0 + 2/2(0))
states are completely ordered and thus an entropy of 0 except for the ( 1 .9, 2.0] state. The
gain in entropy by using the height attribute is thus
=
0.4384 - 2/ 1 5 (0.301)
0.3983
Figure 4. 1 3(a) illustrates a problem in that the tree has multiple splits with identical
results. In addition, there is a subdivision of range (1 .9, 2.0]. Figure 4 . 1 3(b) shows an
optimized version of the tree.
C4.5 and CS.O
The decision tree algorithm C4.5 iwproves ID3 in the following ways:
Missing data: When the decision tree is built, missing data are simply ignored.
That is, the gain ratio is calculated by looking only at the other records that have
a value for that attribute. To classify a record with a missing attribute value, the
value for thai item can be predicted based on what is known about the attribute
values for the other rcords.
Continuous d;:lta: The basic idea is to divide the data into ranges based on the
attribute values for that item that are found irt the training sample.
Rules: C4.5 allows classification via either decision trees or rules generated from
them. In addition, some techniques to simplify complex rules are proposed. One
approach is to replace the left-hand side of a rule by a simpler version if all records
iri the training set are treated identically. An "otherwise" type of rule can be used
to indicate what should be done if no other rules apply.
Splitting: The ID3 approach favors attributes with many divisions and thus may
lead to overfitting. In the extreme, an attribute that has a unique value for each
101
tuple in the training set would be the best because there would be only one tuple
(and thus one class) for each division. An improvement can be made by taking
into account the cardinality of each division. This approach uses the GainRatio as
opposed to Gain. The GainRatio is defined as
.
.
GamRatw(D, S)
Gain (D , S)
(4.22)
Thus, this has the greater gain, and we choose this over gender as the first splitting
attribute. Within this division there are two males, one medium and one tall. This has
occurred because this grouping was too large. A further subdivision on height is needed,
and this generates the DT seen in Figure 4.13(a).
4.4.2
Section 4.4
(lE!J
ID l
(4.23)
..., I D l
For splitting purposes, C4.5 uses the largest GainRatio that ensures a larger than average
information gain. This is to compensate for the fact that the GainRatio value is skewed
toward splits where the size of one subset is close to that of the starting one. Example 4.8
shows the calculation of GainRatio for the first split in Example 4.7.
EXAMPLE 4.8
To calculate the GainRatio for the gender split, we first find the entropy associated with
the split ignoring classes
H
9
-
15
6
-
15
9
- log
15
( )
15
9
6
+ - log
15
( )
15
--
0.292
(4.24)
(4.25)
= 0.332
2
15
'
2 3 4 2
'
'
15 15 lS ' 15
(4.26)
C5.0 (called See 5 on Windows) is a commercial version of C4.5 now widely used
in many data mining packages such as Clementine and RuleQuest. It is targeted toward
use with large datasets. The DT induction is close to that of C4.5, but the rule generation
is different. Unlike C4.5, the precise algorithms used for C5.0 have not been divulged.
C5.0 does include improvements to generate rules. Results show that C5.0 improves on
memory usage by about 90 percent, runs between 5.7 and 240 times faster than C4.5,
and produces more accurate rules [ResO 1].
One major improvement to the accuracy of C5.0 is based on boosting. Boosting
is an approach to combining different classifiers. While boosting normally increases the
time that it takes to run a specific classifier, it does improve the accuracy. The error
rate has been shown to be less than half of that found with C4.5 on some datasets
[ResO l ] . Boosting does not always help when the training data contains a lot of noise.
Boosting works by creating multiple training sets from one training set. Each item in
the training set is assigned a weight. The weight indicates the importance of this item to
the classification. A classifier is constructed for each combination of weights used. Thus,
multiple classifiers are actually constructed. When C5.0 performs a classification, each
classifier is assigned a vote, voting is performed, and the target tuple is assigned to the
class with the most number of votes.
1 02
4.4.3
Chapter 4
Section 4.5
Classification
4.4.4
CART
1 03
Scalable DT Techniques
Classification and regression trees (CART) is a technique that generates a binary decision
We briefly examine some DT techniques that address creation of DTs for large datasets.
criterion. Unlike ID3, however, where a child is created for each subcategory, only two
addresses the scalability issue by ensuring that the CART technique can be applied
tree. As with ID3, entropy is used as a measure to choose the best splitting attribute and
children are created. The splitting is performed around what is determined to be the best
split point. At each step, an exhaustive search is used to determine the best split, where
SPRINT, a gini index is used to find the best split. Here gini for a database D is defined as
"best" is defined by
gini (D)
> (sjt)
2 h PR
L I P (CJ
}= I
(4 .27)
I tL ) - P (CJ I tR) I
L PJ
(4.34)
Dr and D2 is defined by
ginispl it ( D )
!tuples in subtree !
nr
be on the left or nght side of the tree. Thts IS defined as !tuples in training set l " vve assume
.
1 -
This formula is evaluated at the current node, t, and for each possible splitting attribute
and criterion, s. Here L and R are used to indicate the left and right subtrees of the
current node in the tree. P[ , PR are the probability that a tuple in the training set will
.
(gini(D r ) ) +
n2
n
(gini ( D2))
(4.35)
that the right branch is taken on equality. P (CJ I tL ) or P (CJ I tR) is the probability
that a tuple is in this class, C1 , and in the left or right subtree. This is defined as the
The split with the best gini value is chosen. Unlike the earlier approaches, SPRINT
possible criteria. Example 4.9 shows Its use with the height example With Output1 results.
does not need to sort the data by goodness value at each node during the DT induction
process. With continuous data, the split point is chosen to be the midpoint of every pair
By maintaining aggregate metadata concerning database attributes, the RainForest
approach allows a choice of split attribute without needing a training set. For each node
EXAMPLE 4.9
of a DT, a table called the attribute-value class (AVC) label group is used. The table
The first step is to determine the split attribute and criterion for the first split. We again
summarizes for an attribute the count of entries per class or attribute value grouping.
assume that there are six subranges to consider with the height attribute. Using these
Thus, the AVC table summarizes the information needed to determine splitting attributes.
ranges, we have the potential split values of 1 . 6, 1 . 7 , 1 . 8, 1 . 9, 2.0. We thus have a choice
The size of the table is not proportional to the size of the database or training set, but
rather to the product of the number of classes, unique attribute values, and potential
0.224
splitting attributes. This reduction in size (for large training sets) facilitates the scaling of
(4 . 2 8 )
DT induction algorithms to extremely large training sets. During the tree-building phase,
. (4.29)
> (1 .6)
> ( 1 .7 )
2(2/ 1 5) ( 1 3 / 1 5 ) (0 + 8 / 1 5 + 3/ 1 5)
0. 1 69
> (1 .8)
2(5/ 1 5) ( 1 0/ 1 5) (4/ 15 + 6/ 1 5 + 3 / 1 5)
> (1 .9)
> (2.0)
2 ( 1 2/ 1 5 ) (3 / 1 5 ) (4/ 1 5 + 8 / 1 5 + 3/ 1 5)
0.385
The algorithm continues by splitting the training data and constructing the AVC for the
next node.
(4.3 1 )
0.256
(4.32)
0.32
(4.33)
the training data are scanned, the AVC is built, and the best splitting attribute is chosen.
(4.30)
The largest of these is the split at 1. 8. The remainder of this example is left as an exercise.
4.5
classify any given database tuple is constructed. The activation functions typically are
sigmoidal. When a tuple must be classified, certain attribute values from that tuple are
<
F.
input into the directed graph at the corresponding source nodes. There often is one sink
As illustrated with the gender attribute, CART forces that an ordering of the
node for each class . The output value that is generated indicates the probability that
lating the goodness of a split on that attribute. The tree stops growing when no split will
the class with the highest probability of membership. The learning process modifies the
may not be the best for all possible data to be added in the future. The CART algorithm
the labels in the graph, as each tuple in the training set is sent through the network, the
attributes be used. CART handles missing data by simply ignoring that record in calcu
improve the performance. Note that even though it is the best for the training data, it
also contains a pruning strategy, which we will not discuss here but which can be found
in [KLR+98] .
the corresponding input tuple belongs to that class. The tuple will then be assigned to
labeling of the arcs to better classify tuples. Given a starting structure and value for all
projected classification made by the graph can be compared with the actual classification.
Based on the accuracy of the prediction, various labelings in the graph can change. This
1 04
Chapter 4
Classification
Section 4.5
learning process continues with all the training data or until the classification accuracy
technique. Although many approaches can be used, the most common approach is
1. Determine the number of output nodes as well as what attributes should be used as
Stop: The learning may stop when all the training tuples have propagated through
input. The number of hidden layers (between the source and the sink nodes) also
the output prediction to the actual result. If the prediction is accurate, adjust labels
The NN improves its performance by learning. This may continue even after the
to ensure that this prediction has a higher output weight the next time. If the
prediction is not correct, adjust the weights to provide a lower output value for this
class.
There is a low error rate and thus a high degree of accuracy once the appropriate
training has been performed.
1
There are many issues to be examined:
Attributes (number of source nodes): This is the same issue as determining which
Number of hidden layers : In the simplest case, there is only one hidden layer.
NNs are difficult to understand. Nontechnical users may have difficulty understand
ing how NNs work. While it is easy to explain decision trees, NNs are much more
difficult to understand.
Number of hidden nodes: Choosing the best number of hidden nodes per hid
Testing
Verification
training set.
Training data: As with DTs, with too much training data the NN may suffer from
Number of sinks: Although it is usually assumed that the number of output nodes
den layer is one of the most difficult problems when using NNs. There have been
many empirical and theoretical studies attempting to answer this question. The
answer depends on the structure of the NN, types of activation functions, training
algorithm, and problem being solved. If too few hidden nodes are used, the target
function may not be learned (underfitting). If too many nodes are used, overfit
ting may occur. Rules of thumb are often given that are based on the size of the
overfitting, while too little and it may not be able to classify accurately enough.
is the same as the number of classes, this is not always the case. For example, with
two classes there could only be one output node, with the resulting value being the
probability of being in the associated class. Subtracting this value from one would
give the probability of being in the second class.
Interconnections: In the simplest case, each node is connected to all nodes in the
next level.
Weights: The weight assigned to an arc indicates the relative weight between those
two nodes. Initial weights are usually assumed to be small positive numbers and
are assigned randomly.
1 05
Learning technique: The technique for adjusting the weights is called the leing
is adequate.
4.5.1
Propagation
The normal approach used for processing is called propagation. Given a tuple of values
input to the NN, X = (XJ , . . . , Xh ) , one value is input at each node in the input layer.
Then the summation and activation functions are applied at each node, with an output
value created for each output arc from that node. These values are in tum sent to the
subsequent nodes. This process continues until a tuple of output values, Y
(y 1 , . . . , Ym ) ,
i s produced from the nodes in the output layer. The process of propagation i s shown in
Algorithm 4.4 using a neural network with one hidden layer. Here a hyperbolic tangent
activation function is used for the nodes in the hidden layer, while a sigmoid function
is used for nodes in the output layer. We assume that the constant
in the activation
function has been provided. We also use k to be the number of edges corning into
a node.
1 Q6
Classification
Chapter 4
Section 4 . 5
ALGORITHM 4.4
Input :
N
X=
(x1; .
I / neural
. .
,, , .
/ / Input tup l e cori i s t ing of values for
input a t t ributes only
, Xh )
Outp ut :
Y=
network
I /Tup l e
(y1 , . . . , Ym)
:ll
107
Small
for
each
Si
lM-
(I:=1 ( wj i Xj i )) ;
each
Output
for
Tall
( 1- e -5i )
(1 +e csi ) ;
Output Yi =
(1+e
si ) ;
Figure 4. 1 4 shows a very simple NN used to classify university students as short, medium,
or tall. There are two input nodes, one for the gender data and one for the height
data. There are three output nodes, each associated with one class and using a simple
threshold activation function. Activation function h is associated with the short class,
!4 is associated with the medium class, and fs is associated with the tall class. In this
case, the weights of each arc from the height node is 1 . The weights on the gender arcs
is 0. This implies that in this case the gender values are ignored. The plots for the graphs
of the three activation functions are shown.
N
X
D
Outpu t :
4.5.2
NN Supervised Learning
The NN starting state is modified based on feedback of its petformance with the data in
the training set. This type of learning is referred to as supervised because it is known a
priori what the desired output should be. Unsupervised learning can also be performed if
the output is not known. With unsupervised approaches, no external teacher set is used. A
training set may be provided, but no labeling of the desired outcome is included. In this
case, similarities and differences between different tuples in the training set are uncovered.
in this chapter, we examine supervised learning. We briefly explore unsupervised learning
in Chapter 5.
Notice that this algorithm must be associated with a means to calculate the error
as well as some technique to adjust the weights. Many techniques have been proposed
to calculate the error. Assuming that the output from node i is Yi but should be di , the
1 08
Chapter 4
Classification
X ?j
(a) Node j in NN
(Yi - di ) 2
2
F I G U R E 4. 1 5:
t
i=l
(Yi - di ) 2
m
(4.38)
This formula couffl be expanded over all tuples in the training set to see the total error
over all of them. Thus, an error can be calculated for a specific test tuple or for the total
set of all entries.
Tlie Hebb and delta rules are approaches to change the weight on an input arc to a
node based on the knowledge that the output value from that node is incorrect. With both
techniques, a learning rule is used to modify the input weights. Suppose for a given node,
j, the input weights are represented as a tuple (w l j , . . . , Wkj ) , while the input and output
values are (xu . . . . , Xkj ) and YJ respectively. The objective of a learning technique is to
change the weights based on the output obtained for a specific input tuple. The change
in weights using the Hebb rule is represented by the following rule
(4.39)
ALGORITHM 4.6
N
X = (xl , . . . ,
D = (d1 , . . . ,
Inpu t :
The nice feature of the delta rule is that it minimizes the error d1 - YJ at each node.
Backpropagation is a learning technique that adjusts weights in the NN by prop
agating weight changes backwa.td from the sink to the source nodes. Backpropagation
is the most well known form of learning because it is easy to understand and generally
applicable. Backpropagation can be thought of as a generalized delta rule approach.
Figure 4.15 shows the structure and use of one tiode, j, in a neural network graph.
The basic node structure is shown in part (a). Here the representative input arc has
a weight cif W?j . where ? is used to show that the input to node j is corning from
another node shown here as ?. Of course, there probably are multiple input arcs to
a node. The output weight is similarly labeled wJ ? During propagation, data values
input at the input layer flow through the network, with final values corning out of the
network at the output layer. The propagation technique is shown in part (b) Figure 4.15.
Here the smaller dashed arrow underneath the regular graph arc shows the input value
X?j flowing into node j . The activation function fJ is applied to all the input values
and weights, with output values resulting. There is an associated input function that is
applied to the input values and weights before applying the activation function. This
input function is typically a weighted sum of the input values. Here YJ ? shows the output
value flowing (propagating) to the next node from node j . Thus, propagation occurs
by applying the activation function at each node, which then places the output value
on the arc to be sent as input to the next nodes. In most cases, the activation function
produces only one output value that is propagated to the set of connected nodes. The
NN can be used for classification and/or learning. During the classification process, only
propagation occurs. However, when learning is used after the output of the classification
occurs, a comparison to the known classification is used to determine how to change
the weights in the graph. In the simplest types of learning, learning progresses from the
output layer backward to the input layer. Weights are changed based on the changes
that were made in weights in subsequent arcs. This backward learning process is called
backpropagation and is illustrated in Figure 4. 15( c). Weight wJ ? is modified to become
wj ? + 6 wj ? . A learning rule is applied to this 6 wj ? to determine the change at the next
higher level 6 W? j.
Here c is a constant often called the learning rate. A rule of thumb is that c =
A variation of this approach, called the delta rule, examines not only the output
value YJ but also the desired value dj for output. In this case the change in weight is
found by the rule
(4.40)
109
dwj?
(4.37)
This MSE can then be used to find a total error over all nodes in the network or over
only the output nodes. In the following discussion, the assumption is made that only the
final output of the NN is known for a tuple in the training data. Thus, the total MSE
error over all m output nodes in the NN is
!!.w?j
Yj ?
(4.36)
Section 4 . 5
Xh)
dm)
/ / Input
/ /Output
tup l e
tup l e de s i red
Outpu t :
Backpropagat i on algor i t hm :
E=
( N, X) ;
/ / I l lu s t rate backpropagat i on
1/2 L7=l ( di - Yi ) 2 ;
Propagat i on
Gradient
( N, E) ;
ai
wish to find the weight where this slope is zero. Figure 4.16 and Algorithm 4.7 illustrate
the concept. The stated algorithm assumes only one hidden layer. More hidden layers
would be handled in the same manner with the error propagated backward.
110
Chapter 4
Secti on 4.5
Classification
111
Output
wkj
wji
- - -- - - - - - -
Yk
0
.
Y;
'0----->
- -- - --- - - - - -- - - - - -
Yi
4.17
Here node i is at the output layer and node j is at the hidden layer just before it;
-Desired weight
------r--
yJ
is the output of j .
The learning function i n the gradient descent technique is based o n using the
aE = aE ay; as;
/).Wji = awji
ay; as; awji
- 1] --
.ALGORITHM 4.7
wJi
-1] - - --
(4.4 1 )
activation function in the output layer, for the output layer we have
Input :
N
is
F I G U R E 4. 1 6: Gradient descent.
Output :
y;
Also,
as; = YJ
-awji
Gradient algori t hm :
/ / I l l ustrates
layer = previous
(4.44)
::, = a:, ( pdm - Ym )2) = - (d, - y; )
4.7:
1
f).w . = 17(d - y- ) y ( 1 - 1 e-S; ) 1 1e-S = 17(d; - y; ) y1(1 - y; ) y; (4.45)
4.17,
(4.46)
/).Wkj = awaEkj
(4.47)
Jl
This algorithm changes weights by working backward from the output layer to the
approach, the weights are changed once after all tuples in the training set are applied
and a total MSE is found. With the incremental or online approach, the weights are
changed after each tuple in the training set is applied. The incremental technique is
usually preferred because it requires less space and may actually examine more potential
(0, 1 ) ,
although it may be
Applying a learning rule back through multiple layers in the network may be
where
Here the variable m ranges over all output nodes with arcs from j . We then derive
aE
- (dm - Ym )
Overall, however, we are trying to minimize the error at the output nodes, not at each
node in the network. Thus, the approach that is used is to propagate the output errors
-1] --
difficult. Doing this for the hidden layers is not as easy as doing it with the output layer.
For a node j in the hidden layer, calculating the change in the weight for arcs
input layer. There are two basic versions of this algorithm. With the batch or offline
(4.43)
laye r ;
Wj m
(4.48)
(4.49)
(4.50)
1 12
Chapter 4
Classification
Section 4.5
113
For hidden layers, where a hyperbolic tangent activation function is assumed, we have
1 - y2
}
__
Also,
a sJ
a wkj
(4.5 1 )
(4.52)
= Yk
/).Wkj = 11 Yk
1 - (yj ) 2
2
L (dm - Ym )Wjm Ym O - Ym )
(4.53)
aE
B Wj i
+ a f).Wji (t)
(4.54)
Xz
Here the change in weight at time t + 1 is based not only on the same partial derivative
as earlier, but also on the last change in weight. Here
4.5.3
- -
decreases (or increases) with the distance from a central point. The Gaussian activation
function shown in Equation 3 .35 is an RBF with a central point of 0. An RBF has a
Gaussian shape, and an RBF network is typically an NN with three layers. The input layer
is used to simply input the data. A Gaussian activation function is used at the hidden
layer, while a linear activation function is used at the output layer. The objective is to
have the hidden nodes learn to respond only to a subset of the input, namely, that where
the Gaussian function is centered. This is usually accomplished via supervised learning.
When RBF functions are used as the activation functions on the hidden layer, the nodes
can be sensitive to a subset of the input values. Figure 4 . 1 8 shows the basic structure of
!4
{ 1
0
if S > O
otherwise
(4.55)
An alternative way to view this classification problem is shown in Figure 4. 1 9(b). Here
4.5.4
x1 is shown on the horizontal axis and x2 is shown on the vertical axis. The area of the
Perceptrons
The simplest NN is called a perceptron. A perceptron is a single neuron with multiple
inputs and one output. The original perceptron proposed the use of a step activation
'_
function, but it is more common to see another type of function such as a sigmoidal
function. A simple perceptron can be used to classify into two classes. Using a unipolar
activation function, an output of 1 would be used to classify into one class, while an
output of 0 would be used to pass in the other class. Example 4. 1 1 illustrates this.
an MLP used for classifying the height example given in Table 4. 1 . The neurons are
placed in iayers with outputs always flowing toward the output layer. If only one layer
EXAMPLE 4. 1 1
Figure 4 . 19(a) shows a perceptron with two inputs and a bias input. The three weights
are 3, 2, and - 6, respectively. The activation function
!4
needs no more than two hidden layers. Kolmogorov's theorem states that a mapping
1 14
Classification
Chapter 4
Section 4.6
between two sets of numbers can be performed using an NN with only one hidden
layer. In this case, the NN is to have one input node for each attribute input, and
given n input attributes the hidden layer should have (2n + 1) nodes, each with input
from each of the input nodes. The output layer has one node for each desired out
put value.
4.6
f o r each path
a = True
i 60
/ /Rul e s
combined with
label
of
incident
node
node
There are algorithms that generate rules from trees as well as algorithms that generate
rules without first creating DTs.
/ / De c i s ion tree
node do
of
A tree is created based on looking at all classes. When generating rules, only one
class must be examined at a time.
T do
Output :
in
The tree has an implied order in which the splitting is performed. Rules have no
order.
Input :
( l abel
to a leaf
Using this algorithm, the following rules are generated for the DT in Figure 4 . 1 3(a):
ALGORlTHM 4.8
each non - l e af
a = a/\
c = label of leaf
R = R U r = (a, c)
A classification rule, r = (a , c), consists of the if or antecedent, a, part and the then or
consequent portion, c. The antecedent contains a predicate that can be evaluated as true
or false against each tuple in the database (and obviously in the training data). These
rules relate directly to the corresponding DT that could be created. A DT can always be
used to generate rules, but they are not equivalent. There are differences between rules
and trees :
.
from root
f rom a DT
outgoing arc )
One straightforward way to perform classification is to generate if-then rules that cover
all cases. For example, we could have the following rules to determine classification
of grades:
4.6. 1
c l a s s i f i c at i on ru l e s
R=0
for
115
Gen a l go r i thm :
RULE-BASED ALGORITHMS
ule-Based Algorithms
1 16
Classification
Chapter 4
Section 4.6
ALGORITHM 4.9
Input :
D
RX algori t hm :
f rom NN
in terms o f
4.6.3
Generating Rules
iNithout a DT or N N
Height
/ / Classes
/ / Ru l e s
for each A
do
RA = 0 ;
for each p o s s i b l e value ,
Rules
Errors
Total Errors
F Medium
M Tall
(0, 1 . 6] Short
( 1 .6, 1 .7] Short
(1 .7, 1 . 8] Medium
(1 .8, 1 . 9] Medium
(1 .9, 2.0] Medium
(2.0, oo) Tall
3/9
3/6
0/2
0/2
0/3
0/4
1/2
0/2
6/ 15
1 / 15
v,
of A
do
ERRA = numb er of
The objective for the covering algorithms is to replace the "?" in this statement with
predicates that can be used to obtain the "best" probability of being tall.
One simple approach is called 1R because it generates a simple set of rules that are
equivalent to a DT with only one level. The basic idea is to choose the best attribute to
perform the classification based on the training data. "Best" is defined here by counting
the number of errors. In Table 4.4 this approach is illustrated using the height example,
Gender
R = 0;
Attribute
/ / Training data
/ /Att r ibu t e s t o cons ider for rul e s
lR algori t hm :
These techniques are sometimes called covering algorithms because they attempt to
generate rules exactly cover a specific class [WFOO]. Tree algorithms work in a top
down divide and conquer approach, but this need not be the case for covering algorithms.
They generate the best rule possible by optimizing the desired classification probability.
Usually the "best" attribute-value pair is chosen, as opposed to the best attribute with
the tree-based algorithms. Suppose that we wished to generate a rule to classify persons
as tall. The basic format for the rule is then
Option
Output :
input s ;
ALGORITHM 4.10
Input :
terms o f
1 17
Outputl . If we only use the gender attribute, there are a total of 6/15 errors whereas
if we use the height attribute, there are only 1/ 15. Thus, the height would e chosen
and the six rules stated in the table would be used. As with ID3 , 1R tends to choose
attributes w th a large nmber o values leading to overfitting. l R can handle missing
.
dat b addmg an additional attnbute value for the value of missing. Algorithm 4.10,
which 1s adapted from [WFOO], shows the outline for this algorithm.
/ / Training data
N
Output :
tup l e s
incorre c t ly c l as s i f i e d by RA i
4.1
0/9
Gender = M
3/6
0/2
0/2
1 18
Chapter 4
Section 4.7
Classification
0/3
0/4
1 /2
2/2
1 19
ALGORITHM 4.11
Inpu t :
D
/ / Training data
/ / Clas s e s
Outpu t :
R
/ / Rul e s
R = 0;
for e a c h Cj E C do
Since all tuples that satisfy this predicate are tall, we do not add any additional predicates
to this rule. We now need to generate additional rules for the tall class. We thus look at the
remaining 1 3 tuples in the training set and recalculate the accuracy of the corresponding
predicates:
Gender = F
0/9
Gender = M
1 /4
Height < = 1 .6
0/2
0/2
0/3
0/4
1 /2
repeat
T = D;
removed
f ind A = v that
p = p 1\ (A = v) ;
T = {tup l e s in T that s a t i s fy A = v} ;
un t i l a l l tup l e s in T belong to Cj ;
D = D - T;
R = R U r;
Based on the analysis, we see that the last height range is the most accurate and thus
generate the rule:
If 2.0 < height, then class = tall
However, only one of the tuples that satisfies this is actually tall,
another predicate to it. We then look only at the other predicates
tuples. We now see a problem in that both of these are males. The
caused by our "arbitrary" range divisions. We now divide the range
1 . 9 < Height <= 1 . 95
0/ 1
1/1
4.7
so we need to add
affecting these two
.
problem 1s actually
into two subranges:
or
This problem does not exist if we look at tuples individually using the attribte:-value
pairs. However, in that case we would not generate the needed ranges fr class1fymg the
actual data. At this point, we have classified all tall tuples. The algonthm would then
proceed by classifying the short and medium classes. This is left as an exercise.
Cj ;
Given a classification problem, no one classification technique always yields the best
results. Therefore, there have been some proposals that look at combining techniques.
While discussing C5.0, we briefly introduced one technique for combining classifiers
called boosting. 1\vo basic techniques can be used to accomplish this:
A synthesis of approaches takes multiple techniques and blends them into a new
approach. An example of this would be using a prediction technique, such as linear
regression, to predict a future value for an attribute that is then used as input to a
classification NN In this way the NN is used to predict a future classification value.
.
in D that belong to
One approach to combine independent classifiers assumes that there are n inde
pendent classifiers and that each generates the posterior probability Pk ( CJ 1 ti ) for each
class. The values are combined with a weighted linear combination
n
L Wk Pk (Cj I ti )
k= l
(4.56)
1 20
SE!Ction 4.9
Classification
Chapter 4
(a) Classifier 1
121
for both classifiers with respect to class 1 is then: 3/4 + 1 /4. When looking at class 2,
Exercises
Thple in Class 1
and correctly classified
Tuple in Class 1
and incorrectly classified
Thple in Class 2
and correctly classified
Thple in Class 2
and incorrectly classified
(b) Classifier 2
4.8
SUM MARY
No one classification technique is always superior to the others in terms of classification
accuracy. However, there are advantages and disadvantages to the use of each. The
regression approaches force the data to fit a predefined model. If a linear model is
chosen, then the data are fit into that model even though it might not be linear. It
requires that linear data be used. The KNN technique requires only that the data be
such that distances can be calculated. This can then be applied even to nonnumeric data.
Outliers are handled by looking only at the K nearest neighbors. Bayesian classification
assumes that the data attributes are independent with discrete values. Thus, although it is
easy to use and understand, results may not be satisfactory. Decision tree techniques are
easy to understand, but they may lead to overfitting. To avoid this, pruning techniques
may be needed. ID3 is applicable only to categorical data. Improvements on it, C4.5
and CS, allow the use of continuous data and improved techniques for splitting. CART
creates binary trees and thus may result in very deep trees.
When looking at the approaches based on complexity analysis, we see that they
majority is found.
are all very efficient. This is due to the fact that once the model is created, applying
EXAMPLE 4. 1 3
and naive Bayes, require constant time to classify a tuple once the models are built.
The distance-based approaches, simple and KNN, are also constant but require that each
tuple be compared either to a representative for each class or to all items in the training
set. Assuming there are
Recently, a new CMC technique, adaptive classifier combination (ACC), has been
proposed
then the tuples in that neighborhood are classified by each classifier, and finall the
accuracy for each class is measured. By examining the accuracy across all classifiers
for each class, the tuple is placed in the class that has the highest local accuracy. In
effect, the class chosen is that to which most of its neighbors are accurately classified
O (q)
classification techniques, ID3, C4.5, and CART require a number of comparisons that
are (in the worst case) equal to the longest path from a root to a leaf node. Thus, they
performed in constant time as well. The NN approaches again require that a tuple be
propagated through the graph. Since the size of the graph is constant, this can be viewed
as being performed in constant time. Thus, all algorithms are 0 (n) to classify the n items
in the database.
4.9
EXERCISES
1. Explain the differences between the definition of the classification problem found
in Definition 4. 1 and an alternative one with the mapping from C to D .
2 . Using the data i n Table 4 . 1 , draw OC curves assuming that the Output2 column is
the correct classification and Output l is what is seen. You will need to draw three
curves, one for each class.
3. Using the data in Table 4. 1 , construct a confusion matrix assuming Output is the
EXAMPLE 4 . 1 4
4. Apply the method of least squares technique to determine the division between
medium and tall persons using the training data in Table 4. 1 and the classification
shown in the Output1 column (see Example 4.3). You may use either the division
technique or the prediction technique.
1 22
Chapter 4
Section 4. i 0
Classification
6.
7.
8.
9.
10.
11.
12.
13.
14.
Table 4. 1 .
.
hms can be faun onhne. Obtan
18. (Implementation) Various classification algorit
ms to the height examp1e m
code for both CART and C4.5 . Apply these progra
Output2 column. Compare
the
in
shown
ations
c
i
classif
g
Table 4.1 using the trainin
these results to those found in Exercises 1 1 and 12.
in Exercise 18.
19. Generate rules from each of the trees found
that have ben ued for classification
s
dataset
s
20. (Implementation/research) Variou
.
e classificatiOn dataset and gen
real-hf
a
Obtain
online.
found
benchmarks can be
in Exercise 1 8 . Compare these
erate decision trees using the programs you found
two trees. Which is better? Why?
nt guideline that have been proposed for
21. (Research) Compare at least three differe
in an NN.
determining the optimal number of hidden nodes
4.1 0
B I B LIOGRAPHIC NOTES
1 23
B i b li ogra p h i c Notes
There are detailed discussions of linear regression for classification in [HTF0 1].
It was reported that the accuracy for multiclass problems was very poor. A better lin
ear approach is to use linear discriminant analysis (LDA), which is also discussed in
that book.
Chi squared automatic interaction detection (CHAID) was one of the earliest deci
sion tree algorithms, proposed by Hartigan in 1975 [Har75]. CHAID uses the chi squared
statistic to determine splitting. It can use only categorical values. The chi squared test
is used to prevent the tree from growing too big. A modified version of CHAID was
proposed in 1980 by Kass; it reduces the overall time, but does not guarantee the best
result. Kass proposes that pairs of categories (for the predictor variable) be merged
if there is no statistically significant difference between them. The resulting set of
categories defines the split [Kas80]. With exhaustive CHA1D, which was proposed in
199 1 , the split point is determined by merging pairs of categories until only one pair
remains [BdVS91]. The predictor with the best overall prediction is then chosen for
the split.
ID3 was first proposed in the mid 1970s by Quinlan [Qui86] . A thorough investi
gation of C4.5 can be found in the seminal text on the subject [Qui93] .
CART was developed by Breimen in 1984 [BFOS84] . In 1997, another binary
decision tree technique was proposed [LS97] . QUEST (quick unbiased efficient statistical
tree) addresses the problems in CART that it tends to select variables with many values,
which creates a bias iri the model. QUEST handles variable selection and split point
differently. Also, unlike CART, QUEST does not perform an exhaustive search, so it is
more efficient. The approach used by QUEST is to determine the association between
each predictor variable and target variable. The variable with the largest association is
chosen. The split point (for that variable) is then determined.
Many different techniques have been proposed to pnme decision trees. A survey
by Breslow and Aha in 19973 looked at techniques to simplify trees. These included
techniques to control the tree site (pruning), m<;>dify the test space, change the way the
searching of the test space was conducted, reduce the size of the input data set (e.g.,
feature selection), and use alternative data structures. Pruning approaches include pre
pruning algorithms that affect the size of the tree as it is created.4 Post-pruning algorithms
change the tree after it is created. 5
Pattern recognition is a type of classification used in many diverse applications,
including medical diagnosis, assembly line parts inspection, speech recognition, printed
character recognition, military target recognition, robot navigation, and fingerprint analy
sis. While these applications can use the general strategies outlined in this chapter, there
has been much work in the development of algorithms specifically targeted to individual
applications. The interested reader is referred to some of the many texts available on
pattern recognition, including [Bis95], [DHSOO], [Fuk90], [GJJ96], and [TK98] .
3L.
Breslow and D.W. Aha, "Comparing Tree-Simplification Procedures," Proceedings of the Sixth Inter
sification algorithms.2
, Complexity,
2 Tjen-Sien Lim, Wei-Yin Loh, and Yu-Shan-Shih, "A Comparison of Prediction Accuracy
, vol. 40, 2000,
New Classification Algorithms," Machine Learning
and Training Time of Thirty-three Old and
Pruning Decision Trees, "IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.
pp.
203-229.
Research, vol.
15, 2001,
pp.
163-187.
19,
no.
5,
1 24
Chapter 4
Classification
C H A P T E R
Cl u ste ri ng
5.1
INTRODUCTION
5.2
5.3
OUTLIERS
5.4
H I ERARCHICAL ALGORITH M S
5.5
PARTITIONAL ALGORITH MS
5.6
5.7
5.8
COMPARISON
5.9
EXERCISES
5 . 1 0 BIBLIOGRAPHIC NOTES
5.1
INTRODUCTION
Clustering is similar to classification in that data are grouped. However, unlike classifi
cation, the groups are not predefined. Instead, the grouping is accomplished by finding
similarities between data according to characteristics found in the actual data. The groups
are called clusters. Some authors view clustering as a special type of classification. In
this text, however, we follow a more conventional view in that the two are different.
Many definitions for clusters have been proposed:
Set of like elements. Elements from different clusters are not alike.
The distance between points in a cluster is less than the distance between a point
in the cluster and any point outside it.
An international online catalog company wishes to group its customers based o n common
features. Company management does not have any predefined labels for these groups.
Based on the outcome of the grouping, they will target marketing and advertising cam
paigns to the different groups. The information they have about the customers includes
1 25
Cluste r i n g
Chapter 5
1 26
TAB LE 5 . 1 :
..
Income
Age
Children
$25,000
$ 1 5,000
35
25
$20,000
$30,000
40
20
$20,000
Marital Status
Education
Single
Married
High school
High school
0
0
Single
Divorced
High school
High school
25
Divorced
College
$70,000
$90,000
60
30
0
0
Married
Married
College
Graduate school
$200,000
$100,000
45
5
2
Married
Divorced
Graduate school
College
Section 5 . 1
so
Outlier handling is difficult. Here the elements do not naturally fall into any cluster.
They can be viewed as solitary clusters. However, if a clustering algorithm attempts
to find larger clusters, these outliers will be forced to be placed in some cluster.
This process may result in the creation of poor clusters by combining two existing
clusters and leaving the outlier in its own cluster.
Dynamic data in the database implies that cluster membership may change over time.
Interpreting the semantic meaning of each cluster may be difficult. With classifica
tion, the labeling of the classes is known ahead of time. However, with clustering,
this may not be the case. Thus, when the clustering process finishes creating a set
of clusters, the exact meaning of each cluster may not be obvious. Here is where
a domain expert is needed to assign a label or interpretation for each cluster.
There is no one correct answer to a clustering problem. In fact, many answers may
be found. The exact number of clusters required is not easy to determine. Again, a
domain expert may be required. For example, suppose we have a set of data about
plants that have been collected during a field trip. Without any prior knowledge of
plant classification, if we attempt to divide this set of data into similar groupings,
it would not be clear how many groups should be created.
(c) Size-based
income, age, number of children, marital status, and education. Ta le 5 shows some
tuples from this database for customers in the United States. Dependmg on h ty?e of
advertising not all attributes are important. For example, suppose the advertlsmg 1s for
a special s le on children' s clothes. We could target the ad:rising only to the persons
with children. One possible clustering is that shown by the dmswn of the table. The firt
group of people have young children and a high school dgree, whlle the second group 1s
similar but have no children. The third group has both chlldren and a college degree. The
last two groups have higher incomes and at least a college degre' The very last group has
.
.
children. Different clusterings would have been found by exanumng age or mantal status.
1 27
I ntroduction
Another related issue is what data should be used for clustering. Unlike learning
during a classification process, where there is some a priori knowledge concern
ing what the attributes of each classification should be, in clustering we have no
supervised learning to aid the process. Indeed, clustering can be viewed as similar
to unsupervised learning.
We can ilien summarize some basic features of clustering (as opposed to classification):
1 28
Chapter 5
Clustering
Sectio n 5.2
1 29
5.2
hierarchical clustering, a nested set of clusters is created. Each level in the hierarchy has
a separate set of clusters. At the lowest level, each item is in its own unique cluster. At the
highest level, all items belong to the same cluster. With hierarchical clustering, the desired
number of clusters i$ not input. With partitional clustering, the algorithm creates only
one set of clusters. These approaches use the desired number of clusters to drive how the
final set is created. Traditional clustering algorithms tend to be targeted to small numeric
databases that fit into memory. There are, however, more recent clustering algorithms that
look at categorical data and are targeted to larger, perhaps dynamic, databases. Algorithms
targeted to larger databases may adapt to memory constraints by either ampling the
database or using data structures, which can be compressed or pruned to fit mto memory
regardless of the size of the database. Clustering algorithms may also differ base on
whether they produce overlapping or nonoverlapping clusters. Even though we consider
only nonoverlapping clusters, it is possible to place an item in multiple clusters. In turn,
nonoverlapping clusters can be viewed as extrinsic or intrinsic. Extrinsic techniques use
labeling of the items to assist in the classification process. These algorithms are the
traditional classification supervised learning algorithms in which a special input training
set is used. Intrinsic algorithms do not use any a priori category labels, but depend
only on the adjacency matrix containing the distance between objects. All algorithms we
examine in this chapter fall into the intrinsic class.
The types of clustering algorithms can be furthered classified based on the _imle
mentation technique used. Hierarchical algorithms can be categorized as agglomerative
or divisive. "Agglomerative" implies that the clusters are created in a bottom-up fashion,
while divisive algorithms work in a top-down fashion. Although both hierarchical and
partitional algorithms could be described using the agglomerative vs. d viive labl, t
typically is more associated with hierarchical algorithms. Another descnptlve tag mdi
cates whether each individual element is handled one by one, serial (sometimes called
incrementa[), or whether all items are examined together, simultaneous. If a specific tuple
is viewed as having attribute values for all attributes in the schema, then clustering alo
rithms could differ as to how the attribute values are examined. As is usually done with
decision tree classification techniques, some algorithms examine attribute values one at a
time, monothetic. Polythetic algorithms consider all attribute values at one time. Finally,
clustering algorithms can be labeled based on the mathematical formulation given to
the algorithm: graph theoretic or matrix algebra. In this chapter we generally use t e
graph approach and describe the input to the clustering algorithm as an adjacency matnx
. . .
[ZRL96] :
. . .
_L)tm;)
i=l
m - ---N
centroid
(5. 1)
L Ctm i - Cm) 2
i=l
radius
N
N
(5.2)
L L (tm i - lmj) 2
diameter
Dm
i=l j =l
(N)(N - 1 )
(5.3)
Here the centroid is the "middle" of the cluster; it need not be an actual point in the
cluster. Some clustering algorithms alternatively assume that the cluster is represented by
one centrally located object in the cluster called a medoid. The radius is the square root
of the average mean squared distance from any point in the cluster to the centroid, and
the d ameter is the square root of the average mean squared distance between all pairs
.
of pomts m the cluster. We use the notation Mm to indicate the medoid for cluster Km .
130
Chapter 5
Cl ustering
Section 5.4
H i erarchical Algorithms
Many clustering algorithms require that the distance between clusters (rather than
elements) be determined. This is not an easy task given that there are many interpretations
for distance between clusters. Given clusters Ki and KJ , there are several standard
alternatives to calculate the distance between clusters. A representative list is :
Single link: Smallest distance between an element in one cluster and an eiement
in the other. We thus have dis (Ki , K J ) = min(dis(tu , tJm )) Vtu E Ki fj K J and
Vtj m E KJ rj Ki .
Complete link: Largest distance between an element in one cluster and an element
in the other. We thus have dis(Ki , KJ ) = max(dis(tu , tjm ))Vtu E Ki rj KJ and
VtJ m E KJ rj Ki .
0 Three clusters
) 1\vo clusters
131
FIG U RE 5 . 3 :
Medoid: Using a medoid to represent each cluster, the distance between the clusters
can be defined by the distance between the medoids: dis(Ki , Kj)
dis(Mi , MJ ) .
=
5.3
OUTLIERS
As mentioned earlier, outliers are sample points with values much different from those
of the remaining set of data. Outliers may represent errors in the data (perhaps a mal
functioning sensor re.corded an incorrect data value) or could be correct data values that
are simply much different from the remaining data. A person who is 2.5 meters tall is
much taller than most people. In analyzing the height of individuals; this value probably
would be viewed as an outlier.
Some clustering techniques do rit perform well with the presence of outliers. This
problem is illustrated in Figure 5.3. Here if three clusters are found (solid line), the outlier
will occur in a cluster by itself. However, if two clusters are found (dashed line), the two
(obviously) different sets of data will be placed in one cluster because they are closer
together than the outlier. This problem is complicated by the fact that many clustering
algorithms actually have as input the number of desired clusters to be found.
Clustering algorithms may actually find and remove outliers to ensure that they
perform better. However, care must be taken in actually removing outliers. For example,
suppose that the data mining problem is to predict flooding. Extremely high water level
values occur very infrequently, and when compared with the normal water level values
may seem to be outliers. However, removing these values may not allow the data mining
algorithms to work effectively because there would be no data that showed that floods
ever actually occurred.
Outlier detection, or outlier mining, is the process of identifying outliers in a set
of data. Clustering, or other data mining, algorithms may then choose to remove or
treat these values differently. Some outlier detection techniques ate based on statistical
techniques . These usually assume that the set of data follows a known distribution and that
outliers can be detected by well-known tests such as discordancy tests. However, these
B
F I G U R E 5.4:
tests are not very realistic for real-world data because real..world data values may not
follow well-defined data distributions. Also, most of these tests assume a single attribute
value, and many attributes are- involved in real-world datasets. Alternative detection
techniques may be based on distance measures.
5.4
in the dendrogram
Figure 5.5 shows six elements, {A, :S, C, D, E, F}, to be clustered. Parts (a) to (e) of the
figure show five different sets of clusters. In part (a) each cluster is viewed to consist of
Chapter 5
1 32
Clustering
10
9
10
9
8
7
6
5
4
3
2
1
0
Section 5.4
Ex
10
9
8
7
6
5
4
3
2
2 3 4 5 6 7 8
1
0
1
0
2 3 4 5 6 7 8
1 33
Here the adjacency matrix, A, contains a distance value rather than a simple boolean
value: A[i, j]
dis (t; , tj ) . The output of the algorithm is a dendrogram, DE, which we
represent as a set of ordered triples (d, k, K) where d is the threshold distance, k is the
number of clusters, and K is the set of clusters. The dendrogram in Figure 5.7(a) would
be represented by the following:
8
7
6
5
4
3
2
H i erarchical Al gorithms
2 3 4 5 6 7 8
{ (0 , 5 , { { A }, { B }, { C } , { D } , { } } ) , ( 1 , 3 , { {A , B } , {C, D } , { } } )
( 2 , 2 , {{ A , B , C, D } , { } } ) , ( 3 , 1 , { { A , B , C , D, } } ) }
Outputting the dendrogram produces a set o f clusters rather than just one clustering. The
user can determine which of the clusters (based on distance threshold) he or she wishes
to use.
ALGORITHM 5.1
Input :
1 2 3 4 5 6 7 8
a single element. Part (b) illustrates four clusters. Here there are two sets of two-element
dusters. These clusters are formed at this level because these two elements are closer
to each other than any of the other elements. part (c) shows a new cluster formed by
adding a close element to one of the rwo-f!lement clusters. In part (d) the two-element
and three-element clusters are merged to give a five-element cluster. This is done because
these two clusters are closer to each other than to the remote element cluster, {F} . At the
last stage, part (e), all six elements are merged.
The space complexity for hierarchical algorithms is O (n 2 ) because this is the space
required for the adjacency matrix. The space required for the dendrogram is O (kn),
which is much less than O (n 2 ) . The time complexity for hierarchical algorithms is
2
0 (kn ) because there is one iteration for each level in the dendrogram. Depending on
the specific algorithm, however, this could actually be O (maxd n 2 ) where maxd is the
maximum distance between points. Different algorithms may actually merge the closest
clusters from the next lowest level or simply create new clusters at each level with
progressively larger distances.
Hierarchical techniques are well suited for many clustering applications that natu
rally exhibit a nesting relationship between clusters. For example, in biology, plant and
animal taxonomies could easily be viewed as a hierarchy of clusters.
5.4. 1
Agglomerative Algorithms
Agglomerative algorithms start with each individual item in its own cluster and iteratively
merge clusters until all items belong in one cluster. Different agglomerative algorithms
differ in how the clusters are merged at each level. Algorithm 5 . 1 illustrates the typi
cal agglomerative clustering algorithm. It assumes that a set of elements and distances
between them is given as input. We use an n x n vertex adjacency matrix, A, as input.
D = { t 1 , t2 , . . . , tn}
/ / Set o f e l ement s
/ /Adj acency matrix showing dis tance between elements
Output :
DE
Agglomerative
algo r i thm :
d= 0;
k = n;
K = {{ t l }, . . . , { tn}} ;
DE = { (d, k, K) } ;
repeat
o l dk = k ;
d = d+ l ;
Ad = Vertex adj acency mat rix f o r graph with thre s ho l d
di s tance of d ;
( k , K) = Ne wCl u s ters(Ad, D) ;
if oldk =I= k then
DE = DE U (d, k, K) ; I/ New set of clusters added t o dendrogram .
unt i l k = 1
1 34
Chapter 5
Cl ustering
the agglomerative approach is that it is not incremental. Thus, when new elements are
added or old ones are removed or changed, the entire algorithm must be rerun. More
recent incremental variations, as discussed later in this text, address this problem.
Single Link Technique.
.
maxunal connected components m a graph. A connected component is a graph in which
there exists a path between any two vertices. With the single link approach, two clusters
c E
E
1 B
E c
are merged if there is at least one edge that connects the two clusters; that is, if the
tance being considered. For this reason, it is often called the nearest neighbor clustering
_
techmque. Example 5.3 illustrates this process.
D
(d) Graph with threshold of 3
EXAMPLE 5.3
Table 5.2 contains five sample data items with the distance between the elements indicated
in the t ble entries. When viewed as a graph problem, Figure 5.6(a) shows the general
graph With all eqges labeled with the respective distances. To understand the idea behind
c
1
1 B
E <;
c
A
D
(e) Graph with threshold of 4
the hierarchical approach, we show several graph variations in Figures 5 .6(b), (c), (d),
and (e). Figure 5 .6(b) shows only those edges with a distance of
{E}. During the next level of clustering, we look at edges with a length of 2 or less. The
level of the single link clustering algorithm, we merge these two clusters to obtain a total
have an edge (actually three) between the two clusters {A,B} and {C,D}. Thus, at this
last level are mrged into one large cluster that contains all elements. The dendrogram
_
_
f r this smgle link example is shown in Figure 5.7(a). The labeling on the right-hand
?
side shows the threshold distance used to merge the clusters at each level.
graph representing this threshold distance is shown in Figure 5 .6(c). Note that we now
of 3 is shown in Figure 5 .6(d). Here the graph is connected, so the two clusters from the
5
4.5
4
3.5
3
2.5
2
1.5
only two edges. The first level of single link clustering then will combine the connected
clusters (single elements from the first phase), giving three clusters: {A,B}, {C,D}, and
of two clusters : {A,B,C,D} and {E} . The graph that is created with a threshold distance
1 35
D
(c) Graph with threshold of 2
D
(b) Graph with threshold of
D
(a) Graph with all distances
minimum distance between any two points is less than or equal to the threshold dis
Section 5.4
B E
C
D
(b) Complete link
B c
0.5
0
O (n 2) space and time algorithm, is called at each iteration. A more efficient algorithm
Item
c
2
There have been other variations of the single link algorithm. One variation, based
on the use of a minimum spanning tree (MST), is shown in Algorithm 5.2. Here we
D
E
An alternative view to merging clusters in the single link approach is that two
each step. Another problem is that the clustering creates clusters with long chains.
could be developed by looking at which clusters from an earlier level can be merged at
clusters are merged at a stage where the threshold distance is d if the minimum distance
between any vertex in one cluster and any vertex in the other cluster is at most d.
assume that a procedure, MST, produces a minimum spanning tree given an adjacency
matrix as input. The clusters are merged in increasing order of the distance found in the
MST. In the algorithm we show that once two clusters are merged, the distance between
Chapter 5
1 36
Cluster i n g
oo .
ALGORITHM 5.2
Input :
D = { t1 , t 2 , . . . , tn)
I / S e t of e l ements
A
/ /Adj acency matrix showing d i s t ance be tween el ements
Output :
II Dendrogram repre s ented as a set of ordered t r ip l e s
DE
d=O
k=n
K = { { t 1 } , . . . , { tn))
DE = (d, k, K) ; I I Ini t i a l l y dendrogram contains e a c h e l ement in
its own clu s t e r .
M = MST(A) ;
repeat
oldk = k ;
k = oldk - 1 ;
d = dis(Ki , Kj ) i
DE = DEU (d, k, K) ; / / New set of c lusters added to dendrogram .
dis (Ki , Kj ) = oo
un til k = 1
We illustrate this algorithm using the data in Example 5.3. Figure 5.8 shows one
MST for the example. The algorithm will merge A and B and then C and D (or the
reverse). These two clusters will then be merged at a threshold of 2. Finally, E will
be merged at a threshold of 3. Note that we get exactly the same dendrogram as in
Figure 5.7(a).
The time complexity of this algorithm is 0 (n 2 ) because the procedure to create
the minimum spanning tree is O (n 2 ) and it dominates the time of the algorithm. Once
it is created having n
1 edges, the repeat loop will be repeated only n
1 times.
The single linkage approach is infamous for its chain effect; that is, two clusters
are merged if only two of their points are close to each other. There may be points in
the respective clusters to be merged that are far apart, but this has no impact on the
algorithm. Thus, resulting clusters may have points that are not related to each other at
all, but simply happen to be near (perhaps via a transitive relationship) points that are
close to each other.
-
1 37
D = { t l , t2 tn }
/ / Set of e l eme nts
A
/ /Adj acency mat r ix showing distance bet ween e l eme nt s
Outpu t :
DE
/ / Dendrogram repre s ented as a set of ordered trip l e s
Average l ink algor i t hm :
d= 0;
k = n;
K = {{ t l } , . . . , { tn } } ;
DE = (d, k, K) ; I I Init i a l ly dendrogram contains each e l ement
in i t s own c l u s t e r .
repeat
oldk = k;
d= d+
0.5 ;
for each p a i r of Ki , Kj
K do
if ave :S d ,
ti E Ki and tj E Kj ;
then
DE = DE U (d, k, I() ;
c
D
unt i l k =
Note that in this algorithm we increment d by 0.5 rather than by 1. This is a rather
arbitrary decision based on understanding of the data. Cettainly, we could have used
an increment of 1 , but we would have had a dendrogram different from that seen in
Figure 5 .7(c).
1 38
5.4.2
Chapter 5
Section 5.5
Clustering
ALGORITHM 5.4
Input :
D = { t l , t2 , tn}
S(n, k)
k
k i ( 7 )
L(-l)
(5.5)
I=l
There are 1 1 ,259,666,000 different ways to cluster 1 9 items into 4 clusters. Thus, most
algorithms look only at a small subset of all the clusters psing some strategy to identify
sensible clusters. Because of the plethora of partitional algorithms, we will look at only
a representative few. We have chosen some of the fncist well known algorithms as well
as some others that have appeared recently in the literature.
5.5.1
Since we have agglomerative and divisive algorithms based on the use of an MST,
we also present a partitional MST algorithm. This is a very simplistic approach, but it
algorithm :
(5.4)
(i)n
/ /Set of elements
/ /Adj acency matrix showing distance between elements
/ /Number of des ired clusters
Partitional
M = MST(A)
k
Output :
L L
139
illustrates how partitional algorithms work. The algorithm is shown in Algorithm 5.4.
Since the clustering problem is to define a mapping, the output of this algorithm shows
the clusters as a set of ordered pairs (ti , j) where f Cti )
KJ .
Divisive Clustering
With divisive clustering, all items are initially placed in one cluster and clusters are
repeatedly split in two until all items are in their own cluster. The idea is to split up
clusters where some elements are not sufficiently close to other elements.
One simple example of a divisive algorithm is based on the MST version of the
single link algorithm. Here, however, we cut out edges from the MST from the largest
to the smallest. Looking at Figure 5.8, we would start with a cluster containing all
items: { A , B, C, D, E}. Looking at the MST, we see that the largest edge is between
D and E. Cutting this out of the MST, we then split the one cluster into two: { E } and
{ A , B, C, D}. Next we remove the edge between B and C. This splits the one large
cluster into two: {A , B } and {C, D } . These will then be split at the next step. The order
depends on how a specific implementation would treat identical values. Looking at the
dendrogram in Figure 5.7(a), we see that we have created the same set of clusters as
with the agglomerative approach, but in reverse order.
5.5
5.5.2
The squared error clustering algorithm minimizes the squared error. The squared error
for a cluster is the sum of the squared Euclidean distances between each element in the
cluster and the cluster centroid, Ck. Given a cluster Ki , let the set of items mapped to
that cluster be {ti l , ti 2 , . . . , tim } . The squared error is defined as
m
seK; =
L ll ti} - ck f
(5.6)
J=l
k
L se Kj
J= l
(5.7)
In actuality, there are many different examples of squared error clustering algo
rithms. They all follow the basic algorithm structure shown in Algorithm 5.5.
Chapter
140
Section 5.5
ciustering
ALGORiTHM 5.5
k
K
I /Set of element s
/ /Number of desired clusters
{ t1 , t2, . . . , tn)
k
K
// Set of clusters
K-means
algo r i t hm :
Output :
assign each item ti to the cluster which has the closest mean ;
calculate new mean for each cluster ;
unt i l convergence criteria is met ;
assign each item ti to the cluster which has the c losest center ;
calculate new center for each cluster ;
calculate squared error ;
unt i l the difference between successive squared errors
is below a threshold ;
For each iteration in the squared error algorithm, each tuple is assigned to the
cluster with the clsest center. Since there are k clusters and n items, this is an 0 (kn)
operation. Assuming t iterations, this becomes an O (tkn) algorithm. The amount of space
may be only O (n) because an adjacency matrix is not needed, as the distance between
all items is not used.
2
2. We initially assign the means to the first two values: m 1
and suppose that k
{2, 3} and K2
4. Using Euclidean distance, we find that initially Kt
and m2
.
{4, 10, 12, 20, 30, 1 1 , 25 }. The value 3 i s equally close to both means, so we arb1trarlly
choose K 1 Any desired assignment could be used in the case of ties. We then recalculate
16. We again make assignments to clusters to gt
2.5 and m2
the means to get m 1
{ 10, 12, 20, 30, 1 1 , 25 }. Continuing in this fashion, we obtam
{2, 3, 4} and K2
K1
the following:
=
K-means is an iterative clustering algorithm in which items are moved among sets of clus. ters until the desired set is reached. As such, it may be viewed as a type of squared error
algorithm, although the convergence criteria need not be defined based on the squared
error. A high degree of similarity among elements in clusters is obtained, while a high
degree of dissimilarity among elements in different clusters is achieved simultaneously.
, tim } is defined as
{tn , ti2
The cluster mean of Ki
.
. .
mi
tiJ
m L
(5.8)
J=l
This definition assumes that each tuple has only one numeric value as opposed to a
tuple with many attribute values. The K-means algorithm requires that some definition
of cluster mean exists, but it does not have to be this particular one. Here the mean
is defined identically to our earlier definition of centroid. This algorithm assumes that
the desired number of clusters, k, is an input parameter. Algorithm 5.6 shows the K
means algorithm. Note that the initial values for the means are arbitrarily assigned. These
could be assigned randomly or perhaps could use the values from the first k input items
themselves. The convergence criteria could be based on the squared error, but they need
not be. For example, the algorithm could stop when no (or a very small) number of tuples
are assigned to different clusters. Other termination techniques have simply looked at a
fixed number of iterations. A maximum number of iterations may be included to ensure
stopping even without convergence.
ALGORITHM 5.6
Input :
D = { t l , t2 , tn)
// Set of elements
. =
K-Means Cl ustering
(5.9)
m1 , m2 , . . . , mk ;
repeat
5.5.3
141
Output :
Input :
Partitional Algorithms
m1
3
4.75
7
m2
K!
18
19 . 6
25
{2, 3, 4, 1 0 }
{2, 3, 4, 10, 1 1 , 12}
{2, 3, 4, 10, 1 1 ' 12}
K2
{ 1 2, 20, 30, 1 1 , 25}
{20, 30, 25}
{20, 30, 25}
Note that the clusters in the last two steps are identical. This will yield identical means,
{2, 3, 4, 10, 1 1 , 12} and
and thus the means have converged. Our answer is thus Kt
K2 - {20, 30, 25 } .
=
Chapter 5
1 42
5.5.4
Cl uste ring
Section 5.5
An algorithm similar to the single link technique is called the nearest neighbor algorithm.
With this serial algorithm, items are iteratively merged into the existing clusters that are
closest. In this algorithm a threshold, t, is used to determine if items will be added to
existing clusters or if a new cluster is created.
ALGORITHM 5.7
Input :
D = { t1 , t2 , . . . , tn}
/ / Set of e l ements
A
/ /Adj acency matrix showing d i s t ance between el ements
Output :
K1 = f t 1 } ;
K = {Kl } i
k= 1;
for i = 2 to.;
n do
Km = Km U t i
k = k+ l;
Kk = { t i } ;
tj
2. fj
3. lj
fj
4.
We leave it as an exercise to determine the cost of each of these cases. The total impact
to quality by a medoid change T Ch then is given by
Example 5.5 shows the application of the nearest neighbor algorithm to the data
shown in Table 5.2 assuming a threshold of 2. Notice that the results are the same as
those seen in Figure 5 .7(a) at the level of 2.
T Ct h =
EXAMPLE 5 . 5
PAM Algorithm
The PAM (partitioning around medoids) algorithm, also called the K-medoids algorithm,
represents a cluster by a medoid. Using a medoid is an approach that handles outliers
1 43
well. The PAM algorithm is shown in Algorithm 5.8. Initially, a random set of k items
is taken to be the set of medoids. Then at each step, all items from the input dataset that
are not currently medoids are examined one by one to see if they should be medoids.
That is, the algorithm determines whether there is an item that should replace one of the
existing medoids. By looking at all pairs of medoid, non-medoid objects, the algorithm
chooses the pair that improves the overall quality of the clustering the best and exchanges
them. Quality here is measured by the sum of all distances from a non-medoid object
to the medoid for the cluster it is in. An item is assigned to the cluster represented by
the medoid to which it is closest (minimum distance). We assume that K; is the cluster
represented by medoid t; . Suppose t; is a current medoid and we wish to determine
whether it should be exchanged with a non-medoid th . We wish to do this swap only
if the overall impact to the cost (sum of the distances to cluster medoids) represents an
improvement.
Following the lead in [NH94], we use Cji h to be the cost change for an item fj
associated with swapping medoid t; with non-medoid fh . The cost is the change to the
sum of all distances from items to their cluster medoids. There are four cases that must
be examined when calculating this cost:
1.
else
Partitional Algorithms
L Cjt h
(5. 10)
j =i
ALGORITHM 5.8
Input :
D = { t l , t2 , . . . , tn}
/ / S e t of e l ements
A
//Adj acency mat rix showing dis t ance between e l ement s
/ /Number of de s ired clusters
k
Output :
/ / Set of c lusters
P AM algor i t hm :
th not a medoid
medoid t i do
do
for each
c a l culate TCi h ;
find i, h where TCih is the sma l l e s t ;
i f TCih < 0 , then
rep lace medo id ti with th ;
un t i l TCih :::: 0 ;
for each
ti E D
do
144
Cl ustering
Chapter 5
Example 5.6 shows the application of the PAM algorithm to the data shown in
Table 5.2 assuming a threshold of 2.
<c
D
EXAMPLE 5.6
Suppose that the two medoids that are initially chosen are A and B. Based on the
distances shown in Table 5.2 and randomly placing items when distances are identical to
the two medoids, we obtain the clusters { A , C, D} and {B, E} The three non-medoids,
{ C, D , E}, are then examined to see which (if any) should be used to replace A or B .
We thus have six costs to determine: T CA c , T CAD TCA E , TCB c , TCBD, and TCB E
Here we use the name of the item instead of a numeric subscript value. We obtain the
following:
T CA c = CAAc
CBA C
CcA C
CDA C
CEAC = 1
+0-2-1 +0
= -2
Section 5 . 5
145
A
1
B
Here A is no longer a medoid, and since it is closer to B , it will be placed in the cluster
with B as medoid, and thus its cost is CAAC = 1 . The cost for B is 0 because it stays
1
a cluster medoid. C is now a medoid, so it has a negative cost based on its distance to
the old medoid; that is, CcA B = -2. D is closer to C than it was to A by a distance
of 1, so its cost is CDA C = - 1 . Finally, E stays in the same cluster with the same
distance, so its cost change is 0. Thus, we have that the overall cost is a reduction of 2.
Figure 5.9 illustrates the calculation of these six costs. Looking at these, we see that
the minimum cost is 2 and that there are several ways to reduce this cost. Arbitrarily
choosing the first swap, we get C and B as the new medoids with the clusters being
{C, D} and { B , A , E}. This concludes the first iteration of PAM. At the next iteration,
we examine changing medoids again and pick the choice that best reduces the cost. The
iterations stop when no changes will reduce the cost. We leave the rest of this problem
to the reader as an exercise.
(5. 1 1)
PAM does not scale well to large datasets because of its computational complexity.
For each iteration, we have k (n - k) pairs of objects i, h for which a cost, T Cih, should be
determined. Calculating the cost during each iteration requires that the cost be calculated
for all other non-medoids tj . There are n - k of these. Thus, the total complexity per
iteration is n (n - k) 2 . The total number of iterations can be quite large, so PAM is not
an alternative for large databases. However, there are some clustering algorithms based
on PAM that are targeted to large datasets.
CLARA (Clustering LARge Applications) improves on the time complexity of PAM
by using samples of the dataset. The basic idea is that it applies PAM to a sample of the
underlying database and then uses the medoids found as the medoids for the complete
clustering. Each item from the complete database is then assigned to the cluster with the
medoid to which it is closest. To improve the CLARA accuracy, several samples can be
drawn with PAM applied to each. The sample chosen as the final clustering is the one
that performs the best. Because of the sampling, CLARA is more efficient than PAM for
large databases. However, it may not be as effective, depending on the sample size. Five
samples of size 40 + 2k seem to give good results [KR90] .
CLARANS (clustering large applications based upon randomized search) improves
on CLARA by using multiple different samples. In addition to the normal input to PAM,
Partitional Algorithms
B
3
E
(d) TC8c: Change -2
F I G U R E 5.9:
E
(e) TC8v: Change -2
E
(f) TC8E: Change -2
5.5.6
U:
146
Chapter 5
Cl ustering
Section 5.5
Partitional Algorithms
1 47
Algorithm 5.9 shows one possible iterative refinement technique for clustering that
uses a genetic algorithm. The approach is similar to that in the squared error approach iu
that an initial random solution is given and successive changes to this converge on a local
optimum. A new solution is generated from the previous solution using crossover and
mutation operations. Our algorithm shows only crossover. The use of crossover to create
must be used and may be defined based on an inverse of the squared error. Because
of the manner in which crossover works, genetic clustering algorithms perform a global
D = { t 1 , t2 , .
Output :
may be stored at different sites from other fragments. The basic steps of this
clustering
// S e t
algorithm are:
. .
, tn}
// Number
// S et of el ements
of de s i red clusters
of clusters
GA clustering algo r i t hm :
repeat
the two associate attributes. The entries in the similarity matrix are based on the
unt i l
2. The BEA then converts this similarity matrix to a BOND matrix in which the
entries represent a type of nearest neighbor bonding based on probability of co
EXAMPLE 5.7
access. The BEA algorithm rearranges rows or columns so that similar attributes
appear dose together in the matrix.
3. Finally, the designer draws boxes around regions n the matrix with high similarity.
{ A , C, E } , { B ,
F},
and { D , G,
boxes represent the attributes that have been grouped together into two clusters.
Ai
C, D, E, F, G, H}, which
000 1 00 1 1 , respectively. Suppose we choose the first and third individuals as parents and
The resulting matrix, modified from [OV99] , is illustrated in Figure 5 . 10 The two shaded
Two attributes
are to be placed into three clusters. We could mtially place the items into the three clusters
101000 1 1 .
in database applications. At the heart of the BEA algorithm is the global affinity measure.
Suppose that a database schema consists of
affinity measure,
AM, is defined as
AM =
l )bond(A i , A i - l ) + bond(A i , A i + I ) )
i=l
(5 . 1 2)
data that characterize the desired output. They look for clusters of like data.
of NNs are often called
These types
5.5.7
With the
There have been clustering techniques based on the use of genetic algorithms. To deter
b.Wj i
mine how to perform clustering with genetic algorithms, we first must determine how
to. represent each cluster. One simple approach would be to use a bit-map representation
for each possible cluter. So, given a database with four items, { A , B , C , D } , we would
represent one solution to creating two clusters as 1001 and 0 1 10. This represents the two
clusters { A , D } and { B , C}.
With
T/YJYi
(5. 13)
competitive learning, nodes are allowed to compete and the winner takes all.
This approach usually assumes a two-layer NN in which all nodes from one layer are
connected to all nodes in the other layer. As training occurs, nodes in the output layer
become associated with certain tuples in the input dataset. Thus, this provides a grouping
148
Chapter 5
Cl ustering
Section 5.6
149
of these tuples together into a cluster. Imagine every input tuple having each attribute
value input to a specific input node in the NN. The number of input nodes is the same
as the number of attributes. We can thus associate each weight to each output node with
one of the attributes from the input tuple. When a tuple is input to the NN, all output
nodes produce an output value. The node with the weights more similar to the input
tuple is declared the winner. Its weights are then adjusted. This process' continues with
each tuple input from the training set. With a large and varied enough training set, over
time each output node should become associated with a set of tuples. The input weights
to the node are then close to an average of the tuples in this cluster.
Self-Organizing Feature Maps.
or
self
Learning is based on the concept that the behavior of a node should impact only those
nodes and arcs near it. Weights are initially assigned randomly and adjusted during the
learning process to produce better results. During this learning process, hidden features or
patterns in the data ,are uncovered and the weights are adjusted accordingly. SOFMs were
F I G U R E 5. 1 1 : Kohonen network.
developed by obsefving how neurons work in the brain and in ANNs. That is [BS97] :
The firing of neurons impact the firing of other neurons that are near it.
The term
self-organizing
can be calculated by
sim(X,
i)
h
I >j Wj i
j =l
(5. 14)
The competitive node most similar to the input node wins the competitive. Based on this,
clusters based on the similarity between them. Those nodes that are closer together are
more similar than those that are far apart. This hints at how the actual clustering is
the matrix are increased. This is the learning phase. Given a node
performed. Over time, nodes in the output layer become matched to input nodes, and
Ni
D.w
There is one input layer and one special layer, which produces output values that compete.
In this formula,
In effect, multiple outputs are created and the best one is chosen. This extra layer is
kJ
{ C(Xk - Wkj)
0
if
Ni
otherwise
(5. 15)
c indicates the learning rate and may actually vary based on the node rather
not technically either a hidden layer or an output layer, so we refer to it here as the
than being a constant. The basic idea of SOM learning is that after each input tuple in the
seen in Figure 5 . 1 1 . Each input node is connected to each node in this grid. Propagation
the tuple. Over time, a pattern on the output nodes emerges, which is close to that of the
competitive layer.
training set, the winner and its neighbors have their weights changed to be closer to that of
training data. At the beginning of the training process, the neighborhood of a node may be
occurs by sending the input value for each input node to each node in the competitive
defined to be large. However, the neighborhood may decrease during the processing.
layer. As with regular NNs, each arc has an associated weight and each node in the
competitive layer has an activation function. Thus, each node in the competitive layer
produces an output value, and the node with the best output wins the competition and is
determined to be the output for that input. An attractive feature of Kohonen nets is that
the data can be fed into the multiple competitive nodes in parallel. Training occurs by
adjusting weights so that the best output is even better the next time this input is used.
A common approach is to initialize the weights on the input arcs to the com
5.6
tering techniques. When clustering is used with dynamic databases, these algorithms may
not be appropriate. First, they all assume that [because most are
O (n 2) ]
sufficient main
memory exists to hold the data to be clustered and the data structures needed to support
them. With large databases containing thousands of items (or more), these assumptions
petitive layer with normalized values. The similarity between output nodes and input
are not realistic. In addition, performing 1/0s continuously through the multiple itera
, xh) and weights on arcs input to a competitive node i as WJi , . . . , Whi , the
algorithms do not scale up to large databases. Another issue is that some assume that the
vectors is then determined by the dot product of the two vectors. Given an input tuple
X =
(x1 , . . .
tions of an algorithm is too expensive. Because of these main memory restrictions, the
1 50
Section 5.6
Clu steri ng
Chapter 5
data are present all at once. These techniques are not appropriate for dynamic databases.
Clustering techniques should be able to adapt as the database changes.
The algorithms discussed in the following subsections each examine an issue asso
ciated with performing clustering in a database environment. It has been argued that to
perform effectively on large databases, a clustering algorithm should [BFR98]:
151
This algorithm uses a tree called a CF tree a s defined in Definition 5.4. The size o f the
tree is determined by a threshold value, T, associated with each leaf node. This is the
maximum diameter allowed for any leaf. Here diameter is the average of the pairwise
distance between all points in the cluster. Each internal node corresponds to a cluster
that is composed of the subclusters represented by its children.
{
DEFINITION 5.3. A clustering feature (CF) is a triple (N, s, SS), where the
{
number of the points in the cluster is N, s is the sum of the points in the cluster,
2. have the ability to provide status and "best" answer so far during the algorithm
4. be able to update the results incrementally as data are added or removed from the
number of children a node may have) B . Each internal node contains a CF triple
for each of its children. Each leaf node also represents a cluster and contains a CF
database.
entry for each subcluster in it. A subcluster in a leaf node must have a diameter
6. be capable of performing different techniques for scanning the database. This may
include sampling.
the CF tree contains clustering feature information about its subclusters. As points are
added to the clustering problem, the CF tree is built. A point is inserted into the cluster
(represented by a leaf node) to which it is closest. If the diameter for the leaf node is
greater than T, then a splitting and balancing of the tree is performed (similar to that
used in a B-tree). The algorithm adapts to main memory size by changing the threshold
value. A larger threshold, T , yields a smaller CF tree. This process can be performed
without rereading the data. The clustering feature data provides enough information to
perform this condensation. The complexity of the algorithm is O (n).
ALGORITHM 5.10
Inpu t :
D = { t 1 , t2 , . . . , tn}
T
BIRCH
BIRCH (balanced iterative reducing and clustering using hierarchies) is designed for
clustering a large amount of metric data. It assumes that there may be a limited amount
of main memory and achieves a linear 1/0 time requiring only one database scan. It is
incremental and hierarchical, and it uses an outlier handling technique. Here points that
/ / S e t of
e l ements
Outpu t :
K
/ / S e t of cl u s t e r s
ti E D
for each
do
determine correc t
l e a f node
for
ti
ins e r t i on ;
ti
add
5.6.1
then
else
i f room t o i n s e r t
insert
ti
as
ti ,
then
else
sp l i t l e a f node and redist ribute CF fe ature s ;
are found in sparsely populated areas are removed. The basic idea of the algorithm is that
Algorithm 5 . 1 0 outlines the steps performed in BIRCH. Not shown in this algorithm
a tree is built that captures needed information to perform clustering. The clustering is
are the parameters needed for the CF tree construction, such as its branching factor, the
then performed on the tree itself, where labelings of nodes in the tree contain the needed
page block size, and memory size. Based on size, each node has room for a fixed number,
B, of clusters (i.e., CF triples). The first step creates the CF tree in memory. The threshold
value can be modified if necessary to ensure that the tree fits into the available memory
space. Insertion into the CF tree requires scanning the tree from the root down, choosing
the node closest to the new point at each level. The distance here is calculated by looking
1 52
Chapter 5
at the distance between the new point and the centroid of the cluster. This can be easily
calculated with most distance measures (e.g., Euclidean or Manhattan) using the CF
triple. When the new item is inserted, the CF triple is appropriately updated, as is each
triple on the path from the root down to the leaf. It is then added to the closest leaf node
found by adjusting the CF value for that node. When an item is inserted into a cluster at
the leaf node of the tree, the cluster must satisfy the threshold value. If it does, then the
CF entry for that cluster is modified. If it does not, then that item is added to that node
as a single-item cluster.
Node splits occur if no space exists in a given node. This is based on the size of
the physical page because each node size is determined by the page size. An attractive
feature of the CF values is that they are additive; that is, if two clusters are merged, the
resulting CF is the addition of the CF values for the starting clusters. Once the tree is
built, the leaf nodes of the CF tree represent the current clusters.
In reality, this algorithm, Algorithm 5 . 1 0, is only the first of several steps proposed
for the use of BIRCH with large databases. The complete outline of steps is:
1. Create initi
2. The clustering represented by the CF tree may not be natural because each entry
has a limited size. In addition, the input order can negatively impact the results.
These problems can be overcome by another global clustering approach applied
to the leaf nodes in the CF tree. Here each leaf node is treated as a single point
for clustering. Although the original work proposes a centroid-based agglomer
ative hierarchical clustering algorithm to cluster the subclusters, other clustering
algorithms could be used.
3. The last phase (which is optional) reclusters all points by placing them in the clus
ter that has the closest centroid. Outliers, points that are too far from any centroid,
can be removed during this phase.
BIRCH is linear in both space and I/0 time. The choice of threshold value is
imperative to an efficient execution of the algorithm. Otherwise, the tree may have to be
rebuilt many times to ensure that it can be memory-resident. This gives the worst-case
time complexity of O (n 2) .
5.6.2
Section 5.6
Cl ustering
DBSCAN
Minpts
Eps
I /
(
I
I
I
I
\
'
'
'
p I '\
.------!
...... _ _ _
/I
1 53
I
I
I
I
(a) Bps-neighborhood
DB SCAN example.
DBSCAN uses a new concept of density. We fitst must look at some definitions
froin [EKSX96] . Definition 5.5 defines directly density-reachable. The first part of the
definition ensures that the second point is "close enough" to the first point. The second
portion of the definition ensures that there are enough core points close enough to each
other. These core points form the main portion of a cluster in that they are all close to
each other. A directly density-reachable point must be dose to one of these core points,
but it need not be a core point itself. In that case, it is called a border point. A point is
said to be density-reachable from another point if there is a chain from one to the other
that contains only points that are directly density-reachable from the previous point. This
guarantees that any cluster will have a core set of points very close to a large number of
other points (core points) and then some other points (border points) that are sufficiently
close to at least one core point.
DEFINITION 5.5. Given values Eps and MinPts, a point p is directly density
reachable from q if
Figure 5.12 illustrates the concepts used by DBSCAN. This figure shows 1 2 points.
The assumed Eps value is illustrated by the straight line. In part (a) it is shown that there
are 4 points within the neighborhood of point p. As is seen, p is a core point because
it has 4 (MinPts value) points within its neighborhood. Part (b) shows the 5 core points
in the figure. Note that of the 4 points that are in the neighborhood of p, only 3 are
themselves core points. These 4 points are said to be directly density-reachable from p.
Point q is not a core point and is thus called a border point. We have partitioned the
points into a core set of points that are all close to each other; then border points, which
are close to at least one of the core points; and finally the remaining points, which are
not close to any core point. Part (C) shows that even though point r is not a core point,
it is density-reachable from q.
Algorithm 5 . 1 1 outlines the DBSCAN algorithm. Because of the restrictions on
what constitutes a cluster when the algorithm finishes, there will be points not assigned
to a cluster. These are defined as noise.
1 54
Chapter 5
Clustering
Section 5.6
0
0 0
0 0
0
0
0
Input :
//Set of elements
D = { t 1 , t 2 , . . . , tn}
MinPts
/ / Number of points in cluster
Eps
II Maximum distance for density measure
Output :
I /Set
of clusters
DBSCAN algorithm:
k=
1 55
ALGORITHM 5.11
K = {K1 , K2 , . . . , Kk}
for
0 0
0
0
0
0
0
0
0
(b) Three clusters with representative points
"'
'
I
0 .
0 0 /
/ 0 0 /
I
I 0
0(
/
\
- - "' l l} I
0
I
(Q ... ./
/
\I
/ 0 0 o \
t
"'D
0
\
(0
\.
0 0 \
/
/
5.6.3
CURE Algorithm
One objective for the CURE (Clustering Using REpresentatives) clustering algorithm is to
handle outliers well. It has both a hierarchical component and a partitioning component.
First, a constant number of points, c, are chosen from each cluster. These well-scattered
points are then shrunk toward the cluster's centroid by applying a shrinkage factor, a .
When a i s 1 , all points are shrunk t o just one-the centroid. These points represent the
cluster better than a single point (such as a medoid or centroid) could. With multiple rep
resentative points, clusters of unusual shapes (not just a sphere) can be better represented.
CURE then uses a hierarchical clustering algorithm. At each step in the agglomerative
algorithm, clusters with the closest pair of representative points are chosen to be merged.
The distance between them is defined as the minimum distance between any pair of
points in the reprentative sets from the two clusters.
The basic approach used by CURE is shown in Figure 5 . 1 3 . The first step shows
a sample of the data. A set of clusters with its representative points exists at each step
in the processing. In Figure 5 . 1 3(b) there are three clusters, each with two representative
points. The representative points are shown as darkened circles. As discussed in the
following paragraphs, these representative points are chosen to be far from each other as
well as from the mean of the cluster. In part (c), two of the clusters are merged and two
new representative points are chosen. Finally, in part (d), these points are shrunk toward
the mean of the cluster. Notice that if one representative centroid had been chosen for
the clusters, the smaller cluster would have been merged with the bottom cluster instead
of with the top cluster.
CURE handles limited main memory by obtaining a random sample to find the
initial clusters. The random sample is partitioned, and each partition is then partially
clustered. These resulting clusters are then completely clustered in a second pass. The
______
_
_ _ _
,'
F I G U R E 5. 1 3 : CURE exampie.
sampling and partitioning are done solely to ensure that the data (regardless of database
size) can fit into available main memory. When the clustering of the sample is complete,
the labeling of data on disk is performed, A data item is assigned to the cluster with the
closest representative points. The basic steps of CURE for large databases are:
1. Obtain a sample of the database.
Algorithm 5. 12). This provides a first guess at what the clusters should be. The
number of clusters is
for some constant q .
4. Remove outliers. Outliers are eliminated b y the use o f two different techniques.
The first technique eliminates clusters that grow very slowly. When the number of
clusters is below a threshold, those clusters with only one or two items are deleted.
It is possible that close outliers are part of the sample and would not be identified
by the first outlier elimination technique. The second technique removes very small
clusters toward the end of the clustering phase.
Chapter 5
1 56
Clustering
Section 5 . 7
5. Completely cluster all data in the sample using Algorithm 5 . 1 2. Here, to ensure
processing in main memory, the input includes only the cluster representatives from
the clusters found for each partition during the partial clustering step (3).
insert(T,
ALGORITHM 5.12
Input :
D = { t 1 , t 2 , . . . , tn}
k
/ / Des ired
CURE algor i t hm :
T = buil d(D) ;
Q = heapi fy(D) ;
each
repeat
u = min(Q) ;
del e t e (Q, u . cl ose) ;
w = merge( u, v) ;
del e t e(T, u) ;
del e te(T, v) ;
to
x;
Traditional algorithms do not always work with categorical data. Example 5.8 illus
trates some problems that exist when clustering categorical data. This example uses a
hierarchical-based centroid algorithm to illustrate the problems. The problem illustrated
here is that the centroid tends to weaken the relationship between the associated cluster
and other clusters. The problems worsens as more and more clusters are merged. The
number of attributes appearing in the mean increases, while the individual values actually
decreases. This makes the centroid representations very similar and makes distinguishing
between clusters difficult.
EXAMPLE 5.8
Consider an information retrieval system where documents may contain keywords {book,
water, sun, sand, swim, read}. Suppose there are four documents, where the first contains
the word {book}, the second contains {water, sun, sand, swim}, the third contains {water,
sun, swim, read}, and the fourth contains {read, sand}. We can represent the four books
using the following boolean points: ( 1 , 0, 0, 0, 0, 0), (0, 1, 1 , 1 , 1, 0), (0, 1 , 1, 0, 1 , 1),
(0, 0, 0, 1 , 0, 1 ). We can use the Euclidean distance to develop the following adjacency
matrix of distances:
/ /Set of element s
number of clusters
Output :
1 57
w) ;
x E Q do
x . close = find closest cluster
if x is closest to w, then
w. close = x;
insert(Q, w) ;
unt i l number of nodes in Q is k ;
for
6. Cluster the entire database on disk using c points to represent each cluster. An item
in the database is placed in the cluster that has the closest representative point to
it. These sets of representative points are small enough to fit into main memory,
so each of the n points must be compared to ck representative points.
The time complexity of CURE is O (n 2 lg n), while space is O (n). This is worst
case behavior. The improvements proposed for main memory processing certainly
improve on this time complexity because the entire clustering algorithm is performed
against only the sample. When clustering is performed on the complete database, a time
complexity of only O (n) is required. A heap and k-D tree data structure are used to
ensure this performance. One entry in the heap exists for each cluster. Each cluster has
not only its representative points, but also the cluster that is closest to it. Entries in the
heap are stored in jncreasing order of the distances between clusters. We assume that
each entry u in the heap contains the set of representative points, u. rep; the mean of
the points in the cluster, u.mean; and the cluster closest to it, u. closest. We use the heap
operations: heapify to create the heap, min to extract the minimum entry in the heap,
insert to add a new entry, and delete to delete an entry. A merge procedure is used to
merge two clusters. It determines the new representative points for the new cluster. The
basic idea of this process is to first find the point that is farthest from the mean. Sub
sequent points are then chosen based on being the farthest from those points that were
previously chosen. A predefined number of points is picked. A k-D tree is a balanced
binary tree that can be thought of as a generalization of a binary search tree. It is used to
index data of k dimensions where the i1h level of the tree indexes the i1h dimension. In
CURE, a k-D tree is used to assist in the merging of clusters. It stores the representative
points for each cluster. Initially, there is only one representative point for each cluster,
the sole item in it. Operations performed on the tree are: delete to delete an entry form
the tree, insert to insert an entry into it, and build to initially create it. The hierarchical
clustering algorithm itself, which is from [GRS98], is shown in Algorithm 5 . 12. We do
not include here either the sampling algorithms or the merging algorithm.
1
2
3
4
0
2.24
2.24
1 .73
2.24
0
1 .4 1
2
2.24
1 .4 1
0
2
4
1 .73
2
2
0
The distance between points 2 and 3 is the smallest ( 1 .41), and thus they are merged.
When they are merged, we get a cluster containing { (0 1 , 1 , 1 , 1 , 0) , (0, 1 , 1 , 0, 1 , 1 ) }
with a centroid of (0, 1, 1 , 0.5, 1 , 0.5). At this point we have a distance from this new
cluster centroid to the original points 1 and 4 being 2.24 and 1.73, respectively, while the
distance between original points 1 and 4 is 1 .73. Thus, we could next merge these points
,
1 58
Cl ustering
Chapter 5
Section 5.8
1
2
3
4
boolean data and categorical data. A novel approach to identifying similarity is based
on the number of links between items. A pair of items are said to be
neighbors if their
1
0
0
0
0
3
3
3
0
3
3
3
0
3
3
3
similarity exceeds some threshold. This need not be defined based on a precise metric,
Comparison
1 59
set of clustering found with a traditional Euclidean distance, we see that a "better" set
but rather a more intuitive approach using domain experts could be used. The number of
links between two items is defined as the number of common neighbors they have. The
objective of the clustering algoritlun is to group together points that have more links.
The algorithm is a hierarchical agglomerative algorithm using the number of links as the
2. Performing clustering on the data using the link agglomerative approach. A good
coefficient, has been proposed. One proposed similarity measure based on the Jaccard
coefficient is defind as
1
sim(t; , tJ )
I t n t I
'
1
I t; u t1 I
ness measure is used to determine which pair of points is merged at each step.
3. Using these clusters the remaining data on disk are assigned to them.
(5. 1 6)
If the tuples are viewed to be sets of items purchased (i.e., market basket data), then we
g ( Ki , K1 )
look at the number of items they have in common divided by the total number in both.
(ni + n J )
0 and 1 .
The number of links between a pair of points can be viewed as the number of
Ki when the threshold used for the similarity measure is e. The function
/ (8) depends
on the data, but it is found to satisfy the property that each item in K; has approximately
Also note that a hierarchical approach could be used with different threshold values for
n (E>) neighbors in the cluster. Obviously, if all points in the cluster are connected,
/ (8) = 1 . Then n [ is the number of links between points in K; .
The first step in the algorithm converts the adj acency matrix into a boolean matrix
EXAMPLE 5.9
where an entry is
1 if the two corresponding points are neighbors. As the adj acency matrix
2
n , this is an O (n 2) step. The next step converts this into a matrix indicating the
2
links. This can be found by calculating S x S, which can be done in 0 ( n 37 ) [GRS99].
is of size
2
3
4
1
0
0
0
The hierarchical clustering portion of the algorithm then starts by placing each point in
0
1
0.6
0.2
0
0.6
1
0.2
neighbors :
ni and n J are
of links because larger clusters would be expected to have more links simply because
Note that different threshold values for neighbors could be used to get different results.
(5. 17)
the number of points in each cluster. The denominator is used to normalize the number
than similarity (distance) measures provides a more global approach because the similarity
Here link(K; , KJ) is the number of links between the two clusters. Also,
2 between them. The authors argue that the use of links rather
link(Ki , KJ)
-;-:-o;.=--:-;;:::-)
+2/(E>)-n'1 +2/(8) -n 11+ 2/(8
= -
{ (2, 3 ) , (2, 4) , (3, 4) } . Note that in the following we add to these that a point is
{ (1 , 1 ) , (2, 2) , (3 , 3) , (4, 4) }.
The following table shows the number of links (common neighbors between points)
0.6:
q, is created to represent each cluster. Here q contains every cluster that has a nonzero
link to the cluster that corresponds to this cluster. Initially, a cluster is created for each
point,
t; . The heap for t; , q [t; ], contains every cluster that has a nonzero link to { td
The global heap contains information about each cluster. PJl information in the heap is
ordered based on the goodness measure, which is shown in Equation
k clusters are
found. To facilitate this processing, both local and global heaps are used. A local heap,
0
0.2
0.2
5.8
5. 17.
COMPARISON
The different clustering algoritluns discussed in this chapter are compared in Table 5.3.
Here we include a classification of the type of algorithm, space and time complexity,
and general notes concerning applicability.
1 60
Chapter 5
TAB LE 5.3:
Algorithm
Section 5 . 1 0
Cl uste ring
Space
Time
O (n2 )
O (n2 )
O (n2 )
O (n 2 )
O (kn 2 )
O (kn 2 )
O (kn 2)
O (n 2 )
Squared error
K-means
Nearest neighbor
PAM
Hierarchical
Hierarchical
Hierarchical
Hierarchical/
partitional
Partitional
Partitional
Partitional
Partitional
O (n)
O (n)
O (n 2 )
O (n 2 )
O (tkn)
O (tkn)
O (n 2 )
O (tk(n - k) 2 )
BIRCH
Partitional
O (n)
CURE
Mixed
O (n)
O (n)
(no rebuild)
O (n)
ROCK
Agglomerative
O (n 2 )
O (n 2 lgn )
DB SCAN
Mixed
O (n 2 )
O (n 2 )
Single link
Average link
Complete link
MST
Notes
Not incremental
Not incremental
Not incremental
Not incremental
161
the genetic algorithms in this table because their performance totally depends o n the
technique chosen to represent individuals, how crossover is done, and the termination
condition used.
Type
B i b l i ographic Notes
5. 9
EXERCISES
1. A major problem with the single link algorithm is that clusters consisting of long
Iterative
Iterative; No categorical
Iterative
Iterative; Adapted agglomerative; Outliers
CF-tree; Incremental;
Outliers
Heap; k-D tree; Incremental; Outliers;
Sampling
Sampling; Categorical;
Links
Sampling; Outliers
Item
A
B
0
1
4
5
1
0
2
6
4
2
0
3
5
6
3
0
3. Construct a graph showing all edges for the data in Exercise 2. Find an MST for
this graph. Is the MST single link hierarchical clustering the same as that found
using the traditional single link algorithm?
4. Convert Algorithm 5 . 1 to a generic divisive algorithm. What technique would be
used to split clusters in the single link and complete link versions?
5. Trace the results of applying the squared error Algorithm 5.5 to the data from
The single link, complete link, and average link techniques are all hierarchical tech
niques with O (n 2) time and space complexity. While we discussed the agglomerative
versions of these are also divisive versions, which create the clusters in a top-down man
ner. They all assume the data are present and thus are not incremental. There are several
clustering algorithms based on the construction of an MST. There are both hierarchical
and partitional versions. Their complexity is identical to that for the other hierarchical
techniques, and since they depend on the construction of the MST, they are not incre
mental. Both K-means and the squared error techniques are iterative, requiring O (tkn)
time. The nearest neighbor is not iterative, but the number of clusters is not prede
termined. Thus, the worst-case complexity can be O (n 2 ) . BIRCH appears to be quite
efficient, but remember that the CF-tree may need to be rebuilt. The time complexity
in the table assumes that the tree is not rebuilt. CURE is an improvement on these by
using sampling and partitioning to handle scalability well and uses multiple points rather
than just one point to represent each cluster. Using multiple points allows the approach
to detect nonspherical clusters. With sampling, CURE obtains an O (n) time complexity.
However, CURE does not handle categorical data well. This also allows it to be more
resistant to the negative impact of outliers. K-means and PAM work by iteratively reas
signing items to clusters, which may not find a global optimal assignment. The results
of the K-means algorithm is quite sensitive to the presence of outliers. Through the use
of the CF-tree, Birch is both dynamic and scalable. However, it detects only spherical
type clusters. DBSCAN is a density-based approach. The time complexity of DBSCAN
can be improved to O (n lgn ) with appropriate spatial indices. We have not included
Example 5.4 into two clusters. Indicate the convergence threshold you have used.
6. Use the K-means algorithm to cluster the data in Example 5.4 into three clusters.
7. Trace the use of the nearest neighbor algorithm on the data of Exercise 2 assuming
a threshold of 3.
8. Determine the cost Cjih for each of the four cases given for the PAM algorithm.
10.
5.1 0
BIBLIOGRAPH IC NOTES
There are many excellent books examining the concept of clustering. In [JD88], a thor
ough treatment of clustering algorithms, including application domains, and statement
of algorithms, is provided. This work also looks at a different classification of cluster
ing techniques. Other clustering and prediction books include [Har75], [JS7 1], [SS73],
[TB70], and [WI98].
A survey article of clustering with a complete list of references was published
in 1 999 [JMF99] . It covers more clustering techniques than are found in this chapter.
Included are fuzzy clustering, evolutionary techniques, and a comparison of the two.
An excellent discussion of applications is also included. Fuzzy clustering associates a
membership function with every item and every cluster. Imagine that each cluster is
1 62
Chapter 5
Section 5. 1 0
Cl ustering
represented as a vector of membership values, one for each element. Other clustering
surveys have been published in [NH94] and [HKT0 1].
Clustering tutorials have been presented at SIGMOD 1 999 [HK99] and
PAKDD-02. 1
The agglomerative clustering methods are among the oldest techniques. Proposals
include SLINK [Sib73] for single linkage and CLINK [Def77] for complete linkage. An
excellent study of these algorithms can be found in [KR90]. The AGNES and DIANA
techniques are some of the earliest methods. AGNES (AGglomerative NESting) is agglom
erative, while DIANA (Dlvisia ANAlysis) is divisive. B oth are known not to scale well.
Articles on single link clustering date back to 1 95 1 [FLP+ 5 1 ] . The EM algorithm has
frequently been used to perform interative clustering [DLR77] . There have been many
variations of the K-means clustering algorithm. The earliest reference is to a version
by Forgy in 1 965 [For65], [McQ67] . Another approach for partitiomil clustering is to
allow splitting and merging of clusters. Here merging is performed based on the dis
tance between the centroids of two clusters. A cluster is split if its variance is above a
certain threshold! One proposed algorithm performing this is called ISODATA [BH65].
CURE, predominantly a hierarchical technique, was first proposed in [GRS98]. Finding
connected components in a graph is a well-known graph technique that is described in
any graph theory or data structures text such as [Har72] .
The MST partitional algorithm was originally proposed by Zahn [Zha7 1 ] .
A recent work has looked at extending the definition of clustering to be more
applicable to categorical data. CAtegorical ClusTering Using Summaries (CACTUS) gen
eralizes the traditional definition of clustering and distance. To perform the clustering, it
uses summary information obtained from the actual data. The summary information fits
into main memory and can be easily constructed. The definition of similarity between
tuples is given by looking at the support of two attribute values within the database D .
Given two categorical attribute Ai , A j with domains D; , Dj , respectively, the support
of the attribute pair (a; , a j) is defined as:
(5 . 1 8)
The similarity of attributes used for clustering is based on the support [GGR99b] .
Several techniques have been proposed to scale up clustering algorithms to work
effectively on large databases. Sampling and data compressions are perhaps the most
common techniques. With sampling, the algorithm is applied to a large enough sample to
ideally produce a good clustering result. Both BIRCH and CACTUS employ compression
techniques. A more recent data compression approach, data bubbles, has been applied to
hierarchical clustering [BKKS01 ] . A data bubble is a compressed data item that represents
a larger set of items in the database. Given a set of items, a data bubble consists of (a
representative item, cardinality of set, radius of set, estimated average k nearest neighbor
distances in the set). Another compression technique for hierarchical clustering, which
actually compresses the dendrogram, has recently been proposed [XD0 1b].
A recent article has proposed outlier detection algorithms that scale well to large
datasets and can be used to model the previous statistical tests [KN98] .
1 0smar Zaiane and Andrew Foss, "Data Clustering Analysis, from Simple Groupings to Scalable Clustering
with Constraints," Tutorial at Sixth Pacific-Asia Conference on Knowledge Discovery and Data Mining, May
2002.
6,
1 63
Discussion of the bond energy algorithm and its use can be found in [WTMSW72],
[OV99], and [TF82] . It can be used to cluster attributes based on use and then perform
logical or physical clustering.
DB SCAN was first proposed in [EKSX96]. Other density-based algorithms include
DENCLUE [HK98] and OPTICS [] .
ROCK was proposed by Guha in [GRS99].
There have been many approaches targeted to clustering of large databases; includ
ing B IRCH [ZRL96], CLARANS [NH94] , CURE [GRS98], DBSCAN [EKSX96], and
ROCK [GRS99]. Specifics concerning CF tree maintenance can be found in the lit
erature [ZRL96] . A discussion of the use of R*-trees in DBSCAN can be found in
[EKSX96]. It is possible that a border point could belong to two clusters. A recent
algorithm, CHAMELEON, is a hierarchical clustering algmithm that bases merging of
clusters on both closeness and their interconnection [KHK99] . When dealing with large
databases, the requirement to fix the number of clusters dynamically can be a maj or
problem . Recent research into online (or iterative) clustering algorithms has been per
formed. "Online" implies that the algorithm can be performed dynamically as the data
are generated, and thus it works well for dynamic databases. In addition, some work has
examined how to adapt, thus allowing the user to change the number of clusters dynam
ically. These adaptive algorithms avoid having to completely recluster the database if
the users' needs change. One recent online approach represents the clusters by profiles
(such as cluster mean and size). These profiles are shown to the user, and the user has
the ability to change the parameters (number of clusters) at any time during process
ing. One recent clustering approach is both online and adaptive, OAK (online adaptive
clustering) [XDOl b] . OAK can also handle outliers effectively by adjusting a viewing
parameter, which gives the user a broader view of the clustering, so that he or she can
choose his or her desired clusters.
NNs have been used to solve the clustering problem [JMF99] . Kohonen's self
organizing maps are introduced in [Koh82]. One of the earliest algorithms was the leader
clustering algorithm proposed by Hartigan [Har75]. Its time complexity is only O (kn)
and its space is only O (k) . A common NN applied is a competitive one such as an
SOFM where the learning is nonsupervised [JMF99] .
References to clustering algorithms based on genetic algorithms include [JB 9 1 ]
and [BRE9 1 ] .
Section 6 . 1
C H A P T E R
6.1
6.3
6.4
INTRODUCTION
LARGE ITEMSETS
BASIC ALGORITHMS
PARALLEL AN D DISTRIBUTED ALGORITH MS
6.6
COMPARING APPROACHES
INCREM ENTAL RULES
6.7
6.5
6.8
6.9
1 65
jelly. Keeping grocery story cash register transactions in mind, each item represents
an item purchased, while each tuple is the list of items purchased at one time.
In the simplest cases, we are not interested in quantity or cost, so these may be
removed from the records before processing. Table 6. 1 is used throughout this chapter
to illustrate different algorithms. Here there are five transactions and five items:
{Beer, Bread, Jelly, Milk, PeanutButter}. Throughout this chapter we list items in alpha
betical order within a transaction. Although this is not required, algorithms often assume
that this sorting is done in a preprocessing step.
The support of an item (or set of items) is the percentage of transactions in which
that item (or items) occurs. Table 6.2 shows the support for all subsets of items from our
total set. As seen, there is an exponential growth in the sets of items. In this case we
could have 31 sets of items from the original set of five items (ignoring the empty set).
This explosive growth in potential sets of items is an issue that most association
Associ ation Ru l es
6.2
Introduction
TABLE 6 . 1 :
6. 1 0 BIBLIOGRAPHic NOTES
Items
Transaction
6.1
INTRODUCTION
The purchasing of one product when another product is purchased represents an asso
ciation rule. Association rules are frequently used by retail stores to assist in marketing,
advertising, floor placement, and inventory control. Although they have direct applicability
to retail businesses, they have been used for other purposes as well, including predicting
faults in telecommunication networks. Association rilles are used to show the relationships
between data items. These uncovered relationships are not iriherent in the data, as with
functional dependencies, and they do not represent any sort of causality or correlation.
Instead, association rules detect common usage of items. Example 6. 1 illustrates this.
TABLE 6.2:
Set
Beer
Bread
Jelly
Milk
PeanutButter
Beer, Bread
Beer, Jelly
Beer, Milk
Beer, PeanutButter
Bread, Jelly
Bread, Milk
Bread, PeanutButter
Jelly, Milk
Jelly, PeanutButter
Milk, PeanutButter
Beer, Bread, Jelly
EXAMPLE 6 . 1
A grocery store chain keeps a record o f weekly transactions where each transaction
represents the items bought during one cash register transaction. The executives of the
chain receive a summarized report of the transactions indicating what types of items have
sold at what quantity. In addition, they periodically request information about what items
are commonly purchased together. They find that 100% of the time that PeanutButter
is purchased, so is Bread. In addition, 33.3% of the time PeimutButter is purchased,
Jelly is also purchased. However, PeanutButter exists in only about 50% of the overall
transactions.
A database in which an association rule is to be found is viewed as a set
of tuples, where each tuple contains a set of items. For example, a tuple could be
{PeanutButter, Bread, Jelly}, which consists of the three items: peanut butter, bread, and
164
.....
Support
40
80
20
40
60
20
0
20
0
20
20
60
0
20
20
0
Set
Beer, Bread, Milk
Beer, Bread, PeanutButter
Beer, Jelly, Milk
Beer, Jelly, PeanutButter
Beer, Milk, PeanutButter
Bread, Jelly, Milk
Bread, Jelly, PeanutButter
Bread, Milk, PeanutButter
Jelly, Milk, PeanutButter
Beer, Bread, Jelly, Milk
Beer, Bread, Jelly, PeanutButter
Beer, Bread, Milk, PeanutButter
Beer, Jelly, Milk, PeanutButter
Bread, Jelly, Milk, PeanutButter
Beer, Bread, Jelly, Milk, PeanutButter
Support
0
0
0
0
0
0
20
20
0
0
0
0
0
0
0
1 66
Chapter 6
Association Rules
Section 6.2
rule algorithms must contend with, as the conventional approach to generating association
rules is in actuality counting the occurrence of sets of items in the transaction database.
Note that we are dealing with categorical data. Given a target domain, the under
lying set of items usually is known, so that an encoding of the transactions could be
performed before processing. As we will see, however, association rules can be applied
to data domains other than categorical.
=
. . .
EXAMPLE 6.2
A telephone company must ensure that a high percentage of all phone calls are made
within a certain period of time. Since each phone call must be routed through many
switches, it is imperative that each switch work correctly. The failure of any switch
could result in a call not being completed or being completed in an unacceptably long
period of time. In this environment, a potential data mining problem would be to predict
a failure of a node. Then, when the node is predicted to fail, measures can be taken by
the phone company to route all calls around the node and replace the switch. To this
end, the company keeps a history of calls through a switch. Each call history indicates
the success or failure of the switch, associated timing, and error indication. The history
contains results of the last and prior traffic through the switch. A transaction of the type
(success, failure) indicates that the most recent call could not be handled successfully,
while the call before that was handled fine. Another transaction (ERR l , failure) indicates
that the previous call was handled but an error occurred, ERR l . This error could be
something like excessive time. The data mining problem can then be stated as finding
association rules of the type X ::::} Failure. If these types of rules occur with a high
confidence, we could predict failure and immediately take the node off-line. Even though
the support might be low because the X condition does not frequently occur, most often
when it occurs, the node fails with the next traffic.
DEFINITION 6.2. The support (s) for an association rule X ::::} Y is the percentage
DEFINITION 6.3. The confidence o r strength (a) for an association rule X ::::} Y
is the ratio of the number of transactions that contain X U Y to the number of
transactions that contain X.
TA B LE 6.3:
transactions D
s
60%
60%
20%
20%
20%
0%
a
75%
1 00%
50%
33.3%
1 00%
0%
The efficiency of association rule algorithms usually is discussed with respect to the
number of scans of the database that are required and the maximum number of itemsets
that must be counted.
X =} Y
1 67
rule exists only in 20% of the transactions, but when the antecedent Jelly occurs, the
consequent always occurs. Here an advertising strategy targeted to people who purchase
Jelly would be appropriate.
The discussion so far has centered around the use of association rules in the market
basket area. Example 6.2 illustrates a use for association rules in another domain: telecom
munications. This example, although quite simplified from the similar real-world problem,
illustrates the importance of association rules in other domains and the fact that support
need not always be high.
U1 , h . . . , Im } and a database of
items I
Ui ! , liz
, Iik } and liJ E /, an
{tJ , tz , . . . , tn } where ti
association rule is an implication of the form X =} Y where X, Y c I are sets of
items called itemsets and X n Y = 0.
DEFINITION 6. 1 . Given a set of
transactions D
Large ltemsets
6.2
The most common approach to finding association rules is to break up the problem into
two parts:
1. Find large itemsets as defined in Definition 6.5.
1 68
Chapter 6
Section 6.3
Association Ru les
Input :
L
s
Once the large itemsets have been found, we know that any interesting association
rule, X ::::} Y, must have X U Y in this set of frequent itemsets. Note that the subset of any
large itemset is also large. Because of the large number of notations used in association
rule algorithms, we summarize them in Table 6.4. When a specific term has a subscript,
this indicates the size of the set being considered. For example, lk is a large itemset of
size k. Some algorithms divide the set of transactions into partitions. In this case, we use
p to indicate the number of partitions and a superscript to indicate which partition. For
example, D i is the i 1h partition of D .
Finding large itemsets generally i s quite easy but very costly. The naive approach
would be to count all itemsets that appear in any transaction. Given a set of items of size
m, there are 2m subsets. Since we are not interested in the empty set, the potential number
of large itemseti is then 2m - 1 . Because of the explosive growth of this number, the
challenge of solving the association rule problem is often viewed as how to efficiently
determine all large itemsets. (When m = 5 , there are potentially 3 1 itemsets. When
m
30, this becomes 1 073741 823.) Most association rule algorithms are based on
smart ways to reduce the number of itemsets to be counted. These potentially large
itemsets are called candidates, and the set of all counted (potentially large) itemsets is
the candidate itemset (c). One performance measure used for association rule algorithms
is the size of C . Another problem to be solved by association rule algorithms is what
data structure is to be used during the counting process. As we will see, several have
been proposed. A trie or hash tree are common.
When all large itemsets are found, generating the association rules is straightfor
ward. Algorithm 6. 1 , which is modified from [AS94], outlines this technique. In this
algorithm we use a function support, which returns the support for the input itemset.
Output :
R
/ /As sociat i on Rul e s sat i s fying s and
ARGen algori thm :
R = 0;
for each 1 E L do
for each x c 1 such that x i= 0 do
1.
X, Y
X ::::} Y
L
l
c
support(1) ::>
support(x) - a t hen
To illustrate this algorithm, again refer to the data in Table 6 . 1 with associated
supports shown in Table 6.2. Suppose that the input support and confidence are s
30%
and a = 50%, respectively. Using this value of s, we obtain the following set of large
itemsets:
=
We now look a t what association rules are generated from the last large itemset. Here l
{Bread, PeanutButter} . There are two nonempty subsets of l: {Bread} and {PeanutButter}.
With the first one we see:
support({Bread, PeanutButter})
support({Bread})
support({Bread, PeanutButter})
Description
Database of transactions
Transaction in D
Support
Confidence
Itemsets
Association rule
Set of large itemsets
Large itemset in L
Set of candidate itemsets
Number of partitions
60
=
80
0.75
This means that the confidence of the association rule Bread ::::} PeanutButter is 75%,
just as is seen in Table 6.3. Since this is above a, it is a valid association rule and is
added to R. Likewise with the second large itemset
R = R U {x ==> (1 - x)} ;
ti
s
1 69
ALGORITHM 6.1
Term
Basic Algorithms
support({PeanutButter})
60
60
This means that the confidence of the association rule PeanutButter ::::} Bread is 1 00%,
and this is a valid association rule.
All of the algorithms discussed in subsequent sections look primarily at ways to
efficiently discover large itemsets.
6.3
6.3.1
BASIC ALGORITH MS
Apriori Algorithm
The Apriori algorithm is the most well known association rule algorithm and is used
in most commercial products. It uses the following property, which we call the large
itemset property:
Any subset of a large itemset must be large.
1 70
Section 63
Chapter 6
The large itemsets are also said to be downward closed because if an itemset satisfies
the minimum support requirements, so do all of its subsets. Looking at the contrapos
itive of this, if we know that an itemset is small, we need not generate any super
sets of it as candidates because they also must be small. We use the lattice shown in
Figure 6 . 1 (a) to illustrate the concept of this important property. In this case there are
four items { A , B , C, D}. The lines in the lattice represent the subset relationship, so the
large itemset property says that any set in a path above an itemset must be large if the
original itemset is large. In Figure 6 . 1 (b) the nonempty subsets of ACD 1 are seen as
{AC, A D , CD, A , C, D }. If ACD is large, so is each of these subsets. If any one of these
subsets is small, then so is ACD.
the basic idea of the Apriori algorithm is to generate candidate itemsets of a
particular size and then scan the database to count these to see if they are large. During
scan i , candidates of size i, C; are counted. Only those candidates that are large are used
to generate candidates for the next pass. That is L; are used to generate C;+l An itemset
is considered as a candidate only if all its subsets also are large. To generate candidates
of size i + 1 , joins are made of large itemsets found in the previous pass. Table 6.5 shows
the process using the data found in Table 6 . 1 with s
30% and a = 50%. There are no
candidates of size three because there is only one large itemset of size two.
=
An algorithm called Apriori-Gen is used to generate the candidate itemsets for each
pass after the first. All singleton itemsets are used as candidates in the first pass. Here
the set of large itemsets of the previous pass, L; - I , is joined with itself to determine
the candidates. Individual itemsets must have all but one item in common in order to be
combined. Example 6.3 further illustrates the concept. After the first scan, every large
itemset is combined with every other large items et.
CD
AB
Basic Algorithms
171
Pass
Candidates
Large Itemsets
{Beer} , {Bread},
{Milk}, {PeanutButter}
{Bread, PeanutButter}
!1
!2
t3
!4
ts
!6
!7
ts
tg
!JO
Items
Items
Transaction
Blouse
Shoes, Skirt, TShirt
Jeans, TShirt
Jeans, Shoes, TShirt
Jeans, Shorts
Shoes, TShirt
Jeans, Skirt
Jeans, Shoes, Shorts, TShirt
Jeans
Jeans, Shoes, TShirt
tu
!J2
!1 3
! ]4
tls
!J6
!J7
fJ S
!J 9
t2o
TShirt
Blouse, Jeans, Shoes, Skirt, TShirt
Jeans, Shoes, Shorts, TShirt
Shoes, Skirt, TShirt
Jeans, TShirt
Skirt, TShirt
Blouse, Jeans, Skirt
Jeans, Shoes, Shorts, TShirt
Jeans
Jeans, Shoes, Shorts, TShirt
EXAMPLE 6.3
F I G U R E 6. 1 : Downward closure.
A woman's clothing store has 10 cash register transactions during one day, as shown in
Table 6.6. When Apriori is applied to the data, during scan one, we have six candidate
itemsets, as seen in Table 6.7. Of these, 5 candidates are large. When Apriori-Gen is
applied to these 5 candidates, we combine every one with all the other 5. Thus, we
get a total of + 3 + 2 + 1
1 0 candidates during scan two. Of these, 7 candidates
are large. When we apply Apriori-Gen at this level, we join any set with another set
that has one item in common. Thus, {Jeans, Shoes} is joined with {Jeans, Shorts} but
not with {Shorts, TShirt} . {Jeans, Shoes} will be joined with any other iteinset con
taining either Jeans or Shoes. When it is joined, the new item is added to it. There
are four large itemsets after scan four. When we go to join these we must match on
two of the three attributes. For example {Jeans, Shoes, Shorts} After scan four, there
is only one large itemset. So we obtain no new itemsets of size five to count in the
next pass. joins with {Jeans, Shoes, TShirt} to yield new candidate {Jeans, Shoes, Shorts,
TShirt}.
I Following the usual convention with association rule discussions, we simply list the items in the set rather
than using the traditional set notation. So here we use ACD to mean {A, C, D).
AB
ABC
AC
AD
BC
BD
ACD
ABD
BCD
ABCD
(a) Lattice of itemsets for (A, B, C, D)
ABC
AD
AC
ABD
BC
ACD
BD
CD
BCD
ABCD
(b) Subsets of ACD
Chapter 6
172
{Skirt}, {Tshirt}
{Skirt, TShirt}
{Skirt, TShirt}
Lk = 0 ;
for each
ci
Ci
/ / Candidates of
Ck
i t ems .
do
= 0;
The Apriori algorithm assumes that the database is memory-resident. The maximum
number of database scans is one more than the cardinality of the largest large itemset.
This potentially large number of database scans is a weakness of the Apriori approach.
6.3.2
Sampling Algorithm
Ii E
to be the
ALGORITHM 6.2
/ / Large i t emsets of
set
L = L U Lk ;
Li _ 1
are
1 73
pruning step, not shown, could be added at the end of this algorithm to prune away any
candidates that have subsets of size i
1 that are not large.
Outpu t :
/ / I n i t i a l candidates
B a s i c Alg orit hm s
Inpu t :
= I;
repeat
k = k+ l;
Large ltemsets
C1
Apriori-Gen Example
Candidates
Scan
Section 6 . 3
size i - 1
size i
ci = 0 ;
To facilitate efficient counting of itemsets with large databases, sampling of the database
may be used. The original sampling algorithm reduces the number of database scans
to one in the best case and two in the worst case. The database sample is drawn such
that it can be memory-resident. Then any algorithm, such as Apriori, is used to find the
large itemsets for the sample. These are viewed as potentially large (PL) itemsets and
used as candidates to be counted using the entire database. Additional candidates are
determined by applying the negative border function, BD - , against the large itemsets
from the sample. The entire set of candidates is then C
BD- (PL) U PL. The negative
border function is a generalization of the Apriori-Gen algorithm. It is defined as the
minimal set of itemsets that are not in PL, but whose subsets are all in PL. Example 6.4
illustrates the idea.
=
then
Given the large itemset property and Apriori-Gen, the Apriori algorithm itself (see
Algorithm 6.3) is rather straightforward. In this algorithm we use Ci to be the count for
item /i E / .
EXAMPLE 6.4
Suppose the set of items is {A, B, C , D}. The set of large itemsets found to exist in a
sample of the database is PL
{ A , C, D, C D } . The first scan of the entire database,
then, generates the set of candidates as follows: C BD - (PL) U PL
{ B , A C , AD } U
{A, C, D, CD } . Here we add AC because both A and C are in PL. Likewise we add AD .
We could not have added ACD because neither AC nor AD is in PL. When looking at
the lattice (Figure 6.2), we add only sets where all subsets are already in PL. Note that
we add B because all its subsets are vacuously in PL.
=
ALGORITHM 6.3
Input :
I
D
s
/ / Support
/ / Large i t emsets
/ / I tems ets
/ / Databa s e of transact ions
Outpu t :
Apriori algori t hm :
k = 0 ; I I k i s u s e d as the s can number .
L = 0;
Algorithm 6.4 shows the sampling algorithm. Here the Apriori algorithm is shown
to find the large itemsets in the sample, but any large itemset algorithm could be used.
Any algorithm to obtain a sample of the database could be used as well. The set of large
1 74
C h a pter 6
Association Rules
Section 6.3
<P
AB AC AD BC BD
ABC
ABD
ACD
BCD
ABCD
A
CD
(a) PL
<P
AB
BC BD
ABC
ABD
ACD
BCD
ABCD
A
AC
AD
CD
(b) PL U BD-(PL)
Ds
Sample drawn from D;
PL = Apriori(I, Ds, smal l s) ;
C = PL U BD- (PL) ;
L = 0;
for each Ii E C do
Ci = 0 ;
I I Initial count s for each itemset are 0 ;
for each t j E D do
I I First scan count .
for each Ii E C do
if Ii E t j
then
Ci = Ci + 1 ;
for each Ii E C do
if ci ;::: (sx I D I) do
L = L U ii ;
//Mi s sing large itemsets
ML = {x I x E BD- (PL) 1\ x E L} ;
if ML =f. 0 1 then
C = L;
I I Set candidates to be the large itemset s .
=
repeat
C= C U BD- (C) ;
I
D
Output :
/ / Itemsets
/ /Database of transact ions
// Support
//Large itemsets
unt i l
for
0.
each
Ii E C do
E tj
then
Ci = Ci + 1 ;
i f Ci ;::: (sx I D I) do
L = L U ii ;
if
Ii
The algorithm shows that the application of the Apriori algorithm to the sample is
performed using a support called smalls. Here smalls can be any support values less than
s. The idea is that by reducing the support when finding large itemsets in the sample, more
of the true large itemsets from the complete database will be discovered. We illustrate
the use of the sampling algorithm on the grocery store data in Example 6.5.
EXAMPLE 6.5
We use the sampling algorithm to find all large itemsets in the grocery data where
s = 20%. Suppose that the sample database is determined to be the first two transactions:
Ds = {t1
ALGORITHM 6.4
Input :
175
itemsets is used as a set of candidates during a scan of the entire database. If an itemset
is large in the sample, it is viewed to be potentially large in the entire database. Thus,
the set of large itemsets from the sample is called PL. In an attempt to obtain all the
large itemsets during the first scan, however, PL is expanded by its negative border. So
the total set of candidates is viewed to be C = PL U BD-(PL).
During the first scan of the database, all candidates in C are counted. If all candi
dates that are large are in PL [none in BD- (PL)], then all large itemsets are found. If,
however, some large itemsets are in the negative border, a second scan is needed. Think
of B v- (PL) as a buffer area on the border of large itemsets. The set of all itemsets is
divided into four areas: those that are known to be large, those that are known to be
small, those that are on the negative border of those known to be large, and the others.
The negative border is a buffer zone between those known to be large and the others. It
represents the smallest possible set of itemsets that could potentially be large. Because
of the large itemset property, we know that if there are no large itemsets in this area,
then there can be none in the rest of the set.
During the second scan, additional candidates are generated and counted. This is
done to ensure that all large itemsets are found. Here ML, the missing large itemsets,
are those in L that are not in PL. Since there are some large itemsets in ML, there may
be some in the rest of the set of itemsets. To find all the remaining large itemsets in the
second scan, the sampling algorithm repeatedly applies the negative border function until
the set of possible candidates does not grow further. While this creates a potentially large
set of candidates (with many not large), it does guarantee that only one more database
scan is required.
Basic Algorithms
If we reduce s to be smalls = 10%, then for an itemset to be large in the sample it must
occur in at least 0. 1 x 2 transactions. So it must occur in one of the two transactions.
When we apply Apriori to Ds we get:
PL
1 76
Chapter 6
Section 6.3
Association Ru les
We thus use the following set of candidates to count during the first database scan:
Remember that during this scan we use s = 20% and apply it against all five transactions
in the entire database. For an itemset to be large, then, we must have
177
ssing,
with Example 6.5. Since Jelly is
.
Note that this is smaller than what we found
.
ed apphcatwn of B D . The
repeat
during
ts
itemse
of
set
entire
the
te
we will not genera
second application yields
PL =
Basic Algorithms
an
itemset in 20% x 5
or at least one transaction. We then find that both {Beer} and {Milk} are large. Thus,
ML = { {Beer} , {Milk} }. Following the algorithm, we first set C = L, which in this case
is also PL. Applying the negative border, we get
C
Since this has uncovered new itemsets, we again apply it and this time find all itemsets
of size three. A last application then finds all itemsets of size four, and we scan the
database using all remaining itemsets not already known to be large.
Example 6.5 illustrates a potential problem with the use of the sampling algorithm;
that is, a very large set of candidates may be used during the second scan. This is required
6.3.3
Partitioning
in several ways:
to ensure that all large itemsets are found during the second scan. However, the set of
candidates generated by successive applications of the negative border function will not
always generate the entire set of itemsets. This happened in Example 6.5 because we
found that all itemsets in
PL were large and all itemsets in BD- (PL) also were large.
If instead of using a support of 20%, we had used one of 40%, the results would be
EXAMPLE 6.6
itemsets only if they have a support of at least 40%. Looking at Table 6.2, we see that
from the initial scan we obtain the following large itemsets:
border we get
C = BD- (C)
a
itemsets have ee i'opo.sed based .o
Various approaches to generating large
tttlons
P
p
mto
d
this case, D 1s dtvtde
:U
partitioning of the set of transactions. In
.
performan,e of findmg large 1temsets
the
ve
impro
may
oning
Partiti
DP.
,
.
.
.
D L , D2,
t property, we know that a large itm
By taking advantage of the large itemse
ons. This idea can help to destgn
set must be large in at least one of the partiti
on looking at the entire database.
algorithms more efficiently than those based
178
Chapter 6
nt
D2
It
12
Bread, PeanutBptter
13
/4
Beer, Bread
Is
Section 6.4
Association R u l es
L1
L2
Beer, Milk
ALGORITHM 6.5
Input :
/ / I tems e t s
v2, . . . , .oP}
/ / Support
s
Outpu t :
/ / L arge i t ems e t s
Partition algorithm:
C= 0 ;
for i
1
1 to p do
= Ap riori (I, D , s) ;
i
C= CUL ;
for each Ii E C do
0;
/ / In i t i a l count s
for
Count candidates
ci = ci + 1 ;
for each Ii E C do
if ci 2: (sx I D I) do
L = L U ii i
Figure 6.3 illustrates the use of the partition algorithm using the market basket data.
Here the database is partitioned into two parts, the first containing two transactions and
the second with three transactions. Using a support of 1 0%, the resulting large itemsets
L 1 and L 2 are shown. If the items are uniformly distributed across the partitions, then a
large fraction of the itemsets will be large. However, if the data are not uniform, there
may be a large percentage of false candidates.
6.4
6.4.1
One data parallelism algorithm is the count distribution algorithm (CDA). The database
_
is divided into p partitions, one for each processor. Each processor counts the candidates
for its data and then broadcasts its counts to all other processors. Each processor then
determines the global counts. These then are used to determine the large itemsets and to
generate the candidates for the next scan. The algprithm is shown in Algorithm 6.6.
ALGORITHM 6.6
L = 0;
ci =
1 79
large enough to store all candidates at each scan (otherwise the performance will degrade
considerably because 1/0 is required for both the database and the candidate set). The task
parallelism approaches can avoid this because only the subset of the candidates that are
assigned to a processor during each scan must fit into memory. Since not all partitions of
the candidates must be the same size, the task parallel algorithms can adapt to the amount
of memory at each' site. The only restriction is that the total size of all candidates be small
enough to fit into the total size of memory in all processors combined. Note that there are
some variations of the basic algorithms discussed in this section that address these memory
issues. Performance studies have shown that the data parallelism tasks scale linearly with
the number of processors and !lie database size. Because of the reduced memory require
ments, however, the tast p<jfallelism may work where data parallelism may not work.
D = {D1 ,
Most parallel o r distributed association rule algorithms strive to parallelize either the data,
known as dataparallelism, or the candidates, referred to !iS task parallelism. With task par
allelism, the candidates are partitioned and counted sep!ifately at each processor. Obviously,
the partition algorithm would be easy to parallelize using the task parallelism approach.
Other dimensions in differentiating different parallel association rule algorithms are the
load-balancing approach used and the architecture. The data parallelism algorithms have
reduced communication cost over the task, because only the initial candidates (the set of
items) and the local counts at each iteration must be distributed. With task parallelism,
not only the candidates but also the local set of transactions must be broadcast to all other
sites. However, the data parallelism algorithms require that memory at each processor be
Inpu t :
I
/ / I tems e t s
pl , p2 , . . . PP ;
D = D1 , D2 , . . . , .oP;
s
I / Support
/ / Proc e s s o r s
Output :
L
/ / Large i t ems e t s
0;
P1
/ / Count in paral l e l .
L = 0;
c1 = I ;
repeat
k = k+ l ;
Lk = 0 ;
for each Ii E Ck do
ci = 0 ;
for each tj E D1 do
for each Ii E CK do
i:fi Ii E
then
broadca s t
c i = ci + 1 ;
for each Ii E Ck do
Ci
=
'
Ll=l
ci i
for each Ii E Ck dp
if ci 2: (sx I D1 U v2 U
Lk = Lk U Ii ;
L = L U Lk i
U nP I) do
0 .
Association Rules
Chapter 6
1 80
ft,
p3
p2
pl
nt:
Section 6.5
Dz :
tz
D3:
t3, t4
Counts:
Beet O
Beer l
Bread 2
Jelly O
Milk l
Bread 2
Jelly 1
Milk O
PeanutButter 2
ts
1 ::; 1 ::; p do
repeat
k = k+ 1 ;
Lfc = 0 ;
for each Ii
ci = 0 ;
for each
for
E cfc
do
each Ii
if
Ii
E
E tj
for each
local counts are obtained, they are then broadcast to the other sites so that global counts
if
tj E rf'l do
Ii
E
E tj
dates as well as the database are partitioned among the processors. Each processor in
Li
l.
/ /Determine
global count s .
parallel cmmts the candidates given to it using its local database partition. Following our
convention, we use
do
then
ci ::: (sx I D1 U D2 U . . U vP I) do
Lfc = Lfc U Ii i
broadcast Lfc to all other processors ;
Lk = Lk U L U . . U L ;
/ /Global large
k- itemsets .
Ck+l = Apriori- gen(Lk)
/ /Determine next set of
c+l c ck+l ;
local candidates .
unt i l cfc l = 0 ;
+
Task Parallelism
Also,
ci
ci = ci + 1 ;
if
do
then
for each Ii
can be generated.
The
0.
ci = ci + 1 ;
/ /Determine local counts .
broadcast D1 t o all other processors ;
for every other processor m =f: 1 do
Figure 6.4, which is modified from [DXGHOO] , illustrates the approach used by the
6.4.2
tj E D1 do
CDA algorithm using the grocery store data. Here there are three processors. The first
1
two transactions are counted at P , the next two at
181
Counts:
Beer l
Bread 0
Jelly 0
Milk l
Counts:
its database prutition to all other processors. Each processor then uses this to obtain a
global count for its data and broadcasts this count to all other processors. Each processor
then can determine globally large itemsets and generate the next candidates. These candi
dates then are divided among the processors for the next scan. Algorithm 6. 7 shows this
Figure 6.5, which is modified from [DXGHOO], illustrates the approach used by
approach. Here we show that the candidates are actually sent to each processor. How
the DDA algorithm using the grocery store data. Here there are three processors. P 1 is
ever, some prearranged technique could be used locally by each processor to determine
one at
Input :
/ / Itemsets
, . . . , PP ;
p ,p
D = D1 , D2 , . . . , IJP ;
/ /Support
6.5
//Processors
/ /Database divided into partitions
Output :
P 3 When the local counts are obtained, the database paititions are then broadcast
to the other sites so that each site can obtain a global count.
ALGORITHM 6.7
I
its own candidates . This algorithm suffers from high message traffic whose impact can
//Large itemsets
COMPARING APPROACHES
Although we have covered only the major association rule algorithms that have been
proposed, there have been many such algorithms (see Bibliography). Algorithms can be
classified along the following dimensions [DXGHOO] :
Target: The algorithms we have examined generate all rules that satisfy a given
support and confidence level. Alternatives to these types of algorithms are those
that generate some subset of the algorithms based on the constraints given.
1 82
Association Rules
Chapter 6
pl
Counts:
Beer O
Bread 2
Section 6.5
p2
Counts:
Jelly 0
Milk 1
Comparing Approaches
1 83
p3
Counts:
PeanutButter 0
1, 4
2, 5
3, 6
3,4, 6
3, 5, 6
2, 5, 6
2, 4, 6
Type: Algorithms may generate regular association rules or more advanced asso
J
6.7 and Chapters 8 and 9.
6.7. For simplicity we have replaced each item with its numeric value in
4 hash to the first entry;
2 , 5 hash to the second entry; and 3 , 6 hash to the third entry.
in Table
Data type: We have examined rules generated for data in categorical databases.
Rules may also be derived for other types of data such as plain text. This concept
is further investigated in Section
mining.
(as seen in the hash tree example), and the use of bit maps has also been proposed.
for market basket data. This assumes that data are present in a transaction. The
absence of data may also be important.
Itemset strategy: Itemsets may be counted in different ways. The most naive
approach is to generate all itemsets and count them. As this is usually too space
Parallelism strategy: B oth data parallelism and task parallelism have been used.
Table
intensive, the bottom-up approach used by Apriori, which takes advantage of the
large itemset property, is the most common approach. A top-down technique could
also be used.
Transaction strategy: To count the itemsets, the transactions in the database must
Data source: Our investigation has been limited to the use of association rules
ciation rule algorithms we have covered in this chapter. When m is the number of items,
the maximum number of scans is m + 1 for the level-wise algorithms. This applies to
the parallel algorithms because they are based on Apriori. Both sampling algorithms and
Itemset data structure: The most common data structure used to store the can
didate itemsets and their counts is a hash tree. Hash trees provide an effective
technique to store, access, and count itemsets. They are efficient to search, insert,
and delete itemsets . A
Partitioning
Scans
Apriori
m+ 1
Sampling
opposed to comparing key values to branching points in the node. A leaf node in
Partitioning
the hash tree contains the candidates that hash to it, stored in sorted order. Each
CDA
6.6
DDA
internal node actually contains a hash table with links to children nodes. Figure
shows one possible hash tree for candidate itemsets of size 3 , which were shown
2
2
m + l
m+ 1
Data Structure
Parallelism
hash tree
none
not specified
none
hash table
none
hash tree
data
hash tree
task
184
Chapter 6
Section 6.7
185
Food
partitioning algorithms require at most two complete scans of the transaction database.
However, remember that the sampling algorithm must access the database to read the
sample into memory and then many scans of it into memory may be required. Similarly,
for the partitioning algorithm, each partition must be read into memory and may be
Fruit
Vegetables Grain
Bread
6.6
Rice
INCREMENTAL R U LES
Wheat White
All algorithms discussed so far assume a static database. However, i n reality w e cannot
assume this. With these prior algorithms, generating association rules for a new database
Rye
Meat
Dairy
Whole
2%
Skim
state requires a complete rerun of the algorithm . Several approaches have been proposed
to address the issue of how to maintain the association rules as the underlying database
changes. Most of the proposed approaches have addressed the issue of how to modify the
association rules as inserts are performed on the database. These
incremental updating
db where D is a
EXAMPLE 6.7
db
1
are updates to it and where the large itemsets for D , L are known.
- 1,
db
Figure
6.7
shows a partial concept hierarchy for food. This hierarchy shows that Wheat
Bread is a type of Bread, which is a type of grain. An association rule of the form
.
Bread =? PeanutButter has a lower support and threshold than one of the form Gram =?
PeanutButter. There obviously are more transactions containing any type of grain than
the candidate set for scan k to be Lk found in D . The difference is that the num
ber of candidates examined at each iteration is reduced through pruning of the candi
dates. Although other pruning techniques are used, primary pruning is based on the fact
=?
=.>-
PeanutButter.
that we already know L from D. Remember that according to the large itemset prop
erty, an itemset must be large in at least one of these partitions of the new database.
For each scan k of
db, Lk
db,
6.7.2
will be large in the entire database without scanning D. We need not even count any
items in Lk during the scan of
db
database.
6.7
level i + 1 . Large k-itemsets at one level in the concept hierarchy are used as candtdates
to generate large k-itemsets for children at the next level.
Modification to the basic association rule ideas may be changed. We expect that
In this section w e investigate several techniques that have been proposed t o generate
there is more support for itemsets occurring at higher levels in the concept hierarch! .
association rules that are more complex than the basic rules.
6.7.1
Thus, the minimum support required for association rules may vary based on level
Using a concept hierarchy that shows the set relationship between different items, gen
6. 7 illustrates the
6.7. Association
generalized associ
The minimum support for all nodes in the hierarchy at the same level is iqentical.
ation rule, X =? Y, is defined like a regular association rule with the restriction that
no item in Y may be above any item in X. When generating generalized association
rules, all pos sible rules are generated using one or more given hierarchies. Several
algorithms have been proposed to generate generalized rules. The simplest would be
to expand each transaction by adding (for each item in it) all items above it in any
hierarchy.
the hierarchy. We would expect that the frequency of itemsets at higher levels i s much
. .
greater than the frequency of itemsets at lower levels. Thus, for the reduced nurumum
6.7.3
i - 1,
then
a; -1
is the minimum
a; - 1 > ct; .
Chapter 6
1 86
Association Rules
Section 6.7
with a support of 6%. Thus, having only one support value for all association rules may
not work well. Some useful rules could be missed. This is particularly of interest when
looking at generalized association rules, but it may arise in other situations as well. Think
of generating association rules from a non-market basket database. As was seen with
quantitative rules, we may partition attribute values into ranges. Partitions that have a
small number of values obviously will produce lower supports than those with a large
number of values. If a larger support is used, we might miss out on generating meaningful
association rules.
This problem is called the rare item problem. If the minimum support is too high,
then rules involving items that rarely occur will not be generated. If it is set too iow,
then too many rules may be generated, many of which (particularly for the frequently
occurring items) are not important. Different approaches have been proposed to handle
this. One approach is to partition the data based on support and generate association
rules for each partition separately. Alternatively, we could group rare items together and
generate association rules for these groupings. A more recent approach to handling this
problem is to combine clustering and association rules. First we cluster the qata together
based on some clustering criteria, and then we generate rules for each cluster separately.
This is a generalization of the partitioning of the data solution.
One approach, M/Sapriori, allows a different support threshold to be indicated for
each item. Here MIS stands for minimum item support. The minimum support for a rule
is the minimum of all the minimum supports for each item in the rule. An interest
ing problem occurs when multiple minimum supports are used. The minimum support
requirement for an itemset may be met even though it is not met for some of its subsets.
This seems to violate the large itemset property. Example 6.8, which is adapted from
[LHM99], illustrates this. A variation of the downward closure property, called the sorted
downward closure property, is satisfied and used for the MISapriori algorithm. First the
items are sorted in ascending MIS value. Then the candidate generation at scan 2 looks
only at adding to a large item any item following it (larger than or equal to MIS value)
in the sorted order.
The cost quantity has been divided into an interval (much as was done when we looked
at handling numeric data in clustering and classification). In these cases, the items are
not simple literals. For example, instead of having the items {Bread, Jelly} , we might
have the items {(Bread:[O . . . 1]), (Bread: ( l . . . 2]) , (Bread:(2 . . . oo)) , (Jelly: [O . . . 1 .5]),
(Jelly: ( l .5 . . . 3]), (Jelly: (3 . . . oo)) } .
The basic approach to finding quantitative association rules is found in Algo
rithm 6.8 . Here we show the J\priori algorithm being used to generate the large itemsets,
but any such algorithm could
pe used.
.
-
ALGORITHM 6.8
Input :
D=
s
/ / I t emsets
...
, PP ;
,
2
1
D , D , . . ,
/ / Processors
IJP;
/ / S upport
Output :
L
Ij E I do
I / Part i t i on it ems .
if Ij i s to be part it ione d , then
determine number of par t i t i ons ;
map at tribute value s into new part i t i ons creat ing new items ;
rep lace Ij in I wi th the new i t ems Ij 1 , . . . , Ijm ;
Apriori ( I , D , s ) ;
for each
Because we have divided what was one item into several items, the minimum
support and confidence used for quantitative rules may need to be lowered. The min
imum support prolem obviously is worse with a large number of intervals. Thus,
an alternative solution would be to combine adjacent intervals when calculating sup
port. Similarly, when there are a small number of intervals, the confidence thresh
old may need to be lowered. For example, look at X => Y. Suppose there are only
two intervals for X . Then the count for those transactions containing X will be quite
high when compared to those containing Y (if this is a more typical item with many
intervals).
6.7.4
187
attribute that has hundreds of values. It might be more meaningful to find a rule of
the form
SkimMilk ==> WheatBread
A customer buys wine for between $30 and $50 a bottle => she also buys caviar
EXAMPLE 6.8
Suppose we have three items, {A, B, C}, with nummum supports MIS(A)
20%,
MIS(B)
3%, and MIS(C)
4%. Because the support for A is so large, it may be
small, while both AB and AC may be large because the required support for AB
min(M/S (A), MIS (B))
3% and AC
min(MIS (A) , MIS (C))
4%.
=
When looking at large databases with many types of data, using one minimum support
value can be a problem. Different items behave differently. It certainly is easier to obtain
a given support tllreshold with an attribute that has only two values than it is with an
6.7.5
Correlation Rules
A correlation rule is defined as a set of itemsets that are correlated. The motiva
tion for developing these correlation rules is that negative correlations may be useful.
1 88
Chapter 6
Association R u l es
Example 6.9, which is modified from [BMS77], illustrates this concept. In this example,
even though the probability of purchasing two items together seems high, it is much
higher if each item is purchased without the other item. Correlation satisfies upward
closure in the itemset lattice. Thus, if a set is correlated, so is every superset of it.
EXAMPLE 6.9
Suppose there are two items, {A, B} where A ::::} B has a support of 15% and a confidence
of 60%. Because these values are high, a typical association rule algorithm probably
would deduce this to be a valuable rule. However, if the probability to purchase item
B is 70%, then we see that the probability of purchasing B has actually gone down,
presumably because A was purchased. Thus, there appears to be a negative correlation
between buying A and buying B . The correlation can be expresed as
correlation(A ===> B) =
1
P (A , B)
P (A) P (B)
(6. 1 )
255
6.8
Section 6.8
Support and confidence are the normal methods used to measure the quality of an asso
ciation rule:
s (A ===> B) = P (A , B)
(6.2)
and
a (A ===> B) = P (B I A)
interest(A ===> B) =
P (A , B)
P (A) P (B)
Pf1;(k) .
P (A) P (-B)
. .
convtctwn(A ===> B) =
-. )
P (A, B
(6.5)
Conviction has a value of 1 if A and B are not related. Rules that always hold have a
value of oo.
The usefulness of discovered association rules may be tied to the amount of surprise
associated with the rules or how they deviate from previously known rules. Here surprise
is a measure of the changes of correlations between items over time. For example, if you
are aware that beer and pretzels are often purchased together, it would be a surprise if
this relationship actually lowered significantly. Thus, this rule beer ::::} pretzel would be
of interest even if the confidence decreased.
Another technique to measure the significance of rules by using the chi squared
test for independence has been proposed. This significance test was proposed for use
with correlation rules. Unlike the support or confidence measurement, the chi squared
significance test takes into account both the presence and the absence of items in sets.
Here it is used to measure how much an itemset (potential correlation rule) count differs
from the expected. The chi squared test is well understood because it has been used in the
statistics community for quite a while. Unlike support and confidence, where arbitrary
values must be chosen to determine which rules are of interest, the chi squared values are
well understood with existing tables that show the critical values to be used to determine
relationships between items.
The chi squared statistic can be calculated in the following manner. Suppose the
set of items is I = {!1 , h . . . , Im } . Because we are interested in both the occurrence
and the nonoccurrence of an item, a transaction t1 can be viewed as
(6.6)
Given any possible itemset X, it also is viewed as a subset of the Cartesian product. The
chi squared statistic is then calculated for X as
X2
(6.7)
XE/
Here O (X) is the count of the number of transactions that contain the items in X. For
one item Ii , the expected value is E [ Ii ] = 0 Ui ), the count of the number of transactions
that contain Ii . E[Ji]
n - O (li ) . The expected value E [X] is calculated assuming
independence and is thus defined as
=
(6.4)
This measure takes into account both P (A) and P (B). A problem with this measure is
that it is symmetric. Thus, there is no difference between the value for interest(A ::::} B)
and the value for interest(B ::::} A) .
As with lift, conviction takes into account both P (A) and P (B). From logic we
know that implication A B = -.(A 1\ B ) A measure of the independence of
the negation of implication, then, is
To take into account the negation, the
-.
conviction measure inverts this ratio. The formula for conviction is [BMS77]
(6.3)
However, there are some problems associated with these metrics. For example, confidence
totally ignores P (B). A rule may have a high support and confidence but may be an
obvious rule. For example, if someone purchases potato chips, there may be a high
likelihood that he or she would also buy a cola. This rule is not really of interest because
it is not surprising. Various concepts such as surprise and interest have been used to
evaluate the quality or usefulness of rules. We briefly examine some of these in this
section.
With correlation rules, we saw that correlation may be used to measure the rela
tionship between items in a rule. This may also be expressed as the lift or interest
1 89
E[X] = n
m E [ I; ]
i=l
-n
(6.8)
1 90
Section 6. 1 0
Association Rules
Chapter 6
TABLE 6.9:
Total
Total
7. Calculate the lift and conviction for the rules shown in Table 6.3. Compare these
25
75
100
10
20
30
15
55
70
A
A
incrementally.
6.10
Xe/ .;
.
(20 - 22.5) 2
22.5
1 . 587
( 1 0 - 7 . 5) 2
( 1 5 - 17 .5) 2
17.5
7.5
(55 - 52.5) 2
+
52.5
(6.9)
If all values were independent, then the chi squared statistic should be 0. A chi squared
table (found in most statistics books) can be examined to interpret this value. Examining
a table of critical values for the chi squared statistic, we see that a chi squared value
less than 3.84 indicates that we should not reject the independent assumption. This is
done with 95% confidence. Thus, even though there appears to be a negative correlation
between A and B, it is not statistically significant.
6.9
EXERCISES
1. Trace the results of using the Apriori algorithm on the grocery store example with
20% and a = 40%. Be sure to show the candidate and large itemsets for each
database scan. Also indicate the association rules that will be generated.
s =
2. Prove that all potentially large itemsets are found by the repeated application of
3. Trace the results of using the sampling algorithm on the clothing store exam
ple with s = 20% and a = 40%. Be sure to show the use of the negative
border function as well as the candidates and large itemsets for each database
scan.
4. Trace the results of using the partition algorithm on the grocery store example with
s = 20% and a = 40%. For the grocery store example, use two partitions of size
2 and 3, respectively. You need not show all the steps involved in finding the large
itemsets for each partition. Simply show the resulting large itemsets found for each
partition.
5. Trace the results of using the count distribution algorithm on the clothing data with
20%. Assume that there are three processors with partitions created from the
beginning of the database of size 7, 7, and 6, respectively.
s =
6. Trace the results of using the data distribution algorithm on the clothing data
with
191
from the beginning of the database o f size 7 , 7 , and 6 , respectively. Assume that
candidates are distributed at each scan by dividing the total set into subsets of equal
size.
B i b l i ogra p h i c Notes
20%. Assume that there are three processors with partitions created
BIBLIOGRAPHIC NOTES
The development of association rules can be traced to one paper in 1 993 [AIS93].
Agrawal proposed the AIS algorithm before Apriori [AIS93]. However, this algorithm
and another, SETM [HS95], do not take advantage of the large itemset property and thus
generate too many candidate sets. The a priori algorithm is still the major technique used
by commercial products to detect large itemsets. It was proposed by 1994 in [AS94].
Another algorithm proposed about the same time, OCD, uses sampling [MTV94] . It
produces fewer candidates than AIS.
There have been many proposed algorithms that improve on Apriori. Apriori-TID
does not use the database to count support [AS94]. Instead, it uses a special encoding
fo the candidates from the previous pass. Apriori has better performance in early passes
of the database while Apriori-T/D has better performance in later. A combination of the
two, Apriori-Hybrid, has been proposed [Sri96] . The dynamic itemset counting (DIC)
algorithm divides the database into intervals (like the partitions in the partition algorithm)
[BMUT77]. The scan of first interval counts the 1 -itemsets. Then candidates of size 1 are
generated. The scan of the second interval, then, counts the 1-itemsets as well as those
2-itemsets. In this manner itemsets are counted earlier. However, more memory space
may be required. The partition algorithm was first studied in 1 995 by Savasere [SON95],
while the sampling algorithm is attributed to Toivonen in 1 996 [Toi96] . The problem of
uneven distribution of data in the partition algorithm was addressed in [LD98], where
a set of algorithms were proposed that better prune away false candidates before the
second scan.
The CDA and DDA algorithms were both proposed in [AS96]. Other data parallel
algorithms include PDM [PCY95], DMA [CHN+ 96], and CCPD [ZOPL96]. Additional
task parallel algorithms include IDD [HKK97], HPA [SK96], and PAR [ZPOL97] . A
hybrid approach, hybrid distribution (HD), which combines the advantages of each tech
nique has a speed-up close to the data parallelism approach [HKK97] . For a discussion
of other parallel algorithms, see either [DXGHOO] or [Zak99] .
Many additional algorithms have been proposed. CARMA (continuous association
rule mining algorithm) [Hid99] proposes a technique that is dynamic in that it allows
the user to change the support and confidence while the algorithm is running. Some
recent work has examined the use of an AI type search algorithm called OPUS [WebOO].
OPUS prunes out portions of the search tree based on the desired rule characteristics.
However, many scans of the database are required and, thus it assumes that the database
is memory-resident.
An approach to determining if an item should be partitioned when generating
quantitative rules has been proposed [SA96a] . Variations of quantitative rules include
profile association rules where the left side of the rule represents some profile information
1 92
Chapter 6
Association Ru les
about a customer while the right side of the rule contains the purchase information
[ASY98]. Another variation on quantitative rules is a ratio rule [KLKF98]. These rules
indicate the ratio between the quantitative values of individual items. When fuzzy regions
are used instead of discrete partitions, we obtain fUzzy association rules [KFW98] . Recent
research has examined association rules for multimedia data [ZHLH98].
After the initial work in [CHNW96] , much additional work has examined associ
ation rules in an incremental environment. Several improvements on the original FUP
have been proposed [CNT96] [CLK97] . Another technique aims at reducing the number
of additional scans of the original database [TBAR97] .
Generalized association rules were studied in [SA95]. Multiple-level association
rules were proposed in [HF95]. Quantitative association rules were studied in [Sri96].
Algorithm 6.8 does not show how to determine whether an item should be partitioned.
One technique proposed to do this is a metric called partial completeness [SA96a] . The
rare item problem was investigated [Man98]. The MISapriori approach was subsequently
proposed in [LHM99] . Correlation rules were first examined in [BMS77]. Many of the
additional measures for rules were investigated in [BMS77].
A survey of as lociation rules has recently appeared [DXGHOO]. A survey of parallel
and distribution association rule algorithms has also been published [Zak99] . One recent
textbook is devoted to the study of association rules and sequential patterns [AdaOO]. A
recent tutorial [HLPO l ] has examined association rules and sequential patterns.
P A R T
T H R E E
ADVAN CE D TO P I CS
C H A P T E R
We b M i n i n g
7.1
INTRODUCTION
7.2
7.3
7.4
7.6
BIBLIOGRAPHIC NOTES
7.5
7.1
EXERCISES
INTRODUCTION
Determining the size of the World Wide Web is extremely difficult. It 1 999 it was
estimated to contain over 350 million pages with growth at the rate of about 1 million
pages a day [CvdBD99] . Google recently announced that it indexes 3 billion Web docu
ments [Goo0 1 ] . The Web can be viewed. as the the largest database available and presents
a challenging task for effective design and access, Here we use the term database quite
loosely because, of course, there is no real structure or schema to the Web. Thus, data
mining applied to the Web has the potential to be quite beneficial. Web mining is min
ing of data related to the World Wide Web. This may be the data actually present in
Web pages or data related to Web activity. Web data can be classified into the following
classes [SCDTOO] :
Intrapage structure includes the HTML or XML code for the page.
Usage data that describe how Web pages are accessed by visitors.
Web mining tasks can be divided into several classes. Figure 7 . 1 shows one taxo
nomy of Web mining activities [Za199]. Web content mining examines the content of
Web pages as well as results of Web searching. The content includes text as well as
graphics data. Web content mining is further divided into Web page content mining and
search results mining. The first is traditional searching of Web pages via content, while
195
1 96
Chapter 7
Section 7.2
Web M i n i n g
1 97
Web sites (user interface, effective use of graphics, response time, etc.), in this chapter
we cover only techniques that involve data mining.
7.2
the second is a further search of pages found from a previous search. Thus, some mining
activities have beep built on top of traditional search engines, using their result as the
data to be mined. With Web structure mining, information is obtained from the actual
organization of pages on the Web . Content mining is similar to the work performed by
basic IR techniques, but it usually goes farther than simply employing keyword searching.
For example, clustering may be applied to Web pages to identify similar pages. The
intrapage structure includes links within the page as well as the code (HTML, XML) for
the page. Web usage mining looks at logs of Web access. General access pattern tracking
is a type of usage mining that looks at a history of Web pages visited. This usage may be
general or may be targeted to specific usage or users. Besides identifying what the traffic
patterns look like, usage mining also involves the mining of these sequential patterns.
For example, patterns can be clustered based on their similarity. This in turn can be used
to cluster users into groups based on similar access behavior.
There are many applications for Web mining. One application is targeted advertis
ing. Targeting is any technique that is used to direct business marketing or advertising
to the most beneficial subset of the total population. The objective is to maximize the
results of the advertising; that is, send it to all (and only) the set of potential customers
who will buy. In this manner, the cost of sending an advertisement to someone who will
not purchase that product can be avoided. Targeting attempts to send advertisements to
people who have not been to a Web site to entice them to visit it. Thus, a targeted ad
is found on a different Web site. All of the data mining techniques we have seen so far
could be used to target advertising to a subset of the audience. In this manner, advertising
costs can be reduced while either not impacting results or improving results. On the Web,
targeting can be used to display advertising at Web sites visited by persons that fit into
a business' target demographic area. By examining the Web log data to see what source
sites access a Web site, information about the visitors can be obtained. This in turn can
be used to sell advertising space to those companies that would benefit the most.
Although the different Web mining activities may be described separately, they
are intrinsically related. A Webmaster usually wants to create the best set of pages
to accomplish the desired objective for the pages (advertising, marketing, information
dissemination, etc.). The effectiveness of a set of Web pages depends not only on the
content and organization of individual Web pages, but also on the structure of the pages
and their ease of use. Although there are many issues that impact the effectiveness of
WEB CONTENT M I N I N G
Web content mining can b e thought of as extending the work performed by basic search
engines. There are many different techniques that can be used to search the Internet.
Only a few of these techniques are discussed here. Most search engines are keyword
based. Web content mining goes beyond this basic IR technology. It can improve on
traditional search engines through such techniques as concept hierarchies and synonyms,
user profiles, and analyzing the links between pages. Traditional search engines must
have crawlers to search the Web and gather information, indexing techniques to store the
information, and query processing support to provide fast and accurate information to
users. Data mining techniques can be used to help search engines provide the efficiency,
effectiveness, and scalability needed.
One taxonomy of Web mining divided Web content mining into agent-based and
database approaches [CMS97] . Agent-based approaches have software systems (agents)
that perform the content mining. In the simplest case, search engines belong to this class,
as do intelligent search agents, information filtering, and personalized Web agents. Intelli
gent search agents go beyond the simple search engines and use other techniques besides
keyword searching to accomplish a search. For example, they may use user profiles or
knowledge concerning specified domains. Information filtering utilizes IR techniques,
knowledge of the link structures, and other approaches to retrieve and categorize doc
uments. Personalized Web agents use information about user preferences to direct their
search. The database approaches view the Web data as belonging to a database. There
have been approaches that view the Web as a multilevel database, and there have been
many query languages that target the Web.
Basic content mining is a type of text mining. As seen in Figure 7.2, a modified
version of [Za199, Figure 2 . 1 ] , text mining functions can be viewed in a hierarchy with the
simplest functions at the top and the more complex functions at the bottom. Much research
is currently under way that investigates the use of natural language processing techniques
in text mining to uncover hidden semantics, such as question and answer systems. More
Keyword
Term association
Similarity search
Classification
Clustering
FIG U RE 7.2: Text mining hierarchy (modified version of [Za199, Figure 2. 1]).
1 98
Chapter 7
Web Mining
A distiller determines which pages contain links to many relevant pages. These are
called hub pages. These are thus highly important pages to be visited. These hub
1 99
pages may not contain relevant information, but they would be quite important to
facilitate continuing the search.
Crawlers
A robot (or spider or crawler) is a program that traverses the hypertext structure in
the Web. The page (or set of pages) that the crawler starts with are referred to as the
seed URLs. By starting at one page, all links from it are recorded and saved in a queue.
These new pages are in turn searched and their links are saved. As these robots search
the Web, they may collect information about each page, such as extract keywords and
store in indices for users of the associated search engine. A crawler may visit a certain
number of pages and then stop, build an index, and replace the existing index. This
type of crawler is referred to as a periodic crawler because it is activated periodically.
Crawlers are used to facilitate the creation of indices used by search engines. They allow
the indices to be kept relatively up-to-date with little human intervention. Recent research
has examined how to use an incremental crawler. Traditional crawlers usually replace
the entire index or a section thereof. An incremental crawler selectively searches the Web
and only updates the index incrementally as opposed to replacing it.
Because of the tremendous size of the Web, it has also been proposed that a focused
crawler be used. A focused crawler visits pages related to topics of interest. This concept
is illustrated in Figure 7.3. Figure 7.3(a) illustrates what happens with regular crawling,
while Figure 7.3(b) illustrates focused crawling. The shaded boxes represent pages that
are visited. With focused crawling, if it is determined that a page is not relevant or its
links should not be followed, then the entire set of possible pages underneath it are pruned
and not visited. With thousands of focused crawlers, more of the Web can be covered
than with traditional crawlers . This facilitates better scalability as the Web grows. The
focused crawler architecture consists of three primary components [CvdBD99]:
W e b Content Mining
Section 7 . 2
The crawler performs the actual crawling on the Web. The pages it visits are
determined via a priority-based structure governed by the priority associated with
pages by the classifier and the distiller.
A performance objective for the focused crawler is a high precision rate or harvest rate.
To use the focused crawler, the user first identifies some sample documents that
are of interest. While the user browses on the Web, he identifies the documents that
are of interest. These are then classified based on a hierarchical classification tree, and
nodes in the tree are marked as good, thus indicating that this node in the tree has
associated with it document(s) that are of interest. These documents are then used as the
seed documents to begin the focused crawling. During the crawling phase, as relevant
documents are found it is determined whether it is worthwhile to follow the links out
of these documents. Each document is classified into a leaf node of the taxonomy tree.
One proposed approach, hard focus, follows links if there is an ancestor of this node that
has been marked as good. Another technique, soft focus, identifies the probability that a
page, d, is relevant as
R (d)
P (c I d)
(7 . 1 )
good (c)
Here c is a node in the tree (thus a page) and good(c) is the indication that it has been
labeled to be of interest. The priority of visiting a page not yet visited is the maximum
of the relevance of pages that have been visited and point to it.
The hierarchical classification approach uses a hierarchical taxonomy and a naive
B ayes classifier. A hierarchical classifier allows the classification to include information
contained in the document as well as other documents near it (in the linkage struc
ture). The objective is to classify a document d to the leaf node c in the hierarchy
with the highest posterior probability P (c I d) . Based on statistics of a training set,
each node c in the taxonomy has a probability. The probability that a document can
be generated by the root topic, node q , obviously is 1. Following the argument found
in [CDAR98], suppose q , . . . , Ck
c be the path from the root node to the leaf c. We
=
200
Chapter 7
Web M i n i ng
Section 7 . 2
thus know
P (c; I d) = P (ci - 1 I d) P (c; I c; - J , d)
P (c; I c; - J ) P (d I c; )
s
is
2::
sibling of
P (d I s)
(7.3)
c;
P (d I c; ) can be found using the Bernoulli model, in which a document is seen as a bag
of words with no order [CvdBD99] .
More recent work on focused crawling has proposed the use of context graphs. The
contextfocused crawler (CFC) performs crawling in two steps. In the first phase, context
graphs and classifiers are constructed using a set of seed documents as a training set. In
the second phase, crawling is performed using the classifiers to guide it. In addition, the
context graphs are updated as the crawl takes place. This is a major difference from the
focused crawler, where the classifier is static after the learning phase. The CFC approach
is designed to overcome problems associated with previous crawlers:
There may be some pages that are not relevant but that have links to relevant
pages. The links out of these documents should be followed.
Relevant pages may actually have links into an existing relevant page, but no links
into them from relevant pages. However, crawling can really only follow the links
out of a page. It would be nice to identify pages that point to the current page. A
type of backward crawling to determine these pages would be beneficial.
7 .2.2
7.2.3
Level l documents
Level 2 documents
Level 3 documents
F I G U R E 7.4:
Context graph.
Harvest System
The Harvest system is based on the use of caching, indexing, and crawling. Harvest is
actually a set of tools that facilitate gathering of information from diverse sources. The
Harvest design is centered around the use of gatherers and brokers. A gatherer obtains
information for indexing from an Internet service provider, while a broker provides the
index and query interface. The relationship between brokers and gatherers can vary.
Brokers may interface directly with gatherers or may go through other brokers to get to
the gatherers . Indices in Harvest are topic-specific, as are brokers. This is used to avoid
the scalability problems found without this approach.
Harvest gatherers use the Essence system to assist in collecting data. Although not
designed explicitly for use on the Web, Essence has been shown to be a valid technique for
retrieving Web documents [HS93]. Essence classifies documents by creating a semantic
index. Semantic indexing generates different types of information for different types of
files and then creates indices on this information. This process may first classify files
based on type and then summarize the files typically based on keywords. Essence uses
the file extensions to help classify file types.
The CFC approach uses a context graph, which is a rooted graph in which the root
represents a seed document and nodes at each level represent pages that have links to
a node at the next higher level. Figure 7.4 contains three levels. The number of levels
in a context graph is dictated by the user. A node in the graph with a path of length n
to the seed document node represents a document that has links indirectly to the seed
document through a path of length n. The number of back links followed is designated
as input to the algorithm. Here n is called the depth of the context graph. The context
graphs created for all seed documents are merged to create a merged context graph. The
Seed document
201
context graph is used to gather information about topics that are related to the topic being
explored.
Backward crawling finds pages that are not pointed to by relevant documents but
are themselves relevant. These types of pages may be new and may not yet have been
discovered and linked to from other pages. Although backward links do not really exist
in the Web, a backward crawl can be performed relatively easily because most search
engines already maintain information about the back links. This type of information is
similar to that often used by commercial citation servers, which find documents that
cite a given document. The value of the Science Citation Index in performing tradi
tional literature searches is well known. The use of backlinks on the Web can provide
similar benefits.
CFC performs classification using a term frequency-inverse document frequency
(TF-IDF) technique. The vocabulary used is formed from the documents in the seed set
and is shown in the merged context graph. Each document is represented by a TF-IDF
vector representation and is assigned to a particular level in the merged context graph.
(7.2)
W e b Content M i n i n g
One proposed approach to handling the large amounts of somewhat unstructured data
on the Web is to create a multiple layered database (MLDB) on top of the data in
the Web (or a portion thereof). This database is massive and distributed. Each layer of
this database is more generalized than the layer beneath it. Unlike the lowest level (the
Web), the upper levels are structured and can be accessed (and mined) by an SQL-like
query language. The MLDB provides an abstracted and condensed view of a portion
of the Web. A view of the MLDB, which is called a Virtual Web View (VWV), can be
constructed.
The indexing approach used by MLDB does not require the use of spiders. The
technique used is to have the Web servers (masters, administrators) themselves send their
indices (or changes to indices) to the site(s) where indexing is being performed. This
process is triggered when changes to the sites are made. Each layer of the index is smaller
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Section 7 . 2
Web M i n i ng
than that beneath it and to which it points. To assist in the creation of the first layer of the
MLDB, both extraction and translation tools are proposed. Translation tools are used to
convert Web documents to XML, while extraction tools extract the desired information
from the Web pages and insert it into the first layer of the MLDB . Web documents that
use XML and follow a standard format would not need any tools to create the layers. It is
proposed that any translation functions be performed directly by the local administrators .
The layer- 1 data can be viewed as a massive distributed database.
The higher levels of the database become less distributed and more summarized as
they move up the hierarchy. Generalization tools are proposed, and concept hierarchies
are used to assist in the generalization process for constructing the higher levels of the
MLDB . These hierarchies can be created using the WordNet Semantic Network. WordNet
is a database of the English language. Nouns, adjectives, verbs, and adverbs are listed,
divided into groups of synonyms, and linked together using both lexical and semantic
relationships.
A Web data mining query language, WebML is proposed to provide data mining
operations on the
LDB .
203
to purchase something he or she may not have thought about purchasing. Perhaps t e
.
simplest example of personalization is the use of a visitor' s name when he or she VISits
a page. Personalization is almost the opposite of targeting. With targeting, businesses
display advertisements at other sites visited by their users. With personalization, when
a particular person visits a Web site, the advertising can be designed specifically for
that person. MSNBC, for example, allows personalization by asking the user to enter
his or her zip code and favorite stock symbols [msnOO] . Personalization includes such
techniques as use of cookies, use of databases, and more complex data mining and
machine learning strategies [BDH+95] . Example 7 . 1 illustrates a more complex use
ID),
common problem of user identification with any type of Web mining. Mining activities
related to personalization require examining Web log data to uncover patterns of access
behavior by use . This may actually fall into the category of Web usage mining.
using data mining operations and lists of keywords. A major feature of WebML are four
primitive operations based on the use of concept hierarchies for the keywords [ZaY99] :
Wynette Holder often does online shopping through XYZ.com. Every time she visits
as well as what pages she visits. Mining of the sales and Web usage data is performed
"www .engr.smu.edu" that have a keyword that covers the keyword cat:
SE LECT
This
ID is
by XYZ to develop a very detailed user profile for Wynette. This profile in tum is us d
to personalize the advertising they display. For example, Wynette loves chocolate. This
is evidenced by the volume of chocolate she has purchased (and eaten) during the past
year. When Wynette logs in, she goes directly to pages containing the clothes she is
interested in buying. While looking at the pages, XYZ shows a banner ad about some
the link to this page and adds the chocolate to her shopping cart. She then returns to the
page with the clothes she wants.
FROM document in
ID.
special sale on Swiss milk chocolate. Wynette cannot resist. She immediately follows
The following example illustrates WebML. The query finds all documents at the level of
WebML allows queries to be stated such that the WHERE clause indicates selection
based on the links found in the page, keywords for the page, and information about the
domain Where the document is found. Because WebML is an extension of DMQL, data
mining functions such as classification, summarization, association rules, clustering, and
diction. Through classification, the desires of a user are determined based on tho se for
_
the class. With clustering, the desires are determined based on those users to which he
or she is determined to be similar. Finally, prediction is used to predict what the user
really wants to see. There are three basic types of Web page: personalization [MCSOO] :
Perso nalization
demographics.
alization, Web access or the contents of a Web page are modified to better fit the desires
using the desires of a user to determine what Web documents to retrieve.
With personalization, advertisements to be sent to a potential customer are chosen
based on specific knowledge concerning that customer. Unlike targeting, personalization
may be performed on the target Web page. The goal here is to entice a current customer
Another example of Web content mining is in the area of personalization. With person
of the user. This may involve actually creating Web pages that are unique per user or
Web Content M i n i n g
Content-based filtering retrieves pages based on similarity between them and user
profiles.
One of the earliest uses of personalization was with My Yahoo ! [MPROO]. With
My Yahoo ! a user himself personalizes what the screen looks like [YahOO]. He can
io4
Chapter 7
Web M i n ing
provide preferences in such areas as weather, news, stock quotes, movies, and sports.
Once the preferences are set up, each time the user logs in, his page is displayed. The
personalization is accomplished by the user explicitly indicating what he wishes to see.
Some observations about the use of personalization with My Yahoo! are [MPROO] :
7.3 .1
Most users do not seem to understand what personalization means and use only
use the default page.
Any personalization system should be able to support both types of users.
Web structure mining can be viewed as creating a model of the Web organization or a
portion thereof. This can be used to classify Web pages or to create similarity measures
between documents. We have already seen some structure mining ideas presented in the
content mining section. These approaches used structure to improve on the effectiveness
of search engines and crawlers.
Web Structure M i n i ng
205
PageRank
The PageRank technique was designed to both increase the effectiveness of search
engines and improve their efficiency [PBMW98]. PageRank is used to measure the
importance of a page and to prioritize pages returned from a traditional search engine
using keyword searching. The effectiveness of this measure has been demonstrated by
the success of Google [GooOO] . (The name Google comes from the word googol, which
is 10 100 .) The PageRank value for a page is calculated based on the number of pages
that point to it. This is actually a measure based on the number of backlinks to a page. A
backlink is a link pointing to a page rather than pointing out from a page. The measure
is not simply a count of the number of backlinks because a weighting is used to provide
more importance to backlinks coming from important pages. Given a page p, we use
Bp to be the set of pages that point to p, and Fp to be the set of links out of p. The
PageRank of a page p is defined as [PBMW98]
A few users will create very sophisticated pages by utilizing the custorilization
provided.
Section 7.3
PR(p)
P (q )
q
qE Bp
(7.4)
Here Nq = I Fq 1. The constant c is a value between 0 and 1 and is used for normalization.
A problem, called rank sink, that exists with this PageRank calculation is that when
a cyclic reference occurs (page A points to page B and page B points to page A), the
PR value for these pages increases. This problem is solved by adding an additional term
to the formula:
PR(q )
I
(7.5)
PR (p)
c ""
----;:;-- + c E ( v)
q
qEBp
=
where c is maximized. Here E ( v) is a vector that adds an artificial link. This simulates
a random surfer who periodically decides to stop following links and jumps to a new
page. E (v) adds links of small probabilities between every pair of nodes.
The PageRank technique is different from other approaches that look at links. It
does not count all links the same. The values are normalized by the number of links in
the page.
7.3.2
Clever
One recent system developed at IBM, Clever, is aimed at finding both authoritative
pages and hubs [CDK+ 99] . The authors define an authority as the "best source" for
the requested information [CDK+99] . In addition, a hub is a page that contains links to
authoritative pages. The Clever system identifies authoritative pages and hub pages by
creating weights. A search can be viewed as having a goal of finding the best hubs and
authorities.
Because of the distributed and unsupervised development of sites, a user has no way
of knowing whether the information contained within a Web page is accurate. Currently,
there is nothing to prevent someone from producing a page that contains not only errors,
but also blatant lies. In addition, some pages might be of a higher quality than others.
These pages are often referred to as being the most authoritative. Note that this is different
from relevant. A page may be extremely relevant, but if it contains factual errors, users
certainly do not want to retrieve it. The issue of authority usually does not surface in
traditional IR.
Chapter 7
206
Web M i n i n g
Section 7.4
Hyperlink-induced topic search (HITS) finds hubs and authoritative pages [Kle99a] .
The HITS technique contains two components:
Hub and authority measures are associated with these pages. Pages with the highest
values are returned.
q
s
EXAMPLE 7.2
The webmaster at ABC Corp. learns that a high percentage of users have the following
pattern of reference to pages: (A, B , A, C) . This means that a user accesses page A,
then page B, then back to page A, and finally to page C. Based on this observation, he
determines that a link is needed directly to page C from page B. He then adds this link.
Web usage mining can be used for many different purposes. By looking at the
sequence of pages a user accesses, a profile about that user could be developed, thus
aiding in personalization. With site mining, the overall quality and effectiveness of the
pages at the site can be evaluated. One taxonomy of Web usage mining applications has
included [SCDTOO] :
By determining frequent access behavior for users, needed links can be identified
to improve the overall performance of future accesses.
Web usage patterns can be used to gather business intelligence to improve sales
and advertisement.
Gathering statistics concerning how users actually access Web pages may or may
not be viewed as part of mining.
HITS algorithm
R = SE( W, q)
B = R U {pages l inked to from R) U {pages that l ink to pages in
G(B, L) = Subgraph of W induced by B ;
Delete links i n G within same s i te ;
G(B, L1 )
Xp = Lq where (q,p)ELl Yq ;
I I F ind authority weight s ;
I I Find hub we ight s ;
Yp = Lq where (p,q)ELl Xq ;
A = {p I p has one of the highe st xp ) ;
H = {p I p has one of the highes t Yp ) ;
R) ;
7.4
Web usage mining performs mining on Web usage data, or Web logs. A Web log is a
listing of page reference data. Sometimes it is referred to as clickstream data because
each entry corresponds to a mouse click. These logs can be examined from either a client
207
Output :
A
H
Web Usage M i n i n g
Web usage mining actually consists of three separate types of activities [SCDTOO] :
Preprocessing activities center around reformatting the Web log data before pro
cessing.
Pattern discovery activities form the major portion of the mining activities because
these activities look to find hidden patterns within the log data.
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Section 7.4
f the
There are many issues associated with using the Web log for mining purposes:
7.4. 1
With a Web client cache, the exact sequence of pages a user actually visits
ifficult to uncover from the server site. Pages that are rereferenced may be fou
m the cache.
There are many security, privacy, and legal issues yet to be solved. For example, is
the set of pages a person visits actually private information? Should a Web browser
actually divulge information to other companies about the habits of its users? After
all, this information could be valuable to potential advertisers.
Combine all records from the same source site that occur within a time period.
Add records to a session if they are from the same source site and the time between
two consecutive tiinestamps is less than a certain threshold value.
Associated with each session i s a unique identifier, which i s called a session ID.
The length of a session S is the number of pages in it, whilch is denoted as len(S). Let
database D be a set of such sessions, and the total length of D be len(D) = LseD len(S).
There are many problems associated with the preprocessing activities, and most
of these problems center around the correct identification of the actual user. User iden
tification is complicated by the use of proxy servers, client side caching, and corporate
firewalls. Tracking who is actually visiting a site (and where they come from) is diffi
cult. Even though a visit to a Web page will include a source URL or IP address that
indicates the source of the request, this may not always be accurate in determining the
source location of the visitor. Users who access the Internet through an Internet service
provider (ISP) will all have the source location of that provider. It is not unique to the
individual. In addition, the same user may use different ISPs. Also, there will be many
users accessing the Web at the same time from one machine. Cookies can be used to
assist in identifying a single user regardless of machine used to access the Web. A cookie
is a file that is used to maintain client-server information between accesses that the client
makes to the server. The cookie file is stored at the client side and sent to the server
with each access.
Identifying the actual sequence of pages accessed by a user is complicated by the
use of client side caching. In this case, actual pages accessed will be missing from the
server side log. Techniques can be used to complete the log by predicting missing pages.
Path completion is an attempt to add page accesses that do not exist in the log but that
actually occurred. Some missing pages can be easily added. For example, if a user visits
page A and then page C, but there is no link from A to C, then at least one page in
this path is missing. Algorithms are used both to infer missing pages and to generate an
approximate timestamp.
T e Web usage log probably is not in a format that is usable by mining applicatio
ns. As
With any data to be used in a mining application, the data may need to be
reformatted
and cleansed. There are, in addition, some issues specifically related to the use
of Web
logs. Steps that are part of the preprocessing phase include cleansing, user identifi
cation '
session identification, path completion, and formatting [CMS99] .
DEFINITION 7.1. Let P be a set of literals, called pages or clicks, and U be a
Standard log data consist of the following: source site, destination site, and time
stamp, as shown in Definition 7 . 1 . The source and destination sites could be listed as
a URL or an IP address. The definition assumes that the source site is identified by
a user ID and the destination site is identified by a page ID. Additional data such as
Web browser information also may be included. Before processing the log, the data
may be changed in several ways. For security or privacy reasons, the page addresses
may be changed into unique (but nonidentifying) page identifications (such as alphabetic
characters). This conversion also will save storage space. In addition, the data may be
cleansed by removing any irrelevant information. As an example, the log entries with
figures (gif, jpg, etc.) can be removed.
Data from the log may be grouped together to provide more information. All pages
visited from one source could be grouped by a server to better understand the patterns
of page rferences from each user (source site). Similarly, patterns from groups of sites
may be discovered. References to the same site may be identified and examined to better
understand who visits this page.
A common technique is for a server site to divide the log records into sessions.
As shown in Definition 7.2 from [XDO l a], a session is a set of page references from
one source site during one logical period. Historically, a session would be identified by
.
.
a user loggmg mto a c?mputer, performing work, and then logging off. The login and
logoff represent the logical start and end of the session. With Web log data, this is harder
to determine. Several approaches can be used to identify these logical periods :
Preproce ssing
209
NCR uses an approach based on the second concept. Any inactive period of 30 min
utes or more ends a session [SPOO]. Empirical results have shown that 25.5 minutes is
appropriate [CP95] .
Identification of the exact user is not possible from the log alone.
Web Usage M i n i ng
7.4.2
Data Structures
Several unique data structures have been proposed to keep track of patterns identified
during the Web usage mining process. A basic data structure that is one possible alter
native is called a trie. A trie is a rooted tree, where each path from the root to a leaf
represents a sequence. Tries are used to store strings for pattern-matching applications.
Each character in the string is stored on the edge to the node. Common prefixes of strings
are shared. A problem in using tries for many long strings is the space required. This is
Chapter 7
210
Web M i n ing
Section 7.4
Web Usage M i n i ng
21 1
ACCfCGTCf
0
u
TCGTC
T
A
A
ABOU
(a) uie
AT
GORY
EXAMPLE 7.3
Suppose that one session has been identified to be (C, A, C, C, T, C, G, T, C, T). Many
different patterns exist in this session. As a matter of fact, we could identify patterns
starting at the first character, or the second, or any other. The suffix tree created for this
session is shown in Figure 7.6. This tree does ncit contain the special "$" edges.
illustrated in Figure 7.5(a), which shows a standard trie for the three strings {AB OUT ,
CAT, CATEGORY}. Note that there are many nodes with a degree of one. This is a
waste of pace that is solved by compressing nodes together when they have degrees
of one. Ftgure 7 .5(b) shows a compressed version of this trie. Here a path consisting
of nodes with single children is compressed to one edge. Note in both trees the extra
edge labeled $ This symbol (or any symbol that is not in the alphabet and is used to
constmct the strings) is added to ensure that a string that is actually a prefix of another
(CAT is a prefix of CATEGORY) terminates in a leaf node.
The compressed trie is called a suffix tree. A suffix tree has the following
characteristics:
"
."
Each internal node except the root has at least two children.
With the help of a suffix tree, it is efficient not only to find any subsequence in a
sequence, but also to find the common subsequences among multiple sequences. A
suffix tree can also be constructed from a sequence in time and space linear in the
length of the sequence. When given one session of page references, many different
patterns may be found. The exact number of patterns depends on the exact defini
tion of the pattern to be found (discussed in subsection 7.4.3). Example 7.3 illustrates
this idea.
A slight variation on this suffix tree that is used to build a suffix tree for multiple
sessions is called a generalized suffix tree (GST).
7 .4.3
Pattern Discovery
The most common data mining technique used on clickstream data is that of uncovering
traversal patterns. A traversal pattern is a set of pages visited by a user in a session. Other
types of patterns may be uncovered by Web usage mining. For example, association
mles can look at pages accessed together in one session independent of ordering. Similar
traversal patterns may be clustered together to provide a clustering of the users. This is
different from clustering of pages, which tends to identify similar pages, not users.
Several different types of traversal patterns have been examined. These patterns
differ in how the patterns are defined. The differences between the different types of
patterns can be described by the following features:
The pattern of references may or may not be maximal in the session. A frequent
pattern is maximal if it has no subpattern that is also frequent.
Patterns found using different combinations of these three properties may be used
to discovr different features and thus may be used for different purposes. Knowledge
of contiguous page references frequently made can be useful to predict future references
212
Web M i n ing
Chapter 7
TAB LE 7. 1 :
Section 7.4
Duplicates
Consecutive
Maximal
Support
Since there are four transactions and the support is 30%, an itemset must occur in at least
two sessions. During the first scan, we find that L 1
{{A}, { B } , {C}, {E}}, so C2
{ {A , B}, {A, C}, { A , E}, {B, C}, {B, E}, {C, E}}. Counting these is scan two, we find
{{A, C}, {B, C}, {C, E } } and then generate C3
{{A, B , C}, {A , C , E}, { B , C, E}} .
L2
Counting, we find that none of these are large. The large itemsets are then
=
rules
Episodes
freq(X)
# transactions
freg(X)
# time windows
freq(X)
# customers
freg(X)
# forward sequences
freq(X)
# clicks
Sequential
patterns
Forward
sequences
Maximal
frequent
sequences
and thus for prefetching and caching purposes. Knowledge of backward traversals often
followed can be used to improve the design of a set of Web pages by adding new links to
shorten future traversals. The maximal property is used primarily to reduce the number
of meaningful patterns c\iscovered. The use of such performance improvements as user
side caching may actually alter the sequences visited by a user and impact any mining
of the Web log data at the server side.
The different types of traversal patterns that have been studied and how they view
these three features are shown in Table 7 . 1 (from [XDO l a]). Example 7.4 illustrates a set
of sessions to be used throughout this section. The sessions are listed in order, and all
timestamps have been removed.
EXAMPLE 7.4
The XYZ Corporation maintains a set of five Web pages: {A, B, C, D , E}. The following
sessions (listed in timestamp order) have been created: D
{S1
{U1 , (A, B, C ) } , s2
{ U2, (A , C ) } , S3
{U1 , (B, C, E ) } , S4
{U3 , (A , C, D , C, E ) } } . Here we have added
to each session the user Ib. Suppose the support threshold is 30%.
=
Association Rules. Association rules can be used to find what pages are accessed
together. Here we are really finding large itemsets. A page is regarded as an item,
and a session is regarded as a transaction with both duplicates and ordering ignored.
The support is defined to be the number of occurrences of the itemset divided by the
number of transactions or sessions. The application of the Apriori algorithm to the data
in Example 7.2 is shown in Example 7.5.
Serial episodes are ordered, parallel episodes are riot, and general episodes are partially ordered.
213
EXAMPLE 7 . 5
Association
Web Usage M i n i ng
Sequential Patterns. Although initially proposed for use with market basket data,
have also been applied to Web access logs. A sequential pattern (as
applied to Web usage mining) is defined as an ordered set of pages that satisfies a given
support and is maximal (i.e., it has no subsequence that is also frequent). Support is
defined not as the percentage of sessions with the pattern, but rather the percentage of
customers who have the pattern. Since a user may have many sessions, it is possible
that a sequential pattern could span sessions. It also need not be contiguously accessed
pages. A k-sequence is a sequence of length k (i.e., is it has k pages in it).
Algorithm 7.2 outlines the steps needed to find sequential patterns. After the sort
step to put the data in the correct order, the remaining steps are somewhat similar to those
of the Apriori algorithm. The sort step creates the actual customer sequences, which are
the complete reference sequences from one user (across transactions). During the first
scan it finds all large 1 -itemsets. Obviously, a frequent 1 -itemset is the same as a frequent
1-sequence. In subsequent scans, candidates are generated from the large itemsets of the
previous scans and then are counted. In counting the candidates, however, the modified
definition of support must be used. In the algorithm we show that AprioriAll is used to
perform this step.
sequential patterns
ALGORITHM 7.2
Input :
I /Database of sess ions
D = {S1 , S2 , . . . , Sk}
/ /Support .
s
Output : Sequent i a l patterns
Sequential patterns algorithm:
D = sort D on user- ID and t ime of f irst pa9e re ference
in each s e s s ion ;
f ind L1 in D ;
L = AprioriAl l ( D, s , L1 ) ;
f ind maximal reference sequences from L ;
Generating sequential patterns for Example 7.5 is shown in Example 7.6. Here Ci
represents the candidate i -sequences and Li are the large i -sequences.
EXAMPLE 7.6
In this example, user U1 actually has two transactions (sessions). To find his sequential
patterns, we must think of his sequence as the actual concatenation of those pages in
214
Cha pter 7
Web M i n i n g
Section 7.4
S1 and S3 . Also, since support is measured not by transactions but by users, a sequence
is large if it is contained in at least one customer' s sequence. After the sort step, we
have that D = (S1 = { U1 , (A, B, C) } , S3 = { U1 , ( B , C, E) } , S2 = { U2 , (A , C) } , S4 =
{ U3 , ( A , C , D , C, E) }. We find L 1 {{ A } , { B } , { C } , {D} , { E } } since each page is referenced
b y at least one customer. The following table outlines the steps taken by AprioriAll:
There are variations of this algorithm and several techniques used to improve the
Web Usage M i n i n g
215
addition, the individual page accesses must occur within a particular time frame. A serial
episode is an episode in which the events are totally ordered. Note that they need not be
contiguous, however. A parallel episode is a set of events where there need not be any
particular ordering. They still do need to satisfy the time constraint, however. Finally, a
general episode is one where the events satisfy some partial order. Note that even though
these seem similar to the idea of sequential patterns and association rules, the added con
performance. The set of customer sequences is reformatted after L 1 is found. Each trans
straint of a time window does make an episode different from either of these. The original
before counting by removing any candidates that have subsequences that are not large.
to events by one user or across users. In addition, episodes need not be maximal.
action is replaced with one that consists only of pages from L 1 . Candidates may be pruned
C1 = { (A ) , ( B ) , (C) , ( D ) , (E) }
L 1 = {(A) , ( B ) , (C) , ( D ) , (E) }
C2 = { (A,, B ) , (A, C), (A, D ) , (A, E ) , ( B , A ) , (B, C ) , ( B , D ) ,
( B , E)", ( C , A ) , (C, B) , ( C , D ) , (C, E ) , ( D , A ) , (D, B ) ,
( D , C ) , ( D , E ) , (E, A ) , (E, B ) , (E, C) , ( E , D) }
L 2 = {(A, B ) , (A, C) , (A , D ) , ( A , E ) , ( B , C ) ,
( B , E ) , ( C , B ) , (C, D ) , ( C , E ) , (D, C ) , ( D , E) }
C3 = { ( A , B, C) , (A, B, D ) , (A, B , E ) , (A , C, B ) , ( A , C, D ) , (A, C, E),
(A , D, B), (A, D, C), (A, D , E ) , (A, E, B), (A, E , C) , (A , E , D ) ,
( B , C, E ) , (B, E , C ) , ( C , B , D), (C, B , E ) , (C, D , B) , (C, D , E ) ,
( C , E, B ) , ( C , E, D ) , (D, C , B ) , ( D , C, E ) , (D, E , C) }
L3 = { (A , B , C) , (A, B , E ) , (A, C, B ) , (A, C, D ) , (A, C, E ) , (A , D, C ) ,
(A , D , E), (B, C , E ) , ( C , B, E ) , (C, D , E ) , (D, C, E) }
C4 = { (A , B, C, E ) , (A, B, E, C) , (A , C , B , D) , (A , C, B , E ) ,
(A , C , D , B ) , (A , C, D , E ) , (A, C , E , B ) , (A, C, E , D ) ,
(A , D , C , E ) , (A , D , E , C) }
L4 = { (A , B, C, E) , (A, C, B, E ) , (A , C , D, E ) , (A , D, C, E) }
definition has no concept of user, but, of course, the idea of an episode could be applied
Example 7.7 illustrates the concept of episodes applied to the data in Example 7.4.
The XYZ Corporation maintains a set of five Web pages {A, B , C, D, E } . Assume that the
data in Example 7.2 have the following sequence (independent of user): ( A , B , C, A , C,
We assume that the time window is the difference between the last event and the first
event in the episode. To illustrate episodes, Figure 7.7 illustrates the ordering of events
as shown in a DA G (directed acyclic graph) where arcs are used to represent temporal
ordering. The arcs are labeled with the time between successive events. Starting at the
first event and looking at the maximum window size of 3, we see that we have two serial
episodes: AC and AB . B and C occur as parallel episodes. Stmting at the event looking
at time 12, we have the following serial episodes: ACD, ACC, CCC, CCD, AC, CC,
CC, CD. We have two parallel episodes: A and C, and C and D. There also is a general
episode that can be seen as the subgraph from time 12 to time 1 4. When taking the
frequency into account, an episode must occur a certain number of times in all windows.
Cs = 0
Maximal Frequent Forward Sequences.
patterns is to remove any backward traversals [CPY98]. Each raw session is transformed
The WAP-tree (web access pattern) has been proposed to facilitate efficient count
ing. This tree is used to store the sequences and their counts. Once the tree is built, the
into forward reference (i.e., removes the backward traversals and reloads/refreshes), from
which the traversal patterns are then mined using improved level-wise algorithms. For
original database of patterns is not needed. Each node in the tree is associated with an
event (a page found at a particular time by a user). The node is labeled with the event and
a count that is associated with the pattern prefix that ends at that event. Only individual
nication alarm analysis, can also be applied to Web logs. All pages (corresponding to
events) are ordered by their access time, and the users usually need not be identified (i.e.,
Cha pter 7
216
Section 7.4
Web M i n ing
example, for the session (A, B, A, C) in Example 7 . 2, the resulting forward sequences
are (A, B) and (A, C). Looking at Example 7.8 and the sequence (A, B, C, A , C, B ,
C, A, C, D , E) , w e find the following maximal forward references:
W e b Usage M i n i n g
EXAMPLE 7.8
Given D = { (A, B, C, D , E, D, C, F ) , ( A , A, B, C, D , E ) , ( B , G, H, U, V ) , (G, H, W ) } .
The first session has backward traversals, and the second session has a reload/refresh
0.09. This means
22. Let the minimum support be Srnin
on page A. Here len(D)
that we are looking at finding sequences that occur at least two times. There are two
As observed by the authors, the "real" access patterns made to get to the really used pages
would not include these backward references. They assume that the backward reference
is included only because of the structure of the pages, not because they really want to do
this. The resulting set of forward references are called maximal forward references. They
Algorithm 7.4 (from [XDOl a] ) shows the OAT (Online Adaptive Traversal Pat
terns) algorithm designed to find MFS. It utilizes a suffix tree to store patterns. One
suffix tree is created for all sessions. Counts of patterns are maintained in the tree. A
unique feature of OAT is its ability to adapt to the availabe amount of main memory. If
the suffix tree is too big to fit into memory, it is compressed and the algorithm contin
ues. Details concerning the exact techniques used for compression can be found in the
literature [XDOl a].
reference sequences that are not subsequences of other large reference sequences are
found in step 3 and become the large reference sequences.
ALGORITHM 7.4
Input :
81 , 82 , . . ,
.
ALGORITHM 7.3
Smin :
/ / S upport
Al l maximal
Output :
s e s s ions
f requent
s equenc e s
(MFS s )
OAT algorithm:
Sn :
M: m a i n memory s i z e
Output :
s e s s i ons
of
maximal frequent sequences: (A, B, C, D, E) and (G, H ) . Both sequences occur two
times.
are called maximal because they are not subsequences of other forward references. The
set of important reference patterns are those that occur with frequency above a desired
Input :
D
217
reference
for i from 1 to
/ / compre s s
if
freq(X)
len(D)
freq
=
---
# clicks
do
the
sequenc e s ;
then
ST = OAT_compre s s (S T) ;
endif
sequences (A , B ) and (A, C), we could not tell whether a direct link to page C from
page B is needed, as shown in Example 7.2.
With maximal frequent sequences (MFS), all four properties in Table 7.1 are re
quired. Since an MFS could potentially start with any page (click) in any session, the
definition of support assumes that the number of clicks is in the denominator. Thus, the
support of a sequence X is defined to be
//if
l arge one s ;
---
s c an
/ / f i rst
if interrupted by the u s e r ,
then
endif
end for
/ / second s c an
sc an ,
endi f
output
the MFSs
i n the
suf f ix t re e .
then
the MFS s .
218
7.4.4
Chapter 7
Web M i n ing
Once patterns have been identified, they must be analyzed to detennine how that infor
mation can be used. Some of the generated patterns may be deleted and determined not
to be of interest.
Recent work has proposed examining Web logs not only to identify frequent types
of traversal patterns, but also to identify patterns that are of interest because of their
uniqueness or statistical properties [WUMOO]. Patterns found need not have contiguous
page references. A Web mining query language, MINT, facilitates the statement of inter
esting properties. The idea of a sequence is expanded to the concept of what the authors
call a g-sequence. A g-sequence is a vector that consists not only of the pages visited
(events) but also of wildcards. For example, the g-sequence b * c stands for a sequence
consisting of b, any number of pages, then c. With the use of wildcards, it is indicated
that the events need not be contiguous. More complicated g-sequences can indicate spe
cific constraints on the number of events that replace the wildcard. With MINT, selection
of patterns that satisfy a g-sequence template are accomplished. The selection constraints
may also include restrictiOJlS on support.
Some of the thrust 6f this work has been in comparing the differences between
traversal patterns of the customers of an e-business site and those that are not cus
tomers [SPFOO] . Visitors to a site have been classified as short-time visitors, active
investigators, and customers [BPW96] . Preprocessing first filters out the visitors who are
short-time. Using concept hierarchies, the contents of the Web pages are then abstracted
to more general concepts. The log is then divided into those for customers and those for
noncustomers . Each log is then examined to find patterns based on any desired require
ments (such as frequency). The patterns found across the two logs are then compared
for similarity. Similarity is detennined using the following rule [SPFOO] :
Two patterns are comparable if their g-sequences have at least the first n pages the
same. Here n is supplied by the user.
In addition, only fragments of patterns that occur frequently are considered. The goal
of this work is to increase the number of customers. Noncustomer patterns with no
comparable customer patterns indicate that some changes to the link structure or Web
page designs may be in order. The project proposes rules and the use of a proxy server
to dynamically change the link structures of pages.
7.5
Section 7.6
Pattern Analysis
EXERCISES
1. (Research) Find and describe two different approaches used by Web sites to per
Describe the results. Your description should include the number of documents
retrieved. Compare the differences of the five top pages found by each. Hypothesize
why these differences exist.
3. Construct a trie for the string ( A , B, A , C ) .
4 . Construct a suffix tree for the string ( A , B , A , C) .
5. Given the following sessions, { ( A , B, A , C ) , (C, B , D, F) , ( A , B, A ) } , indicate the
B i b l iogra p h i c Notes
219
and have t e
of frequent sequence statistics. Suppose that two users use one proxy
following sessions:
User 1: ( 1 , 3, 1 , 3, 4, 3, 6, 8, 2, 3, 6)
User 2: (2, 3, 4, 3, 6 , 8, 6, 3, 1 )
When these are viewed together by the Web server (taking into account the time
stamps), one large session is generated:
( 1 , 2, 3 , 3, 4, 1 , 3, 6, 3, 8, 4, 3, 6, 3, 6, 1 , 8, 2, 3, 6 )
Identify the maximal frequent sequences assuming a rninimum support of 2. What
are the maximal frequent sequences if the two users could be separated?
1. (Research) Perform a literature survey concerning current research into solutions
BIBLIOGRAPH IC NOTES
an xcellent
A recent survey of Web mining has been published [KBOO] and contain
art1cl ro
Th1s
ons.
publicati
bibliography and a survey of hundreds of Web mining
mmmg.
content
Web
of
v1ew
database
a
and
vides both an information retrieval view
.
activities
mining
Web
of
es
taxonomi
contain
Both [Za199] and [CMS97]
tal crawlers
There have been many published articles exploring crawlers. Incremen
], [CDI98],
[CDAR98
in
studied
were examined in [CGMOO] . Focused crawlers are
] . The
[CGMOO
in
ted
investiga
was
crawler
periodic
[CvdBD9 9], and [DCL+oo] . The
were
Essence
and
Harvest
.
+oo]
[DCL
in
ted
investiga
first
was
crawler
context focused
prowas
MLDB
described in [BDH+95 ] and [HS93]. The Virtual Web View with the
posed in [Za199] .
.
.
.
of the
An excellent introduction to personalizatwn appeared m a spec1al 1ssue
issue is aimed
same
this
within
section
special
A
[RieOO].
ACM
the
f
o
cations
Communi
s automatic perspecifically at personalization using Web usage mining [SpiOO] . Firefly'
[SM95].
in
sonalization approach was examined
.
.
.
.
eb mm
A recent doctoral dissertation has examined many 1ssues associated w1th
compansons of
ing [Zai:99] . In addition to providing an excellent bibliography . ":ith
laguage,
query
g
rrurun
Web
a
proposes
also
work
this
various Web mining activities,
.
red
multllaye
a
mto
ed
transform
been
have
which
data,
Web
the
accesses
WebML. WebML
database. Information about WordNet is available on the Web [WorOO].
ed for
The suffix tree data structure is actually a PATRICIA trie [Bay74] construct
.
cC76]
[
in
shown
were
trees
suffix
for
s
algorithm
ient
c
the relevant suffixes. Effi
frequent
Sequential patterns as applied to Web log were studied in [AS95] . MaXImal
.
220
Chapter 7
Web M i n i n g
Recently there have been several proposals for query languages aimed at the Web.
Most of these are extensions of SQL (WebSQL, W3QL, WebOQL), while others are
based on deductive type rules (WebLog). Instead of using relations, WebSQL uses virtual
relations, which are viewed as abstractions of Web documents [MMM96]. When using
WebSQL, the actually work of accessing the Web is performed by a traditional search
engine. WebSQL queries are converted into search engine queries, and results of these
queries are compiled and returned to the user. Similarly, W3QL [KS95] takes advantage
of a search engine to access the Web. W3QL, however, allows the use of external code or
Unix commands to be embedded within the query. WebLog uses deductive rules rather
than an SQL-like syntax [LSS96]. It is considered to be a second-generation language
because it actually can generate new Web documents. Based on OQL, WebOQL views
data as consisting of trees, while groups of trees are called webs [GM98].
There are several ongoing research prototypes examining Web mining. The WEB
MINER system being developed at DePaul University [MobOO] consists of both Web log
preparation steps (cleaning, transaction identification, and integration) and mining func
tions. An ongoing researo)l project at the University of Minnesota, WebSIFT, has produced
a comprehensive system design, including preprocessing, knowledge discovery, and pat
tern analysis steps [CTS97] . The data mining functions performed include classification
of Web pages, identification of sequential patterns of Web usage data, clustering of both
pages and users, generation of association rules, and creation of usage statistics.
IBM has a Web mining product called SurfAid Analytics [IBMOO] . SurfAid per
forms traversal pattern analysis, referral analysis, and otqer data mining activities. Refer
ral analysis determines where visitors came from when they entered a Web page.
C H A P T E R
S pati a l M i n i n g
8.1
INTRODUCTION
8.2
8.3
8.1
8.4
8-5
SPATIAL RULES
8.6
8.7
8.8
EXERCISES
8.9
BIBLIOGRAPHIC NOTES
I NTRODUCTION
Spatial data are data that have a spatial or location component. Spatial data can be
viewed as data about objects that themselves ar't located in a physical space. This may
be implemented with a specific location attribute(s) such as address or latitude/longitude
or may be more implicitly included such as by a partitioning of the database based
on location. In addition, spatial data may be adcessed using queries containing spatial
operators such as near, north, south, adjacent, and contained in. Spatial data are stored in
spatial databases that contain the spatial data and nonspatial data about objects. Because
of the inherent distance information associated with spatial data, spatial databases are
often stored using special data structures or indices built using distance or topological
information. As far as data mining is concerned, this distance information provides the
basis for needed similarity measures.
Spatial data are required for many current information technology systems. Geo
graphic information systems (GIS) are used to store infomtation related to geographic
locations on the surface of the Earth. This includes applications related to weather,
community infrastructure needs, disaster management, and hazardous waste. Data min
ing activities include prediction of environmental catastrophes. Biomedical applications,
including medical imaging and illness diagnosis) also require spatial systems.
Spatial mining, often called spatial data mining or knowledge discovery in spatial
databases, is data mining as applied to spatial databases or spatial data. Some of the
applications for spatial data mining are in the areas of GIS systems, geology, environ
mental science, resource management, agriculture, medicine, and robotics. Many of the
techniques discussed in previous chapters are applied directly to spatial data, but there
also are new techniques and algorithms developed specifically for spatial data mining.
221
222
Spatial M i n i ng
Chapter 8
Section 8.2
We investigate these issues in this chapter. Before investigating spatial mining, we first
provide a brief introduction to spatial data and databases.
8.2
8.2.2
Spatial Queries
Because of the complexity of spatial operations, much work has been performed to
examine spatial query processing and its optimization.
A traditional selection query accessing nonspatial data uses the standard comparison
operations: >, <, :::; , ::=:, :j=. A spatial selection is a selection on spatial data that may use
other selection comparison operations. The types of spatial comparators that could be
used include near, north, south, east, west, contained in, and overlap or intersect. The
following are exampes of several spatial selection queries:
'
Find the nearest fire station to 963 1 Moss Haven Drive in Dallas.
A special join operation applied to two spatial relations is called a spatial join. In
some ways, a spatial join is like a regular relational join in that two records are joined
together if they have features in common. With a traditional join, two records must
have attributes in common that satisfy a predefined relationship (such as equality in an
equijoin). With a spatial join, the relationship is a spatial one. The type of relationship is
based on the type of spatial feature. For example, the nearest relationship may be used
for points, while the intersecting relationship is used for polygons.
In GIS applications, it is common to have different views of the same geographic
area. For example, city developers must be able to see where infrastructure facilities are
located, including streets, power lines, phone lines, and sewer lines. At another level,
they might be interested in actual elevations, building locations, and rivers. Each of these
types of information could be maintained in separate GIS files. Merging these disparate
data can be performed using a special operator called a map overlay.
A spatial object usually is described with both spatial and nonspatial attributes .
Some sort of location type attribute must be included. The location attribute could identify
a precise point, such as a latitude or longitude pair, or it may be more logical such as
a street address or zip code. Often, different spatial obj ects are identified by different
location s, and some sort of translation between one attribute and the other is needed to
perform spatial operations between the different objects. As in SAND, the nonspatial
attributes may be stored in a relational database, while each spatial attribute is stored in
some spatial data structure. Each tuple in the relationship represents the spatial obj ect,
and a link to the spatial data structure is stored in the corresponding position in the
nonspatial tuple.
Many basic spatial queries can assist in data mining activities. Some of these
queries include:
A region query or range query is a query asking for objects that intersect a given
region specified in the query.
223
A nearest neighbor query asks to find objects that are close to an identified object.
A distance scan finds objects within a certain distance of an identified object, but
the distance is made increasingly larger.
Accessing spatial data can be more complicated than accessing nonspatial data. There
are specialized operations and data structures used to access spatial data.
8.2.1
Because of the unique features of spatial data, there are many data structures that have
been designed specifically to store or index spati:;tl data. In this section, we briefly exam
ine some of the more popular data structures. Many of these structures are based on
extensions to conventional indexing approaches, such as B-trees or binary search trees.
Nonspatial database queries using traditidmal indexing structures, such as a B
tree, access the data using an exact match query. However, spatial queries may use
proximity measures based on relative locations of spatial objects. To efficiently perform
these spatial queries, it is advisable that objects close in space be clustered on disk. To
this end, the geographic space under consideration may be partitioned into cells based
on proximity, and these cells would then be related to storage locations (blocks on disk).
The corresponding data structure would be constructed based on these cells.
A common technique used to represent a spatial object is by the smallest rectangle
that completely contains that object, minimum bounding rectangle (MBR). We illustrate
the use of MBRs by looking at a lake. Figure 8 . 1 (a) shows the outline of a lake. If we
orient this lake in a traditional coordinate system with the horizontal axis representing
east-west and the perpendicular axis north-south, we can put this lake in a rectangle
(with sides parallel to the axes) that contains it. Thus, in Figure 8 . 1 (b) we show an MBR
that can be used to represent this lake. Alternatively, in Figure 8. l (c) we could represent
it by a set of smaller rectangles. This option can provide a closer fit to the actual object,
but it requires multiple MBRs. An MBR can easily be represented by the coordinates for
two nonadjacent vertices. So we could represent the MBR in Figure 8 . 1 (b) by the pair
{(X J , Y J ) , (x2, Y2) } . There are other ways to store the MBR values, and the orientation of
the MBRs need not be with the axes.
We use the triangle shown in Figure 8.2(a) as a simple spatial object. In
Figure 8.2(b) we show an MBR for the triangle. Spatial indices can be used to assist
in spatial data mining activities. One benefit of the spatial data structures is that they
cluster objects based on location. This implies that objects that are close together in the
(a) Lake
<XJ> Yt>
F I G U R E 8. 1 : MBR example.
224
C h a pter 8
Section 8.2
Spatial Mi ning
(a) Triangle
FIG U R E 8.2:
11 1J.
7 8
18 17
15 16
225
level in a counterclockwise direction starting at the upper right quadrant (as shown in the
figure). Square 0 is the entire area. Square 1 is the upper right at level one. Square 15
is the square in the lower left comer at the second level. In this figure, the triangle is
represented by squares 1 2, 13, and 14 because it intersects these three regions. The quad
tree for this triangle is shown in Figure 8.3(b). Only nodes with nonempty quadrants are
shown. Thus, there are no nodes for quadrants 1 and 4 and 1their subquadrants.
MBRs are similar to the quadrants in the quad tree except that they do not have to
be of identical sizes. If hierarchies of MBRs exist, they do not have to be regular as in
the quadrant decompositions.
Ll
L3l
L:C=::::J
10 9
19 20
4
3
(a) Representing triangle with quadrants
F I G U R E 8.3:
n-dimensional space tend to be stored close together in the data structure and on disk.
Thus, these structures could be used to reduce the processing overhead of an algorithm
by limiting its search space. In effect, filtering is performed as you traverse down a tree.
In addition, spatial queries can be more efficiently answered by use of these structures.
A
D
B
Quad Tree. One of the original data structures proposed for spatial data is that of
a quad tree. A quad tree represents a spatial object by a hierarchical decomposition of the
space into quadrants (cells). This process is illustrated in Figure 8. 3(a) using the triangle
in Figure 8.2. Here the triangle is shown as three shaded squares. The spatial area has
been divided into two layers of quadrant divisions. The number of layers needed depends
on the precision desired. Obviously, the more layers, the more overhead is required for
the data structure. Each level in the quad tree corresponds to one of the hierarchical
layers. Each of the four quadrants at that layer has a related pointer to a node at the
next level if any of the lowest level quadrants are shaded. We label the quadrants at each
c
(b) R-tree
R-tree example.
226
Chapter 8
Section 8.3
Spati al M i n i n g
n
u
tree is used to index one of the attributes. We illustrate the use of the k-D tree assunring
a two-dimensional space. Each node in the tree represents a division of the space into
two subsets based on the division point used. in addition, the division alternates between
the two axes.
In Figure 8.5 we show a k-D tree using the same data we used for the R-tree. As
with the R-tree, each lowest level cell has only one object in it. However, the divisions
are not made using MBRs. Initially, the entire region is viewed as one cell and thus the
toot of the k-D tree. The area is divided first along one dimension and then along another
dimension until each cell has only one object in it. In this example, we see that the entire
region, A, is first divided into two cells (B, C) along the horizontal axis. Then, looking
at B, we see that it is divided into D and E. D is finally divided into H and I.
8.2.3
Thematic Maps
Image Databases
In image databases the data are stored as pictures or images. These databases are used
in many applications, including medicine and remote sensing.
Some early classification work performed using large image databases looked at
ways to classify astrononrical objects. One of the applications of this work is to identify
227
volcanos on Venus from images taken by the Magellan spacecraft [FWD93]. This system
consisted of three parts: data focusing, feature extraction, and classification. The first
component deternrines which of the areas of the images is the most likely to contain
volcanos. Here the intensity of a central point of a region is compared with that of the
background. The important features of these areas are extracted and stored in the second
part. The focusing portion compares the intensity of a central point of a region with
that of the background. During the second phase, interesting features are identified and
extracted. Finally, these features are classified based on classifiers built using training
data provided by domain experts. The third portion uses a decision tree to perform the
actual classification. The tree is created using ID3 and training examples provided by
experts. An accuracy of 80% was achieved.
A related work also used decision trees to classify stellar objects [FS93]. As with
the volcano work, the first two steps were to identify areas of the images of inter
est and then to extract information about these areas. Multiple trees were created, and
from these sets of rules were generated for classification. Accuracy was found to be
approximately 94% . When compared to several neural network approaches, the decision
tree/rules approach was found to be much more accurate. Both of these studies found the
need to normalize the extracted features to compensate for differences between different
images. For example, two images could differ based on the angle at which the image
was taken.
A
H
8.3
Operations needed to support spatial data nrining involve those required for spatial
databases. We review some of these in this section. In these discussions, we assume
that A and B are spatial objects in a two-dimensional space. Each object can be viewed
as consisting of a set of points in the space: (xa , Ya ) E A and (xb , Yb) E B .
As defined in [EFKSOO], there are several topological relationships that can exist
between two spatial objects. These relationships are based on the ways in which two
objects are placed in a geographic domain:
Disjoint:
A is disjoint from
Overlaps o r intersects:
is also in
A overlaps with
B.
Equals:
A equals
that
B.
Covers o r contains:
A contains
A is contained in
that are not in A .
B iff B
is contained in A.
While data nrining tasks may not specifically address these relationships, the similarity
between spatial objects certainly can be defined based partially on these relationships.
Based on the placement of the objects in the space, relationships with respect
to direction may be defined. These usually are defined by adding the traditional map
orientations to the space. Thus, we have the relationships such as north, south, east,
west, and so on. What makes these relationships difficult to identify is the irregular
shape of spatial objects and the fact that they may overlap.
Chapter 8
228
Minimum:
dis (A , B) =
min
(8 . 1 )
EXAMPLE 8.1
Maximum:
dis (A , B) =
Average:
Center:
max
(Xa ,ya)EA,(xb,Yb)EB
Suppose that a computer science student wishes to identify apartments close to the SMU
Computer Science and Engineering (CSE) Department. A given database listing available
apartments in the Dallas metroplex will contain many apartments nowhere near the
SMU campus. An initial filtering of the inappropriate elements can be made by finding
apartments that are "generalized close" to the CSE Department. This can be performed
at any of the levels in the concept hierarchy, Figure 8.6 shows the idea. The closest
apartments to SMU probably would be in the Park Cities. By filtering out all apartments
in all subtrees other than those for the Park Cities, apartments that are fairly close to
SMU would be found. Suppose that a lower level in the concept hierarchy existed that
included zip code. If apartments in the same zip code as the CSE Department were found,
an even finer estimate of close could be used. This process quickly filters out apartments
that could not possibly be used to answer the question. Here a coarser predicate is first
used to filter out potential answers. This predicate can be recursively refined until the
precise answers are found. Note that when looking at the concept hierarchy, the coarser
predicates can be applied to the MBRs at the higher levels, while the finer predicates are
applied at the lower levels.
(8.2)
(8.4)
where (Xea , Yea) is a center point for object A and (Xeb , Yeb) for B .
Note th si larity to distance measures used i n clustering. I n fact, you can think of
t e spatial obJect as a clu ter f the points within it. The center points used for the last
distance formula can be Identified by finding the geometric center of the ob' ect . por
.
.
exap1e, I' f an MB R IS used, the distance between objects could be found using the
.
Euclidean distce between the center of the MBRs for the two objects.
.
Spatial
bJects may be retrieved based on selection, aggregation, or join-type opera.
I
ns.
A
selectiOn
ay b performed based on the spatial or nonspatial attributes. Retriev
mo
as on satlal attnbutes could be performed using one of the spatial operators. A
spatial JOin retneves based on the relationship between two spatial objects.
8.4
The us of a concept hierchy shows le"els of relationships among data. When applied
.
to satlal data chactenstics, concept hierarchies allow the development of rules and
.
relatwships at differnt levels in the hierarchy. This is similar to the use of roll up
and nll down opertlns in O AP.
e have also seen this idea used in generalized
.
.
assocmtwn rules. A Siillllar Idea
IS used m the generalization and specialization concepts
.
.
found m macne learning In these cases, however, the hierarchy is not necessarily
:
related t sati l data. Spatial data mining techniques have involved both generalization
and specializatiOn type approaches.
8.4. 1
8.4.2
Generalization
"W_
Forth Worth
Dallas
Arlington
Mid-cities
Progressive Refinement
229
higher levels in the tree structure, and lower-level entries provide more precise descrip
tions of the spatial objects. Progressive refinement can be viewed as filtering out data
that are not applicable to a problem.
With progressive refinement, the hierarchical levels are based on spatial relation
ships. Example 8 . 1 illustrates the idea of progressive refinement. Here spatial relationships
can be applied at a more coarse (move up the hierarchy ) or more fine (move down the
hierarchy) level.
As mentioned in Chapter 3, the Euclidean and Manhattan measures are often used
to measure the d1stance b etween two pomts. The distance between two spatial obects
.
can be defined as extensiOns to these two traditional definitions:
Section 8.4
Spatial M i n ing
Preston Hollow
M Streets
Lakewood
East
Northern suburbs
Park cities
University Park
Highland Park
230
Chapter 8
Spatial Mining
Section 8.4
hierarchy. Generalization can be petformed using either of these two hierarchies. When the
spatial data are generalized, the nonspatial data must be appropriately changed to reflect
the nonspatial data associated with the new spatial area. Similarly, when the nonspatial
data are generalized, the spatial data must be appropriately modified. Using these two types
of hierarchies, generalization as applied to spatial data can be divided into two subclasses:
spatial data dominant and nonspatial data dominant [LH093]. Both of these subclasses can
be viewed as a type of clustering. Spatial data dominant does the clustering based on spatial
locations (so that objects close together are grouped), whereas nonspatial data dominant
clusters by similarity of nonspatial attribute values. These approaches are referred to as an
attribute-oriented induction because the generalization process is based on attribute values.
With spatial data dominant generalization, generalization is first applied to the
spatial data, and then the related nonspatial attributes are modified accordingly. General
ization is petformed until a threshold number of regions is reached. For example, deter
mining the average rainfall in the southwestern United States could be done by finding
the mean average rainfall for all states shown to be in the Southwest by a spatial hier
archy. Thus, the spatial hiearchy determines which lower-level regions are found in the
higher-level region being queried. Determining how to apply the generalization to the
nonspatial data is, however, not always a straightforward aggregation operation. Deter
mining the average rainfall in this case actually treats each state the same. However, a
weighting by geographic area might be used to provide a more accurate average rainfall
for the higher-level region being queried.
An alternative approach is to generalize the nonspatial attribute values as well.
Generalization is based on grouping of data. Adjacent regions are merged if they have
the same generalized values for the nonspatial data. Suppose that instead of average
rainfall values, we simply returned values that represented the southwestern cluster. We
could assign values of heavy, medium, light, and so on to describe the rainfall rather than
providing actual numeric values . Algorithm 8 . 1 shows the spatial-dominant approach. A
threshold that indicates the maximum number of regions may be given. Based on this
threshold, the correct level in the hierarchy is chosen, and thus the number of regions is
determined.
ALGORITHM 8.1
Input :
D
/ / Spat i a l database
/ / Sp a t i a l hierarchy
/ / Concept hierarchy
/ / Query
Output :
R
I / Ru l e that s t a t e s the general chara c t e r i s t i c s reque s t e d
SPATIAL - data- dominant algorithm:
d = s e t o f d a t a obtained f rom
Fo l l owing the
D b a s e d on
structure o f H,
s e l e c t i on c r i t e r i a i n q ;
the
requested l evel
is
found
in H i s obt a i ne d ;
a rule that
summa r i z e s
the resul t s
found ;
231
Nearest Neighbo r
STI NG
232
C h a pter 8
Spati al M i n i ng
Section 8.5
(b) Level 2
(a) Level l
F I G U R E 8.7:
(c) Level 3
ALGORITHM 8.3
Inpu t :
/ / Tree
ALGORITHM 8.2
/ / Data
to be p l a c e d in
/ /Number of
de s i red c e l l s
at
the
l owest
s t ructure
level
i = 1
/ / Tree
Create
i = 1;
for
init ial i z e d ;
un t i l
each node
k;
in l evel
i do
with
each
i t em in D do
determine
l e a f node j a s s o c i at e d w i t h the p o s i t i on of
update values o f j based o n a t t ribute values in i t em
'
:= l og4 (k) ;
i := i - 1 ;
D;
repeat
f o r each node
J ln level i
in i t s
unt i l
do
4 chi l dren ;
update values of
this
i = 1;
The actual STING algorithm is shown in Algorithm 8.3. The algorithm assumes that
a query, q, that can be answered from the stored statistical information in the constructed
tree, T , is requested. Such a query might be to find the range of price of apartments
near SMU . The statistics (minimum and maximum) of the apartment rental prices for
.
the appropnate
cells should be determined. The cell that SMU is in would determine
the actual values for those closest to SMU. In addition, the query might retrieve the
do
cell
i s rel evant
to
all
layers
in the
rel evant
cells
to
create
cell s ;
Calculating the likelihood that a cell is relevant to a query is based on the percentage
of the objects in the cell that satisfy the query constraints. Using a predefined confidence
interval, if this proportion is high enough, then that cell is labeled as relevant. The
statistical information associated with these relevant cells is used to answer the query. If
this approximate answer is not good enough, then the associated relevant objects in the
database may have to be examined to provide a more exact response. The cells found by
STING approximate those found by DBSCAN. Cells that are found to be close enough
to relevant cells are included in the regions of cells that are found by the algorithm.
i n i t i a l value s ;
i = i + 1;
4
in l evel
if
ident i f y neighboring c e l l s
regions o f
create
cells
i = i + 1
/ / I n i t i a l ly only
repeat
unt i l
each node
det ermine
root node
for
r e l evant
repeat
T = root
/ / Regi ons o f
STING algor i t hm :
Output :
for
/ / Query
Output :
Input :
//
233
information for the cells surrounding this cell or perhaps at the next highest level in the
tree that contains the cll where SMU is located. Th nearby cells could be determined
using some distance function. The crucial concept here is that the appropriate cells must
be determined and then the information from those cells, in the constructed tre must be
retrieved. A breadth-first tree traversal is used to examine the tree. However, a complete
traversal of the tree is not performed. Only children of relevant nodes are examined.
Here the concept of relevance is much like that with IR queries except that relevance is
determined by estimating the proportion of the objects in that cell that meet the query
conditions. The complexity of the STING algorithm is O (k) where k is the number of
cells at the lowest level. Obviously, this is the space taken up by the tree itself. When
used for clustering purposes, k would be the largest number of clusters created.
is divided into two parts. The first part creates the hierarchy and the second part fills in
the values. Since the number of nodes in the tree is less than the number of items in the
database, the complexity of STING BUILD is O (n ) .
D
k
Spatial Rules
8.5
SPATIAL RU LES
Spatial rules can be generated that describe the relationship between and structure of
spatial objects. There are three types of rules that can be found during spatial data
mining [KAH96] . Spatial characteristic rules describe the data. Spatial discriminant
rules describe the differences between different classes of the data. They describe the
features that differentiate the different classes. Spatial association rules are implications
of one set of data by another. The following examples illustrate these three types of rules:
Discriminant rule: In Dallas the average family income is $50,000, while in Plano
the average family income is $75,000.
Chapter 8
234
Spatial M i n i n g
Association rule: In Dallas the average family income for families living near
White Rock Lake is $1 00,000.
Nonspatial antecedent and spatial consequent: All elementary schools are located
close to single-family housing developments.
Spatial antecedent and spatial consequent: Any house that is near downtown is
south of Plano.
Section 8.5
Spati a l Rules
235
be defined by a hierarchy that shows that g_close_to contains close_to as well as other
predicates (such as contains and equal). A first step in d termining satisfiability of the
close_to predicate would be to look at a coarse evaluation of g_close_to. The cose
evaluation is used as a type of filter to efficiently rule out objects that could not posstbly
satisfy the true predicate. The coarse predicate coarse_g_close_to is satisfied by objects
if their MBRs satisfy g_close_to. Only those objects that satisfy coarse_g_close_to are
examined to see if they satisfy g_close_to.
The five-step algorithm is outlined in Algorithm 8.4. It is assumed that a ata
mining query is input. The query contains selection informat on that s used to rtneve
the objects from the database that are of interest. The topologtcl predtcates e ng the
spatial relationships of interest are also input. Using these predtcates, P, an tmttal table
.
is built C p that identifies which pairs of objects satisfy P at a coarse level. The mput
minim m s pports are actually a set of support values to be used at different levels in
the processing. s [ l] is the support level to be used at the coarse filtering level. Af er th s
filtering, the pairs of objects that satisfy the coarse predicates are counted to see tf therr
support is above the minimum. In effect, this frequent coarse predicate (FCP) database
.
is the set of large one-itemsets. The predicates in FCP are then exammed to find the
.
frequent predicates at a fine level (FFP). The last step expands these frequ nt pre tates
of size 1 to all arbitrary predicate sizes and then generates the rules as wtth tradttlonal
association rules. This is performed similarly to Apriori. By finding the FCRs first, the
number of objects to be examined is reduced at the last step.
ALGORITHM 8.4
Input :
D
c
s
Ci
/ / Data ,
/ / Concept hierarchies
/ / Minimum support for leve l s
/ / Conf idence
q
p
of
int e r e s t
Output :
SPATIAL associat ion ru l e a l gorithm:
CP
11
q(D) ;
CP
to d ;
FFP
FCP
from
by f i nding
FCP ;
236
Chapter 8
Section 8.7
Spatial M i n ing
the algorithm can be used to generate multilevel association rules if desired or rules at a
coarse level rather than a fine leveL
8.6
103 Extension
The concept of neighborhood graphs has been applied to perform classification of spatial
objects using an ID3 extension [EKS 97]. A neighborhood graph is a graph constructed
from the objects in the space. Each object becomes a node in the graph. The edges
are constructed from the neighbors; that is, two nodes are connected by an edge in the
neighborhood graph if 9ne is a neighbor of the other. "Neighbor" can be defined based
on any relationship between the spatial objects such as distance less than a particular
threshold, satisfiability of a topological relationship between the objects, or direction
relationship. Note that some of the relationships are order relationships and others are not.
The idea of the algorithm is to take into account the objects that are near a given
object. A max-length indicator is input that specifies the maximum length of a neighbor
hood path starting at the node. This then identifies a set of nodes that are associated with
the target hade. ID3 then considers for classification purposes not only the nonspatial
attributes of the target object, but also those in neighboring objects.
8.6.2
ALGORITHM 8.5
Input :
D
/ / Da ta ,
/ / Concept h i e rarchi es
Output :
/ / B inary de c i s i on tree
samp l e S of
data
from
ident i fy t h e be s t pred i c a t e s
D
p
One spatial classification technique builds decision trees using a two-step process similar
to that used for association rules [KHS98]. The basis of the approach is that spatial objects
can be described based on objects close to them. A description of the classes is then
assumed to be based on an aggregation of the most relevant predicates for objects nearby.
To construct the decision tree, the inost relevant predicates (spatial and nonspatial)
are first determined. It is hoped that this process will create smaller and more accurate
decision trees. These relevant predicates are the ones that will be used to build the
decision tree. It is assumed that a training sample is used to perform this step and that
weights are assigned to attributes and predicates. Initial weights are 0. Two corresponding
objects are examined for each object. The nearest miss is the spatial object closest to the
target object that is in a different class. The nearest hit is the closest target in the same
class. For each predicate value in the target object, if the nearest hit object has the same
value, then the weight of that predicate is increased. If it has a different value, then the
weight is decreased. Likewise, the weight is decreased (increased) if the nearest miss
has the same (different) value. Only predicates with positive weights above a predefined
threshold are then used to construct the tree. It is proposed that, because of the complexity
of finding the relevant predicates, relevant predicates be found first at a coarse level and
then at a finer leveL MBRs, instead of actual objects, and a generalized coarse close_to
relationship are first used to find the relevant predicates. Then these relevant predicates
and the true objects are used during the second pass.
237
For each object in the sample, the area around it, called its buffer, is examined.
A description of this buffer is created by aggregating the values of the most relevant
predicates of the items in the buffer. Obviously, the size and shape of the buffer impact
the resulting classification algorithm. It is possible, although unrealistic, to perform an
exhaustive search around all possible buffer sizes and shapes. The objective would be to
choose the one that results in the best discrimination between classes in the training set.
This would be calculated using the information gain. Other approaches based on picking
a particular shape were examined, and the authors finally used circles (equidistance
buffers).
To construct the tree, it is assumed that each sample object has associated with it a
set of generalized predicates that it satisfies. Counts of the number of objects that satisfy
(do not satisfy) each predicate can then be determined. This is then used to calculate
information gain as is done in ID3. Instead of creating a multiway branching tree, a
binary decision tree is created. The resulting algorithm to construct the decision tree is
shown in Algorithm 8.5.
Spatial classification problems are used to partition sets of spatial objects. Spatial objects
could be classified using nonspatial attributes, spatial predicates (spatial attributes), or
spatial and nonspatial attributes. Concept hierarchies may be used, as may sampling. As
with other types of spatial mining, generalization and progressive refinement techniques
may be used to improve efficiency.
8.6. 1
us ing
8.7
and
Spatial clustering algorithms must b e able to work efficiently with large multidimen
sional databases. In addition, they should be able to detect clusters of different shapes.
Figure 8.8 illustrates what we mean. This figure shows clusters in a two-dimensional
space. Obviously, by looking at this figure it is easy to see that there are four different
clusters, each of a fairly irregular shape. A good spatial clustering algorithm should be
able to detect these four clusters even though the shapes are not regular, and some points
in one cluster may actually be closer to some points of other clusters rather than to points
in its own cluster. An algorithm that works using centroids and simple distance measures
probably will not be able to identify the unusual shapes.
Other desirable features for spatial clustering are that the clusters found should be
independent of the order in which the points in the space are examined and that the
clusters should not be impacted by outliers. In Figure 8.8 the outliers in the lower right
part of the figure should not be added to the larger cluster close to them.
Many of the clustering algorithms discussed in Chapter 5 may be viewed as spa
tial. In the following sections, we evaluate additional algorithms specifically targeted to
spatial data.
238
Chapter 8
Spat i a l M i n i ng
=-- . . . .
. > ...,
..
Section 8.7
, .... , ... . ..
. .. , ..
. ';
}/;.': ; ')J-:':.
'
r
:: . . ,:
":' .:...
I.
W :
239
'
),:
.
. . . . : ! : n ; ::,: : :: . .
;:/:;mmv.;. :
...
8.7.2
SD(CLARAN S)
Spatial dominant CLARANS [SD(CLARANS)] assumes that items to be clustered contain
both spatial and nonspatial components. It first clusters the spatial components using
CLARANS and then examines the nonspatial attributes within each cluster to derive a
description of that cluster. For example, clustering of vegetation in remote areas may
8.7.1
CLARAN S Extensions
The main memory assumption of CLARANS is totally unacceptable for large spatial
databases. Two approaches to improve the performance of CLARANS by taking advan
tage of spatial indexing structures have been proposed [EKX95] .
The first approach uses a type of sampling based on the structure of an R*-tree (an
R-tree variant). To ensure the quality of the sampling, the R *-tree is used to guarantee
that objects from all areas of the space are examined. The most central obj ect found in
each page of the R *-tree is used to represent that page in the search. The most central
object is the object (of all objects stored on that page) with the smallest distance from
it to the center of the page. Remember that the page is actually the MBR that contains
all the objects in that page. So the center of that MBR can be defined as the geometric
center of the bounding rectangle. CLARANS is then used to find clusters for these central
objects. The k medoids found in this step represent the k clusters to be found for the
database as a whole. Since the R *-tree clusters objects that are spatially near on a node
in the tree (and thus page), it is reasonable to believe that this approach to sampling finds
good medoids.
The second technique improves on the manner in which the cost for a medoid
change is calculated (see Formula 5 . 1 0 in Chapter 5.). Instead of examining the entire
database, only the objects in the two affected clusters must be examined. A region
query can be used to retrieve the needed objects. An efficient technique to retrieve only
the objects in a given cluster is based on the construction of a polyhedron around the
cluster medoid. The constructed polyhedron is called the Voronoi polyhedron or Voronoi
This polyhedron is created by constructing perpendicular bisectors between
diagram.
pairs of medoids. This process is illustrated in Figure 8.9. This then defines the cluster.
The objectives within a Voronoi diagram are closer to the medoid of that polyhedron
than to any other.
find that one area (cluster) is predominantly a forest of pine trees, while another contains
massive open plains and grassy areas. SD(CLARANS) assumes that some learning tool,
such as DBLEARN [HCC92], is used to derive the descliption of the cluster. This
description can be viewed as a generalized tuple; that is, by using a concept hierarchy,
the attribute values for the set of tuples in a cluster can be generalized to provide summary
values at a higher level in the hierarchy. The learning tool performs this task. Algorithm
8.6 outlines the SD(CLARANS) algorithm. Note that it is a combination of CLARANS,
DBLEARN, and the spatial-dominant algorithm discussed earlier in this chapter. It also
assumes that in the first step an initial filtering of the da1ta using a relevance based
on the nonspatial data is performed. Any clustering algorithm could be used in place
of CLARANS in this algorithm. In our algorithm we show that the number of desired
clusters is input. However, the authors of the original version propose an approach to
determine the "most natural number of clusters" [NH94] .
ALGORITHM 8.6
Input :
D
/ /Data to be c l u s t e red
k
/ / Number o f de s i red c e l l s at the l owe s t l eve l
Output :
K
/ / S e t of c lu s t e r s
SD ( CLARANS ) algori thm :
I I Find s e t of tup l e s that s at i s fy s e l e c t ion c r i t e r i a .
d = s e l e c t tup l e s from D based on nonspat i a l s e l e c t ion c r i t e r i a ;
/ / App ly CLARANS to
K=
b a s e d on s p a t i a l
att ribut e s .
CLARANS(d) ;
for each k E K do
app l y DBLEARN to
in
k;
240
Chapter 8
Spatia l M i n i ng
Section 8.7
m>
ALGORITHM 8.7
Input :
/ / Spat ial
obj ects
Outpu t :
to be c l ustered
DBCLASD algori t hm :
k = 0 ; I I Ini t i a l ly there a r e n o c l u s t e r s .
/ /Set
of
c l usters
expand
c;
C;
BANG
The BANG approach uses a grid structure similar to a k-D tree. The structure adapts to
the distribution of the items so that more dense areas have a larger number of smaller
grids, while less dense areas have a few large ones. The grids (blocks) are then sorted
based on their density, which is the number of items in the grid divided by its area. Based
on the number of desired clusters, those grids with the greatest densities are chosen as
the centers of the clusters. For each chosen grid, adjacents grids are added as long as
their densities are less than or equal to that of the current cluster center.
(8.5)
Here N is the number of points in the cluster and A is its area. The added points then
become new candidates.
The area of the cluster is estimated by using grids that enclose the cluster with a
polygon. When a point is added to a cluster, the grid containing that point is added to the
polygon. The closeness of the polygon to the real shape of the cluster depends on the size
of the grids. If the grids are too large, the shape may not approximate the cluster well.
If they are too small, the cluster could actually be estimated by disconnected polygons.
The grid length is chosen to be the largest value in the nearest neighbor distance set.
The algorithm DBCLASD is shown in Algorithm 8.7. Since the x2 test usually
requires at least 30 elements, the authors assumed that 29 neighboring points are initially
added to each cluster [XEKS 98] . The last step expands a cluster based on the expected
distribution of the nearest neighbor distance set of C using the candidates found in c. Each
candidate is added one at a time to C, and the distribution of the nearest neighbor distance
set is estimated. If it still has the desired distribution, the points in the neighborhood of
this candidate are added to the set of candidates; otherwise the candidate is removed
from C. This process continues until c is empty. The points in the neighborhood of a
given point are determined based on the radius value stated above.
241
each p oint
been pro c e s s e d to
A recent spatial clustering algorithm based on DB SCAN has been proposed that is called
DBCLASD (Distribution Based Clustering of LArge Spatial Databases). DBCLASD
assumes that the items within a cluster are uniformly distributed and that points out
side the cluster probably do not satisfy this restriction. Based on this assumption, the
algorithm attempts to identify the distribution satisfied by the distances between nearest
neighbors. As with DB SCAN, a cluster is created around a target element. Elements are
added to a cluster as long as the nearest neighbor distance set fits the uniform distribution
assumption. Candidate elements are determined and then are added to the current cluster
if they satisfy a men\bership criteria. Candidate elements are determined by executing a
region query \,Ising a circle of radius m centered around a point p that was just added to
the cluster; m is chosen based on the following formula:
0;
if
DBCLASD
for
8.7.5
WaveCiuster
The WaveCluster approach to generating spatial clusters looks at the data as if they were
signals like STING, WaveCluster uses grids. The complexity of generating clusters is
O (n) and is not impacted by outliers . Unlike some approaches, WaveCluster can find
arbitrarily shaped clusters and does not need to know the dtesired number of clusters. A
set of spatial objects in an n-dimensional space are viewed! as a signal. The boundaries
of the clusters correspond to the high frequencies. Clusters themselves are low-frequency
with high amplitude. Signal processing techniques can be used to find the low-frequency
portions of the space. The authors propose that a wavelet transform be used to find the
clusters. A wavelet transform is used as a filter to determine: the frequency content of the
signal. A wavelet transform of a spatial object decomposes it into a hierarchy of spatial
images. They can be used to scale an image to different sizes.
8.7.6
Approximation
Once spatial clusters are found, it is beneficial to determine why the clusters exist; that
is, what are the unique features of the clusters? Approximation can be used to identify
the characteristics of clusters. This is done by determining the features that are close
to the clusters. Clusters can be distinguished based on features unique to them or that
are common across several clusters . Here, features are spatial objects such as rivers,
oceans, schools, and so on. For example, some clusters may be unique partly because
they are close to the ocean or close to good schools. It usually is assumed that features
and clusters are represented by more complex closed polygons than by simple MBRs.
242
Chapter 8
Spatial M i n i n g
Section 8.9
8.8
Isothetic rectangle: MBR containing a set of points where the sides of the rectangle
are parallel to the coordinate axes.
Encompassing circle: Circle that contains a set of points; found by using the
diagonal of the ti.sothetic rectangle as its diameter.
What makes these shapes efficient is that given a set of n points, the first two points
can be found in 0 (n) time and the last in 0 (n lg n) time. Each type of geometric shape
is viewed as a bounding structure around a feature. These three types of encompassing
geometric shapes are used as multiple levels of filtering of the possible close features.
These are used in order of increasing accuracy and decreasing efficiency. The concept of
using these three types of bounding polygons is shown in Figure 8. 10, which illustrates
a school. The school is fairly accurately represented by a convex hull, but less accurately
represented by a rectangle and a circle. The objective is to obtain a balance between
accuracy and efficiency in identifying the relationships.
The first step in CRH is to apply the encompassing circle. The features (using
the circular approximation) that are ranked the highest (those that are viewed to be the
I
I
I
Circle
Rectangle
Convex hull
School
F I G U R E 8 . 1 0:
...
',
',
',
- - - - - - :
CRH polygons.
243
closest) to a given cluster are then sent to the filter at the next level. At this level the
isothetic rectangle is used to represent the features, and the features are again ranked
based on proximity to the cluster. The highest ranking features at this level are examined
at the final level, where a convex hull bounding polygon is used to estimate each feature.
This approach is used for each cluster. The desired number of features identified at each
level is indicated as input to the algorithm. Although different techniques can be used to
rank the features, intersection may be used or actual distances may be calculated. The
CRH algorithm uses various optimization features to reduce the overall complexity and
to eliminate redundant computation of distances.
::
B i b l iographic Notes
EXERCISES
1. (esearch) Compare the R-tree to the R*-tree.
2. (Research) Another commonly used spatial index is the grid file. Define a grid
file. Compare it to a k-D tree and a quad tree. Show the grid file that would be
used to index the data found in Figure 8.5.
8.9
BI BLIOGRAPH IC NOTES
Most spatial data structures were proposed many years ago. Quad trees were first intro
duced to handle queries on composite keys [FB74] . The k-D tree was proposed in
[Ben75] . There have been many variation of the k-D tree for use with spatial data
data [OSDH93]. The grid file was proposed in [NH84].
There have been many excellent surveys examining spatial data and spatial data
structures. A survey of spatial and multimedia data, including indexing, can be found in
[ZCF+97] . One unpublished survey of spatial indexing techniques not only provides a
taxonomy of the approaches, but also identifies the strengths and weaknesses of the vari
ous techniques [OSDH93]. Nievergelt and Widmayer have written an extremely easy-to
read yet thorough survey of spatial data structures with an excellent bibliography [NW97].
Other surveys of spatial data structures are available [Sam95a, GG98]. This last survey
[GG98] is an extensive examination of multidimensional indexing techniques, including
spatial and nonspatial. It includes a comparison of the various techniques. Additional sur
veys look at query processing of spatial data [Sarn95b]. A more general survey [Gtit94]
covered spatial data modeling, querying spatial databases, spatial indexing, and archi
tectural approaches. Spatial relationships based on direction are examined in [EFKSOO].
The original proposal for R-trees can be found in [Gut84] . The R*-tree is a more efficient
improvement on the R-tree [BKSS90]. Many extensions to the basic R-tree have been
proposed [OSDH93] . The STING approach was proposed in [WYM97].
There also exist some surveys of spatial data mining. [EFKSOO] contains a survey
of the algorithms, relationships, and operations needed to support spatial data mining.
The concept of progressive refinement has been studied extensively in a recent doctoral
dissertation [Kop99]. A recent book [MHOl ] is a collection of many different spatial data
mining articles.
Articles that provide an overview of clustering in spatial databases can be found
in [EKSX98], [HKTO l], and [NH94]. In fact, many of the clustering techniques intro
duced in Chapter 5 can be viewed as spatial: K-means and K-medoids and CLARANS.
DBCLASD was proposed in [XEKS98]. WaveCluster was examined in [SCZ98]. Aggre
gate proximity is defined in [KN96] . However, the authors of the original version
244
Chapter 8
Spatial M i n ing
C H A P T E R
proposed an approach to determine the "most natural number of clusters" [NH94] based
on a concept called silhouette coefficients [KR90].
Some of the earliest work on spatial classification was found in [FWD93] . Here
decision tree techniques were used to categorize objects in the sky. Specifically, stars
Te m po ra l M i n i ng
9.1
DBSCAN (GDBSCAN), which clusters objects using both spatial and nonspatial attributes
9.2
[SEKX98]. It was examined in astronomy, biology, earth science, and geography appli
cations.
A spatial data mining query language based on SQL, GMQL (geo-mining query
9.3
language), has been proposed [Kop99]. This is based on DMQL and is used in DBMiner
9.5
9.4
9.6
and GeoMiner.
9.7
9.8
9.1
INTRODUCTION
MODELING TEMPORAL EVENTS
TIME SERIES
PATIERN DETECTION
SEQUENCES
TEMPORAL ASSOCIATION RULES
EXERCISES
BIBLIOGRAPHIC NOTES
INTRODUCTION
Databases traditionally do not contain temporal data. Instead, the data that are stored
reflect data at a single point in time. Thus, it may be called a snapshot database. For
example, an employee database normally contains only the current company employ
ees rather than all employees who have ever worked for the company. However, many
questions cannot be answered by this snapshot data. A company CEO might wish to
determine trends in the hiring and firing of etiJ.ployees, or he might wish to obtain infor
mation about the ethnic diversity of employees and how it has changed over time. These
types of data mining questions require temporal data. In a temporal database, data are
maintained for multiple time points, not just one time point. Example 9. 1 illustrates the
use of a temporal database that stores employee data. Obviously, storing three separate
tuples for one employee with so much redundant information is not efficient, and tech
niques can be used to eliminate this redundancy. However, this illustrates the concept.
Each tuple contains the information that is current from the date stored with that tuple
to the date stored with the next tuple in temporal order.
EXAMPLE 9.1
XYZ Corp. uses a temporal database to store employee information. I t maintains the
Social Security number (SSN), employee name, address, and salary for each employee.
When a new tuple is stored in the database, the current date is added to this infor
mation. Joe Smith is hired on 2/12/02 at a salary of $50,000. On his six-month per
formance evaluation, he is given a $2,000 raise and a promotion. On 12/10/02, he
moves to a new address. At the end of 2002, there are three tuples in the database
for Joe Smith:
245
246
Chapter 9
Date
Temporal M i n i ng
Section 9. 1
Name
SSN
Address
Salary
Joe Smith
Joe Smith
Joe Smith
123456789
123456789
123456789
10 Moss Haven
10 Moss Haven
13 Chesterton
$50,000
$52,000
$52,000
I ntroduction
247
the last example, these values were 1 / 1 /0 1 and 12/3 1/01, Jrespectively. The time range
then exists for the query as well as for the data. Suppose that y d
[tf , t1J is the valid
=
2/12/02
811 2/02
1 2/10/02
time range for a tuple. Special temporal queries then can involve various combinations
of these two ranges :
Intersection query: A tuple is retrieved only i f its valid time range intersects that
of the query: yd n yq f= 0.
Analysis of temporal (or time-varying) data presents many interesting challenges .
For example, there may be many different interpretations for time. In Example 9 . 1 the
date stored in the record is the date representing when that information becomes current.
This is often called the valid time. The valid time for information is the time during
which the information is true in the modeled world. This usually consists of a start time
and an end time. The end time in the example is implied by the start time of the next
temporal record for the same employee. Another time that could have been used is the
transaction time. The transaction time is the timestamp associated with the transaction
that inserted this record. This could be different from the start time for the valid time
interval. The transaction time interval is the time the tuple actually existed in the database .
For example, Joe Smith may have indicated on 1 1 / 15/02 that he would have the new
address effective 1 2/ 1 0/02. The start valid time for the new address was 12/ 1 0/02, but
the transaction time was 1 1 / 15/02. Other types of times may be used as well. When the
employee information changes, a new tuple is inserted. Changes and deletes may occur
only to change data that was incorrectly inserted .
So far we have seen that temporal data often involve a duration of time; that
is, a start time and an end time. In this interpretation, the range of values [ts , te] is
associated with each record. Here ts is the start time and te is the end time. Different
temporal interpretations may be used. A timestamp of a specific time instance may be
used instead of a range. This is common with time series data where specific values
are associated with times. For example, a common time series is to show the price of a
specific stock at the stock market close each day. This stock quote is the price at that
specific point in time.
Many different examples for temporal data exist. Satellites continually collect
images and sensory data. This information is temporal and is associated with specific
points in time (when the data were obtained). In a hospital, printouts of heartbeats may
be kept for patients. This represents a continuous view of temporal data. When an EEG
is taken for a patient, several different brain waves are measured in parallel. Each wave
represents a continuous set of data over time.
Temporal databases usually do not accept the same types of updates and queries
as traditional snapshot databases. The only updates that are allowed are corrections and
versions. Actual modifications of tuples usually are not allowed. Instead, a new tuple with
a different valid time would be added. Temporal queries may involve fairly complicated
temporal selection criteria. For example, it would not make sense to ask for the salaries
of all employees. Instead, a temporal value would be needed such as: Find the salaries
for all employees on 7/9/0 1 . Or a more complicated range query could be asked: Find
the names of all employees who had a salary greater than $ 1 00,000 between 1 / 1 /0 1
and 12/3 1 /0 1 . A temporal query q involves a valid time range y q
[t.i , t%] i n the
request. Here t.i is the start time and ti is the end time of the query's time range. In
=
Inclusion query: A tuple is retrieved only if its valid time range is completely
:S
tf
:S
t1
:S
t% .
Containment query: A tuple is retrieved only if its valid time range contains that
of the query:
t.i
tf
:S
t.i
:S
t%
:S
t1 .
tf
ti
t%
t1 .
Snapshot: The database system provides no support for any temporal attribute.
The stored data usually are assumed to represent data that are valid at the current
time.
Transaction time: The only temporal data supported by the database is the time
associated with the transaction that inserted the data. This could be a timestamp
for when the transaction was committed (or perhaps was requested) or it could be
a range.
Valid time: This database supports a valid time range for the data. It may be
Bitemporal: A bitemporal databases supports both transaction time and valid time.
With temporal data, the concept of a key is complicated as well. In the salary
database, the employee SSN can no longer determine a unique tuple. Temporal informa
tion is needed as well.
As with spatial mining, several specialized data structures have been proposed
to assist in temporal mining. There are many specialized data structures that have been
proposed to index temporal databases that we do not discuss here. These usually are gen
eralizations of B+ -trees and are similar to those structures we saw with spatial databases.
One difference, of course, is that time is usually one dimension where as space may
be two or three dimensions. These structures usually assume that a valid time range is
associated with each tuple. One complicating factor is the use of the current time. Unlike
spatial data, the temporal dimension keeps expanding. Thus, for items that are currently
valid, it is not possible to have the current end time for the range. One solution to this
problem is to use a special time value called now, which is the current time. Thus, a
range that ends in "now" means that it is valid up to the current time. The resulting effect
is that time ranges that end in "now" actually keep expanding in size.
248
Cha pter 9
Temporal M i n ing
Mining of temporal data involves many of the conventional data mining activities
0.7
but, of course, is complicated by the temporal aspect and the more complicated types
0.3
0.5
of queries. For example, time series data may be clustered based on similarities found.
predicted. Association rules may involve temporal aspects and relationships. Web usage
mining discussed in Chapter 7 involved temporal data. The combination of spatial and
0.1
the string of characters "the." This can be viewed as a temporal sequence of events. Each
event recognizes one character. One of the earliest techniques used to model a sequence
of events was a finite s;ate recognizer (FSR) or finite state machine (FSM) . Figure 9. 1
illustrates an FSR for the sequence "the." The temporal aspect is implied by the arcs. It
can be viewed that the individual events (or characters) occur at specific time intervals.
While FSRs can be used to recognize a known sequence, they do not scale well
when the vocabulary is large. They also do not work well to model transitions between
states that are not precisely defined. Mmkov models and their variant, hidden Markov
models, extend the basic idea of an FSR but scale well and are more general. Figure 9.2
249
0.3
However, determining the similarity between two different sets of time series data is
difficult, as was shown in Chapter 1. Given a time series, a future value also may be
9.2
Section 9 . 2
0.5
0.6
always one state that is designated as the current state. A major property of a Markov
model is the Markov property, which states that, given the current state, the transition
probability is independent of any previous states. Thus, an MM is memoryless. A more
of transitioning from
state
v1
Vi to
v1 .
v1 and
shows a simple Markov model. Notice the similarity and differences between it and the
FSR in Figure 9. 1 . One of the major differences is that the transitions (arcs) are not
associated with specific input values. Just as with an FSM, a Markov model (MM) is a
directed graph that can be used to recognize a pattern. Each node is associated with a
and natural language processing are very common applications for MMs. Suppose that
state in recognizing a sequence of events (or pattern). Although our example shows a
start and end node, these nodes need not be present. One of the major differences is that
a transition (arc) is associated with a probability, transition probability. The probability,
or words. A sequence of these would be a phrase. Given a phrase, the probability that
that phrase occurs is the product of the probabilities from the start state to the end
Markov models have been used in many different applications. Speech recognition
an MM is created to model certain phrases. The individual nodes could represent sounds
state using the transitions associated with each word in sequence. In this manner, the
PiJ on an arc (i, } } is the probability that a transition will be made from state i to state
j. In Figure 9 .2, the probability of transitioning from state 1 to state 2 is 0.3, while that
most likely sequences can be found, and the most likely sequence is the one that is
is 1 . Any arcs not shown are assumed to have a probability of 0. The probabilities can
can be determined. You also could determine the probability of being in a particular state
at a particular time. Another application is in the area of system reliability. Here an MM
of staying in state 1 is 0.7. The sum of the weights on the edges coming out of a node
is used to model the system operation. The transition probabilities can be determined by
domain experts or learned from training data. The resulting model can be used to do
such things as determine system availability and predict the mean time between failures.
An extension to the MM that still satisfies the Markov property, is the hidden
Markov model (HMM) . A major difference between the MM and HMM is the fact that the
states in an HMM need not correspond to observable states. An HMM models a process
that produces as output a sequence of observable symbois. The HMM will actually output
these symbols. Given a sequence of symbols, the HMM can be constructed to produce
these symbols. What is hidden is the state sequence that produced these symbols. There is
no relationship between states and real-world observable values. An observation sequence
As with the MM, the HMM consists of a set of states with transition probabilities. In
addition, the HMM has associated with each state an observation probability distribution.
250
Chapter 9
Section 9.2
Temporal M i n ing
0.5
P(H )
P(T)
=
=
0.5
0.5
0.5
P(H)
P(T)
=
=
251
0.3
0.7
ALGORITHM 9.1
0.5
Input :
I IHMM
Output :
Size: Determining the number of states is not obvious. They need not be associated
with a real-world observable event.
Transition probabilities: Determining what the transition probabilities are is dif
ficult. Domain experts and/or learning algorithms can be used to determine the
probabilities, much as is done with NNs.
Hidden observation probabilities: As with the transition probabilities, these prob
1. Initial state distribution used to determine the starting state at time 0, vo.
2. Each arc (i.j) is labeled with a probability PiJ of transitioning from
This value i s fixed.
3. Given a set of possible observations, O { o1 , oz , . . . , ok } , each state,
Pi k } .
tains a set of probabilities for each observation, { Pi l , Pi 2
. ,
Vi
to vJ .
Vi ,
con
I I Output s e quence
S = (so, s1, . . . , sm- 1 )
HMM obs e rvation sequence algorithm :
t= O
Based on
initial
s t at e di s t r ibut i o n ,
de t e rmine
Vt i
repeat
Output
Choo s e
t = t+l;
unt i l t = k ;
1. Given a sequence of observed elements and an HMM, what is the probability that
the HMM actually produced the sequence? Note that this is associated with the
recognition problem. If the probability is low, then this model probably did not
produce it. As a result, the system that is modeled by the HMM probably did not
produce it.
2. Given a sequence of observed values and an HMM, what is the most likely state
sequence that produced this sequence?
3. How can the model parameters (transition probabilities, observation probabilities,
and starting state distribution) be improved? This problem is similar to that of how
learning is accomplished for a NN.
Relatively efficient algorithms have been proposed to solve all three of these problems.
Traditional feedforward neural networks cannot easily be used to model tempo
ral events because there is no mechanism of time. However, there are advanced neural
network (NN) architectures that can be used for both recognition problems and predic
tion problems. In a recurrent neural network (RNN) a neuron can obtain input from any
other neuron, including those in the output layer. Specifically, the outputs from nodes
in the hidden or output layers are fed back as input to an earlier layer. As RNNs store
information about time, they can be used for temporal prediction applications. How
ever, they are quite difficult to use and to train. Unlike traditional feedforward NNs, the
time that it takes for a recurrent NN to produce output is not known. This is because
the hidden and output layer nodes will continually be activated until the model sta
bilizes. Recurrence implies that the current state of the network depends not only on
the current input values but also on those of the previous cycle (from the previous
outputs).
Figure 9.4 shows the basic structure of an RNN. In Part (a), the structure for a
feedforward NN is shown. Part (b) shows an RNN. In (b), output from the hidden layer
is fed not only to the output layer but also into a new input layer referred to as a context
252
Chapter 9
Input
Temporal M i n i n g
Hidden
Section 9.3
- -I
- -I
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I
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I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I_ _ _I
1_ _ _1
I_ _ _ I
, - -I
(a) Feedforward
Hidden
Input
Output
__..I
I
I
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I
I
Output
- -I
I
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1_ _ _1
1_ _ _1
- -
- -
Context
NN
9.3
TIME SERIES
A time series is a set of attllibute values over a period of time. Alternative definitions
exist. Some investigators view a time series as consisting only of numeric values. Some
investigators assume that the values are at specific, evenly spaced time intervals. Time
series data may be continuous or discrete. In this text we take a general view that
encompasses all of these. As seen in Definition
n values:
{ (t1 , a1 ) , (t2 , a2 ) , . . . , (t11 , a11 ) } . Here there are n time values and for each a corre
sponding value of A. Often the values are identified for specific well-defined points
in time, in which case the values may be viewed as a vector (a 1 , az , . . . , an ) .
(y 1 , . . . , y11 ) if,
such that Yi j = Yk .
Y
'v'l :::;
j :::; m -
1 , ij
(y; 1 ,
<
layer. In this structure the input to the hidden layer then comes from nodes in the input
(b) RNN
Seasonal: Here the detected patterns may be based on time of year or month or
Christmas.
I
r---:-----+1
I
I
I
I_ _ _I
9.3.2
- Values
10
- o- -
Moving average
Typical data mining applications for time series include determining the similarity
between two different time series and predicting future values for an attribute, given a
time series of known values. Obviously, the prediction is a type of classification, while
the similarity can be thought of as either clustering or classification. Given several time
series, we may want to determine which time series are like each other (clustering).
Alternatively, we may be given a time series to find which time series from a set are
like this one (classification). A special type of similarity analysis is that of identifying
patterns within time series.
9.3.1
253
day. As an example, the sales volumes from department stores always jump around
:
I
Time Series
254
C h a pter 9
Tem poral M i n i n g
Sectio n 9.3
correlation may be found between every twelfth value (in monthly sales data). The time
difference between the related items is referred to as the lag. With the sales data, the lag
is 1 2. Autocorrelation functions can be generated to determine the correlations between
data values at different lag intervals. A correlogram graphically shows the autocorrelation
values for different lag values.
The covariance measures how two variables change together. It can be used as the
basis for determining the relationship between either two time series or seasonal trends
in one time series. An autocorrelation coefficient, rk . measures the correlations between
time series values a certain distance, lag k, apart. Several different approaches have been
used for autocorrelation. A correlation coefficient, introduced in Chapter 3, measures the
linear relationship between two variables (or that between the same variable at a given
time lag). Positive values indicate that both variables increase together, while negative
values indicate that as one increases the other decreases. A value close to zero indicates
that there is little correlation between the two variables. One standard formula to measure
correlation is the correlation coefficient r, sometimes called Pearson 's r . Given two time
series, X and Y, with II\eans X and Y, each with n elements, the formula for r is
1
L (x; - X) (y; - Y)
(9. 1)
Applying this to find the correlation coefficient with lag of k, rk . on a time series X
(xi , x2, . . . , Xn ) is straightforward. The first time series is X' (x1 , x2, . . , Xn-k ) , while
the second time series is X"
(xk+ 1 , Xk+2, . . . , Xn ) . Example 9.2 illustrates the use of
autocorrelation coefficients.
=
9.3.3
255
Transformation
To assist in pattern detection, the actual time series data may be transformed in some
manner. A logarithmic transformation can be used to stabilize the variance and to make
seasonal effects constant over years. Transformation is also used to solve the dimension
ality curse. The dimensionality curse is the fact that many problems are caused by data
sets with many dimensions. Data mining on time series data with many variables is not
only difficult but also expensive. Data structures to store high-dimensional data are not
very efficient. Transformation can be used to reduce the number of dimensions. Note that
feature extraction also reduces the number of dimensions.
9.3.4
Sim ilarity
We saw examples for examining the similarity between patterns in Web usage min
ing. Indeed, these applications are temporal data mining. Given a target pattern X
(XJ , x2 , . . . , Xn ) and a sequence Y
(y, , y2, . . . , Ym ) . the problem is that of determin
ing sim(X, Y) . Here n may or may not be the same as m. Matching may be based on
matching the two series completely, matching subseries thereof, or more advanced types
of matching. One series may be scaled or shifted to match the other. Gaps or "don 't
care" values may have to be added to one series to match the second.
Some common distance measures that may be used are Euclidean, linear correlation,
and discrete Fourier transform. We have already seen the Euclidean distance metric.
There are problems with these common distance measures:
=
Length: X and
Scale: While the general shape of X and Y may be identical, the scale may be
somewhat different. For example, one may use a log scale. Different metrics may
be used (Fahrenheit vs. Centigrade) .
EXAMPLE 9.2
By looking at the graph in Figure 9.6, it is obvious that several patterns exist. One is the
fact that the values rise linearly for two time units and then drop and restart. Thus, there
is an obvious autocorrelation with a lag of 3. In this case we find that rk
1 because
there is a perfect positive relationship.
Time Series
Gaps: One of the series may be missing some of the values that exist in the other
series.
Outliers: This is similar to the problem to gaps, except that it is assumed that the
Baseline: The actual baseline values may differ. This means the time between
3.
256
Chapter 9
Section 9.4
Temp ora l M i n i ng
DEFINITION 9.5. Given integer value 8 > 0, real number E < 1, and linear
function function f, and two time series X, Y with the longest one of length n.
(yj1 , yh , . . . , yj11. ) be the longest subseries
, Xi , ) and Y'
(xi1 , Xi2 ,
Let X'
in X and Y, respectively, where:
=
. .
V1 ::: k ::: m ,
ciE)
(9.3)
where ai is a shock. For a moving average, then, the time series value is predicted based
on a weighted average of a set of previous shock values.
Autoregression and moving average can be combined to created a model of a time
series that is called ARMA (Autoregressive Moving Average). In practice, this combined
model is sufficient to represent many real-world time series. When the model is not
stationary, an extension of ARMA, ARIMA (Autoregressive Integrated Moving Average)
may be used. The ARIMA model has become quite popular, but it is relative complex
and requires an expert to use it effectively.
max f (m j n)
The longest common subseries between two given series can be found in O(n 2 ) .
Thus, the most difficulf part o f finding simE, o (X, Y) i s finding f. Several algorithms
have been proposed to find the function. An exact algorithm is O (n 3 ), while approximate
algorithms with better behavior are also proposed [BDGM97].
9.3.5
257
may be viewed as a weighted sum of previous deviations from the mean. Autoregression
models may be stationary or nonstationary.
Another dependency that may exist between values in a time series is that of
a moving average. Here a future value, Xn+ 1 , can be found using a moving average
model applied to a set of previous consecutive talues. There are many different mov
ing average models, and any model could be used. In addition, there may be a lag
between the point where the moving average is applied and the prediction value. For
example, a seasonal forecast for sales could be based on an average of the sales for
the prior season 1 2 months earlier. A future time seties value, Xn+ l . can be predicted
using:
issues. The baseline issue is also addressed by allowing a slight difference, up to 8, in the
time values used. The resulting similarity function sim<,o (X, Y) is shown in Definition
9.5 (modified from [BDGM97]). The maximum is taken over all possible values for f.
The closer si. o (X , Y) is to 1 , the more similar X and Y are.
Pattern Detection
Prediction
The prediction (or forecasting) of time senes data can use some of the techniques dis
cussed earlier, such as regression. However, in practice, time series data are replete with
errors and noise. Using simple regression often is not sufficient. Given a discrete time
series over equally spaced time intervals, the forecasting problem is to predict a value
at time t, x1 (l) , and a lead time of l. It is assumed that previous time series values,
(x 1 , x2 , . . . , x1 ) , are known. The objective is to minimize the mean square of the devia
tions x1+t - x1 (l) . Various models may be used to represent the time series values and
thus predict future values. We briefly review some of these models here.
Studies of time seties prediction often assume that the time series is stationary.
This means that the values come from a model with a constant mean. More complex
prediction techniques may assume that the time series is nonstationary. A time series
usually represents values that are dependent on each other, but they may be viewed
as being generated from a series of independent values called shocks. The shocks are
randomly drawn from a normal distribution with a zero mean. A sequence of these
random values is thought of as representing a white noise process. This white noise
process is transformed into the time seties by a linear filter, which may be viewed as a
simple weighted sum of previous shocks.
A special case of the linear filter model is one that assumes that time series values
are dependent on earlier ones. Autoregression, then, is a method of predicting a future
(x 1 , x2 , . . . , Xn ) .
time seties value by looking at previous values. Given a time series X
a future value, Xn+ 1 , can be found using
=
(9.2)
Here en+ I represents a random error, at time n + 1. In addition, each element in the time
series can be viewed as a combination of a random error and a linear combination of
previous values. Here the <Pi are the autoregressive parameters. Alternatively, the value
9.4
PATTERN DETECTION
Given a set of data values (d1 , d2 , . . . , dn ) where di is collected at time ti and ti < tj
iff i < j , the pattern detection problem is to deterrnine a given pattern that occurs in
this sequence. This can be viewed as a type of classification problem where the pattern
to be predicted is one found in a given set of patterns. Typical pattern detection appli
cations include speech recognition and signal proceS;sing. Spelling correctors and word
processors also use simple pattern detection algorithms. Although these simpler cousins
of the true data mining pattern detection problems are precise, the more general pattern
detection problems are fuzzy with no exact matches. Approximations are needed. While
humans are good at detecting such patterns, machines are not.
9.4. 1
String Matching
The string matching problem assumes that both a long text document and a short pattern
are given. The problem is to determine where the pattern is found in the text. Example 9.3
illustrates the pattern detection problem when it is applied to string matching. This prob
lem is a common one, with many applications in word processing.
EXAMPLE 9.3
Martha Holder is editing her resume using a popular worq processor. She has just gotten
married and wishes to change the name Holder to her new last name of Laros, where
approptiate. Not all occurrences of Holder, however, should be changed. For example,
she does not want to change the author's names of previous publications that were made
under her maiden name. Using the word processor, she repeatedly finds all occurrences
of Holder in the vita. She then must examine the context to determine whether it should
be changed to Laros. In this case, the pattern being matched is (H, o, l, d, e, r ) . Only
258
Chapter 9
Section 9 . 4
Temporal M i n i ng
words that are an exact match to this pattern should be found. Note that here each letter
is viewed as if it occurred at a later point in time. In actuality, it is a later point in the
document.
One of the earliest string matching algorithms is the Knuth-Morris-Pratt or KMP
algorithm. KMP creates a finite state machine (FSM), which is used to recognize the
given pattern. The FSM represents all possible states that exist when scanning a string to
match the given pattern. Each node in the FSM relates to one of these states. Figure 9.7
shows an FSM created to recognize the pattern "ABAABA." Here there are seven states .
State i represents the fact that the first i characters in the pattern match the most recent i
characters in the string. State six is designated as the recognizer state with two concentric
circles . The arcs in the graph are labeled with the character from the pattern that causes a
transition between the two states as indicated. Transitions labeled with "*" indicate that
this transition is taken with any other character found in the string. The KMP algorithm
creates the FSM for a given pattern . The FSM can then be applied to the string by
starting at the first charadter in the string. From a given state, the next character in the
string determines which transition is taken. The accepting state of the FSM is reached
only when the pattern is found in the string. The worst-case behavior of the application
of the FSM is O (m + n), where m is the length of the pattern and n is the length of the
string. The preprocessing phase to create the FSM is O (m) in space and time.
Another algorithm that builds on the KMP approach is called the Boyer-Moore, or
BM, algorithm. The same FSM is constructed to recognize the pattern, but the pattern is
applied to the string in a right-to-left pattern. For example, when looking for the string
"ABAABA," if the sixth character in the string is not A, then we know that the pattern
is not found in the string starting at the first character in the string. We also know that
if the sixth character is neither an "A" nor a "B," then the pattern does not exist in the
string starting at any of the first six characters. The BM needs only one comparison to
determine this, while the KMP would have to examine all of the first six characters.
Again, the BM is O (m + n) in the worst-case scenario, but the expected and best cases
Pattern Detection
259
are better than this. The actual performance depends (of course) on both the pattern and
the string.
Even though KMP and BM are pattern recognition algorithms, they usually are
not thought of as data mining applications. The identification of patterns in these earlier
techniques is precise. Most data mining pattern matching applications are fuzzy; that
is, the pattern being compared to (i.e., the class representative) and the object being
classified will not match precisely. However, as we will see, there are more advanced
pattern recognition algorithms that are similar in that graphical structures are built to
specifically recognize a pattern. In effect, these true data inining applications build on
these earlier non-data mining algorithms.
When examining text strings, it often is beneficial to determine the "distance"
between one string and another. For example, spelling checkers use this concept to
recommend corrections for misspelled words. Again, these usually are not thought of
as data mining activities, but the distance measure technique we discuss here is often
the basis for more advanced distance measure approaches. Suppose that we wish to
convert A = (a1 , az , . . . , an } to B = (b 1 , bz , . . . , bm } . The basic idea is to determine the
minimum cost of steps that are needed to convert one string to another. There are three
operations that can be performed to convert string A to string B. Starting at the first
character in each string, each operation identifies what operation should be performed
on A and B to change A to B . Each operation not only indicates specific functions to be
performed but also associates a cost for it. The following assume that we are currently
examining ai in A and bj in B :
is 1 .
The distance between string A and B i s then determined by the minimum total cost for
all operations needed to convert A to B. For example; tHe distance from catch to cat is
2 because the c and h have to be deleted. Similarly, the distance from cat to hat is 2
because c must be deleted and h must be inserted. Example 9.4 illustrates the process.
EXAMPLE 9.4
Suppose that we wish to determine the distance between a string "apron" and "crayon."
By looking at the strings, we see that we can match at most three characters : either
a , o, n or r, o, n. Figure 9.8 illustrates the use of the first matching. Here the cost is 5
because we have to insert c, r, y and delete p, r. The figure shows that we can view the
problem as a shortest path between two points: the top left comer and the bottom right
corner.
FIGURE 9.7:
260
Temporal M i n i n g
Chapter 9
Section 9 . 5
n
i M;
D
e
t
e
Insert
DEFINITION 9.8. Given a set of customers and transactions for each customer, the
sequence contains S.
F I G U R E 9.8:
9.5
DEFINITION 9.9. The confidence (a) for a sequence association rule S =} T is the
ratio of the number of customers (customer-sequences) that contain both sequences
S and T to the number that contain S.
SEQU ENCES
A sequence is an ordered list of itemsets [AS95] . Definition 9.6 gives the definition of a
sequence.
DEFINITION 9.6. Let I = { h , h . . . , Im } be a set of items. A sequence,
S, is:
As with a time series, we find that there are many different definitions for a
sequence. In Chapter 7 the sequence was a list of web pages. A sequence is some
times viewed as an ordered list of attribute values from any domain. The individual
members of the sequence are sometimes viewed to be sets of items from some under
lying domains (alphabets). One common difference is that the sequence may not have
explicit relationships with time. The only requirement is that the entries be totally ordered.
As a matter of fact, the terms sequence and time series are often used interchangeably.
In this text, we use the two definitions as shown in Definition 9.3 and Definition 9.6.
The basic difference between the two concepts, then, is that a series is an ordered list
of values, while a sequence is an ordered list of sets of items or values. The length
of a sequence is the sum of the cardinalities of all itemsets in the sequence. A subse
quence of a given sequence is one that can be obtained by removing some items and any
resulting empty itemsets from the original sequence. We briefly examined the concept of
sequential patterns in Chapter 7. These are specific type of subsequences in that they
are maximal.
DEFINITION 9.7. Let I = { h , h . . . . , Im } be a set of items. One sequence
T = (ti1 ,
m - 1 , ij
<
contains T .
261
Seque nces
We assume that items are grouped together into transactions. The temporal feature
is added by assuming that a customer may obtain different items at different times.
Each set of items purchased at one time by a customer is a transaction. Example 9.5
Let I = A, B , C, D, suppose that there are three customers, Ct . Cz , and C3 , who purchase
these items at different times. The following table shows purchases made by these three
customers:
Customer
Time
Ct
Ct
10
20
30
15
20
15
c1
Cz
Cz
C3
Itemset
AB
BC
D
ABC
D
ACD
(In this table we have removed commas and set notation.) Given S = ( { A } , { C } ) , we
see that the support is s (S) = 1/3 because it is contained only in the sequential pattern
found for customer CJ . A second sequence T = ({A}, { D }) has a support of s ( T ) = 2/3,
while U = ( { B , C}, {D}) has a support of s ( U) = 2/3. Figure 9.9 shows the lattice with
frequent sequences, assuming a minimum support of 2/3, only for this data.
As with frequent itemsets, a frequent sequence follows a large sequence property.
This means that any subsequence of a large (frequent) sequence is also frequent.
262
Chapter 9
Temporal M i n i ng
Section 9 . 5
Sequences
263
A
Customer
Time
Customer
Time
Customer
Time
Customer
Time
10
15
15
Cr
10
20
15
C1
c2
c3
20
15
15
C1
c2
C3
30
20
15
Cr
c2
c3
9.5.1
9.5.2
AprioriAII
sequenial
Algorithm AprioriAll in Chapter 7 contained a simple algorithm .for finding
relacig
then
ets,
Items
frequent
all
patterns. AprioriAll works in three parts by first finding
.
This
patterns.
quenttal
s
finding
finally
and
itemsets,
original transactions with frequent
would
also
It
step.
twn
transforma
the
of
because
partly
well,
scale
not
does
algorithm
be difficult to incorporate extensions such as sliding windows.
SPADE
The algorithm that we now introduce, SPADE (Sequential PAttern Discovery using Equiv
.
alence classes), identifies patterns by traversing the lattice top-down. To Improve
pro
cessing, SPADE uses an id-list that associates the customers and transactions associated
with each item. Table 9 . 1 illustrates this concept for the data in Example 9.5. Here we
see the id-lists for sequences of length 1. These can be viewed as the atoms to construct
support counts for larger sequences. The support for a k-sequence can be detened y
looking at the intersection of any two of its (k - I )-subsequences. To accomplish this,
temporary id-lists are generated from the starting id-lists. To illustrate this process, look
at the sequence T
A
Looking at Table 9 . 1 , we see that the count for
is 3, as is that for
As seen in Example 9. 5, T
A ,
count of 2. To denve
this, an id-list for T is created by determining the intersection for the two subsequences:
({ } {D}).
({D}).
=
Customer
({ })
({ } {D})
Time
Note that intersection must take times into account. Thus, its count is 2 and its support
is 2/3 . This observation is used in SPADE to count the sequences and determine their
support. The lattice can be traversed to construct id-lists for higher- evel squences by
.
intersecting two subsequences at the next lower level. The problem with this IS that there
may not be enough memory to do this all in memory.
Ct
c2
To address the space issue, th lattice i s divided into partitions and these partitions
are traversed independently. This reduces the memory requirements by reducing the
number of id-lists that must be kept at one time. An equivalence class concept is used
to accomplish this. A k length prefix for a sequence is determined by looking at the
first k items (and associated ordering) for the sequence. Given a sequence S, the k
length prefix of S is denoted by p(S, k), In Example 9.5 , we looked at the sequence
U
This sequence is a 3-sequence because it is of length 3. It has a
length 2 prefix of
as does another 3-sequence W
ek is an
equivalence relation. As Set<n in Definition 9. 10, two sequences are ek equivalent if they
have identical prefixes of length k. Thus, we see that U is equivalent to W, written
as U = W (mod 82). If we had the id-lists for U and W with their counts, we could
determine the count.
=
({B, C, D}).
p(S, k)
p(T, k).
T (mod ek ) iff
264
Section 9 . 5
Tempora l M i n i n g
Chapter 9
Sequences
265
together. When transactions are grouped together, a sequence is said to exist in a customer
sequence if it exists in any of the transactions in a window. The effect of this is to increase
the support of sequences.
One last extension proposed in [SA96b] is to aPd a time constraint that indicates
the allowed tie between successive elements in the sequence. The time constraint is
a pair Umin . tmax ) that indicates the minimum and maximum distances that are allowed
to exist in a sequence. These are allowable time gaps. The time qifference between
transactions with consecutive elements in the sequence must be greater than tmin but no
greater than tmax .
The three extensions also may be combined. With n o concept hierarchy, a window
size of 0, and time constraints of (0, 09), there is the regular concept of sequences.
One algorithm has been proposed specifically to handle generalized sequential
patterns. This algorithm generalized sequential pattern (GSP) has been shown to outper
form an extended version of AprioriAll by up to 20 times [SA96b] . As with Apriori,
GSP scans the database several times. The support for all items is determined during
the first scan. The input to the next scan is the frequent items (sequences of length one)
found during the first traversal. The algorithm works iteratively in this fashion. Dur
ing each scan, candidate sequences are generated from the frequent sequences of the
prior scanned and then counted. As with Apriori, the size of each candidate during a
ALGORITHM 9.2
Input :
D
/ / ID - l i s t s
for
database scan is the same. GSP terminates when no candidates at that pass are found to
be frequent.
To assist with the time constraint issue, the concept of contiguous subsequences is
used. A sequence will always contain all contiguous subsequences, but with time con
straints added, it may not contain noncontiguous subsequences. The definition of con
tiguous subsequence is found in Definition 9 . 1 2 [SA96b]. For example, (A , C , DE , D),
/ / Suppor t
output :
I I Frequent sequences
SPADE algoritlun :
D e t e rmine f r e quent
i t ems ,
F1 ;
;
Det ermine f r e quent 2 - sequen c e s , F2
1 - s equenc e s
l
l
a
for
E
s
e
s
s
a
l
c
lence
Find equiva
for each
[ S]
E E do
Find frequent
9.5.3
[S]e1 ;
sequences F ;
Generalization
concept hierarchies and
The concept of subsequence has been generalized to include
conce?t of sequences
the
make
can
ations
generaliz
more temporal information. These
t would b to
constram
one
,
example
For
ons.
applicati
of
range
applicable to a wider
, you nught
example
. For
include a maximum time between elements in the sequence
within
printer
a
purchase
then
and
want to see customers who purchase a digital camera
types
many
are
There
problem.
hierarchy
three months. This also illustrates the concept
the
requires
This
type.
and
brand
any
for
be
should
sequence
of digital cameras. This
rules.
on
associati
ed
generaliz
in
as
ies
use of taxonom
.
.
orward. The tdea 1s
Adding concept hierarchies to sequences is relatively straightf
9. 1 1 .
to change the definition of subsequence as seen in Definition
, t;111 ) is a subsequence of another
DEFINITION 9.11. One sequence T = (t;1 ,
Vl j m , 3 1 k_ n such
and
s = (s 1 , . . . , sn ) if Vl j m - l , ij < ij+l
_
hierarchy for some 1tem m Sk .
concept
a
in
ancestor
that t; j ; sk or Vx E t;j x is an
one item is dropped from it then that particular itemset in the subsequence is
dropped effectively making the lfmgth of T n - 1 .
Another technique that has been proposed is to look at a sliding window aroun the
data [SA96b]. A sliding window is a maximum time difference used to group transactiOns
The generation of candidate sequences must be handled differently than with can
didate itemsets in Apriori. For example, suppose it is found that ( { A } ) and ({B}) are
frequent during the first scan. These two sequences can be used to generate three can
didates: ( {AB } ) , ( { A } , { B } ) , and ( { B } , {A}) . In addition, the generalization constraints
266
Chapter 9
Section 9.6
Tempora l M i n in g
the database, different association rules can be found for different times or time ranges.
differently. Two sequences T = ( tJ , . . . , t111 ) and S = (SJ , . . . , sn ) are joined if the sub
sequence of T obtained by dropping the first item in t1 is the same as a subsequence of
mining area. The analogy in temporal mining is to cluster the data based on time and then
This is similar to the idea of combining clustering with association rules in the spatial
determine the association rules. This can be done to examine the change in association
S obtained by dropping the last item in s, . When T and S are joined, the new sequence
obtained is either U = ( tJ , . . . , tm , x) or U = (tJ , .
, tm U x ) . The first sequence is
obtained if x = sn ; otherwise the second sequence is obtained.
rules over time, to detect seasonal association rules, and to identify rules that may not
be found if looking at larger sets of data. For example, a grocery store could look at
. .
association rules for an entire year. However, this would not identify frequent items sold
at particular times of the year. This would not allow the store to take advantage of some
of the most frequently sold items over short petiods of time. The importance of this
Feature Extraction
concept is demonstrated by the fact that many supermarket:. now have aisles dedicated
The feature extraction problem is to extract k features from every sequence ai:J.d to
represent that sequence by those features. This approach may make it easier to perform
When time is added to the concept of association rules, different types of rules
space. R-trees or other multidimensional data structures can then be used to store and
search the time series data. The problem, of course, is how to extract the features.
As with time series clustering, identifying features that describe classes of sequences
9.6.1
lntertransaction Rules
is beneficial. One algoqthm, FEATUREMINE, has been proposed to extract features for
The basic association rule approaches look only at items occurring together within one
certainly are situations in wp_ich rules generated across transactions would be of interest.
features [LZ099] :
computer software after they purchase a computer. These purchases could occur in trans
classifier maps each sequence into a class based on features. There are four goals for
For example, an electronics store manager might want to lmow if customers purchase
actions at two differeqt times. To define these new rules, the concept of a window
is applied to the transaction database. Recall that the basic association rule problem
transaction t; has associated with it a value d; which could be time, location, or other
information desctibing the tansaction. We assume here tht the value is time, so that d;
The last item indicates how the feature extraction should not be performed. FEATUREMINE
is the time that t; xcuted: Although the original proposal in [TLHF99] viewed that d
could be any ordinal attributes, to simplify discussion here we look at specific integers
uses SPADE and integrates a pruning technique into the algorithm. The approach is to
traverse the sequence lattice in a depth-first manner to find the frequency sequences. Observe
SPADE, to ensure that processing in main memory, the lattice is partitioned into equivalence
in the window,
that sequences at the root of the lattice are more general than those beneath it. As with
class sections and the traversal actually is performed in each partition separately.
9.6.2
9.6
267
joining frequent sequences from the prior scan. Here, however, joining is defined slightly
9.5.4
Temporal Association R u l es
(TID , CID , l] , In , . . . , Im )
where TID is the ID for the transaction, CID is the ID for the customer, and I1 , . . . , Im
are the items. When considered part of a temporal database, a transaction could be
viewed as
(TID , CID , /1 , In
. . .
, I111 , ts , te )
w,
is an input parameter.
Episode Rules
An episode rule is a generalization of association rules applied to sequences of events.
An event sequence S is an ordered list of events, each one occurring at a particular time.
Thus, it can be viewed as a special type of time series. An episode is a set of event
predicates, A , and a partial order, ::;:, on the events in A : { A , ::;: } . An event predicate
is a predicate that can be evaluated as true or false when applied to an actual event
occurrence. It could be as simple as to check the type or severity of an event. An episode
can be viewed as a directed graph where the vertices are the events and the arcs represent
the partial order. An episode B is a subepisode of an episode A if the graph of B is
where [ts , te] is the valid time range for the transaction. If this were a grocery store
transaction, fs = te could be the point in time that the transaction was completed. Alter
time the order was placed and te might be the time the actual delivery was made. Thus,
natively, if the transaction represented products ordered over the Web, ts might be the
[ts , te] would be the range of time the transaction was active. Once time is added to
definition for episode rule is found in Definition 9 . 1 3 . As with association rules, we may
268
Chapter 9
Section 9.6
Temporal M i n i ng
A correlation pattern is then used to match to the sequences that have been found in the
alarm data. This pattern may be compared to alarms that have occurred in a recent time
window. If the sequence of alarms that have occurred matches a correlation pattern, then
the associated correlation action is taken.
Two different approaches have been proposed to find episode rules. One approach,
WINEPI, applies a window to the events. Given an event sequence, S, the window is a
time span (ts , te ) that defines a subseries of S, namely, those events (in order) that occur
in the window. Given an episode B , the subseries of B that occur in all windows of size
W is referred to as Bw . The window can be used to define support and confidence as
seen in Definition 9.14 and Definition 9 . 1 5 . The support is the percentage of windows
in which the target episode occur.
ij .
9.6.3
Trend Dependencies
Trend dependencies are like association rules in that they compare attribute values, but
they do so over time [WM97] . For example, we might observe that an employee's
salary always increases over time. A formal definition (as found in [WM97]) is found in
Definition 9. 18. Note that the definition does not explicitly indicate that the two database
states mtist differ in time. Of course, this is our assumption here, but in general it is not
269
necessary. To add this temporal aspect to it, we assume that the pattern on the left-hand
side is from a relation state valid at an earlier time than the pattern on the right-hand
side of the trend dependency.
DEFINITION 9.19. Given two relations, fr , [z over schema R , the support (s) for
a trend dependency X ::::} Y is the percentage of tuple pairs in It x [z that satisfy
both patterns X and Y. If I /1 x /z I = 0, then s = 0.
DEFINITION 9.20. Given two relations, I1 , /z over schema R , the confidence (a)
for a trend dependency X ::::} Y is the ratio of the number tuple pairs in It x /z
that satisfy both patterns X and Y to the number that satisfy X. If the number that
satisfy X is 0, then a = 0.
Example 9.6, which is adapted from [WM97], illustrates a trend dependency. In
this example, there are two database states: I It I= 6 and I /z I= 6. Thus, I h x lz I= 36.
Here X = (SSN, =) AND Y = (Salary, ::: ) . The number of tuple pairs in h x /z that
satisfy both patterns is 4. The number that satisfies X is 5. Thus, the a = 4/5 = 80%
and s = 4/36 = 1 1 %.
EXAMPLE 9.6
Imagine having the data in Example 9. 1 for all employees at XYZ. Instead of viewing
it as one table, however, we want to look at it as three different instances: It , /z, and
h It contains the valid data at time 2/12/02, /z has the valid data for 8 / 1 2/02, and h
has the valid data for 12/10/02. A trend that can be observed (at least for Joe Smith)
is that an employee' s salary always increases with time. This trend detection can be
stated as
(SSN, =) =::::} (Salary, :S)
Given two tuples t1 E /r and tz E h if ft (SSN) = tz (SSN) then ft (Salary) ::: tz (Salary) .
This holds for any two database states where the second state is at a later time. The
following tables show It and h
270
Chapter 9
Temporal M i n i n g
Name
Joe Smith
Mary Jones
Bill Adams
Selena Shepherd
Paul Williams
Martha Laros
Name
Mary Jones
Joe Smith
Bill Adams
Selena Shepherd
Paul Williatbs
Bob Holder
SSN
123456789
111111111
222222222
876543298
908734124
873659365
SSN
111111111
123456789
222222222
876543298
908734124
838383838
Section 9.6
Address
10 Moss Haven
1 0 Main
2 1 5 North
25 Georgetown
13 East
1 0 1 0 Fox
Address
1 0 Main
10 Moss Haven
215 North
25 Georgetown
13 East
22 South
Salary
confidence. There are many applications that could use sequence association rules. In
the market basket area, the buying behavior over time can be used to predict future
buying behavior. This could be used to do targeted advertising to customers with the
first type of buying behavior. Note that we are predicting buying patterns over time,
not just within one transaction. An example of sequence association rules is found
in Example 9.7. The SPADE algorithm discussed earlier ca:n be used to find frequent
sequences, and these sequences can then be used to solve the sequence association
rule problem.
Salary
EXAMPLE 9.7
85,000
52,000
90,000
1 5 ,000
270,000
20,000
4/5
271
As with conventional association rules, we can state the sequence association rule
problem to be that of finding sequence association rules with minimum support and
50,000
75 ,000
100,000
1 5 ,000
250,000
150,000
ct =
Using the data introduced for Example 9.5, we can construct the following sequence
association rules:
80% and
Rule
Support
Confidence
( { A } , {C} ) ==> ( { A } , { D } )
( { B , C} , { D }) ==> ( { A } , {C})
113
113
112
9.6.5
Trend dependencies are defined over only two database states. They are not easily
generalized to more states. As with association rules, we c n state a trend dependency
. .
problem as that of finding all trend dependencies with a gten rmruum suppo and
confidence over two identified database states. The complexity of this problem m the
worst-case scenario is quite high. There are 1 e ID possible combinations of attributes
and operations. Here e is the set of operators (we assume there are six in this case), and
D is the number of possible attribute pairs. In the example this becomes 6 1 6. Obviously,
an exhaustive search is not advisable. In fact, it has been shown that the general problem
is NP-complete [WM97]. When the set of operators is restricted to { < , = , > } , it becomes
polynomial and an efficient algmithm has been proposed [WM97].
9.6.4
EXAMPLE 9.8
Suppose that a grocery store wishes to obtain information about purchases for a particular
day. In this case, the time unit is a day (24-hour period). The manager is interested in
finding all association rules in this time frame that satisfy a given minimum support and
confidence. The manager also is interested in association rules that satisfy the support and
confidence for all but five days in a given season. During the year 200 1 , the manager
defines two time intervals by looking at days in winter as defined by the calendar:
272
Chapter 9
Temporal M i n ing
Section 9.8
{ ( 1 , 79) , (355, 365) }. There are 90 time units, or days, in this calend. With a mismatch
threshold of 5 , he is then interested in only association rules that sattsfy the support and
confidence on at least 85 of the days.
One could imagine a regular association rule algorithm, such as Apriori, applied
to a subset of D created by finding all transactions that occur in the given time intral.
.
However, the problem may be more general, such as finding all calendric assoctatwn
rules that occur over any time interval (or some set of intervals). This could be used
to determine important association rules over any day or period of time (n ot just o?e).
.
It is assumed that a particular calendar is defined with potentially many dtfferent ttme
granularities. The more general problem, then, is to find all calendric associ tion les
.
that hold given this calendar. Given a calendar of time intervals and a tme ut vanous
occurrences of the time unit can be defined in each. In Example 9.8, the ttme urut ts a day,
but the intervals are the seasons consisting of days in the four seasons. An association rule
may satisfy the support and confidence for soe of t e t e uits. Thus, an additional
threshold, m, is used td indicate the number of ttme uruts m the mtervals of the c lendar
in which the association rule does not hold. A calendar belongs to a rule X => Y tf there
are at most m mismatches. A calendric association rule algorithm has been proposed
that is given as input a set of possible calendars and a time unit [RMS98] . It first fids
large itemsets over all time units and then determines which calendars belong to which
association rules.
9.7
EXERCISES
1. Using the time series data in Example 9.2, determine the autocorrelation with a lag
273
One recent text has examined the impaCt of time on databases, logic, and data min
ing [BJW98]. There are several excellent surveys and tutorials concerning temporal data
mining, including a recent textbook that examines sequential patterns [AdaOO] [HLP0 1 ] . 1
Markov models and hidden Markov models have been extensively studied. An
excellent introduction to the topic of HMM can be found [RJ86].
Time series have been extensively studied in the literature with several introduc
tory books and surveys. One online statistics textbook contains a survey, [Sta0 1 ] that is
quite simple to understand and very complete. Many time series textbooks are available,
including [And7 1 ] , [BJR94], and [BD96]. A recent dissertation examined the telecom
munication network alarm issues in detail [Kle99b] . One of the earliest proposals for
a recurrent network was by Jordan in 1 986 [Jor86]. Jordan proposed feedback from
output units to special context input nodes. Note that this allows conventional NN back
propagation learning techniques. Elman proposed that R.Nl\Ts allow a feedback form the
hidden layer to a separate context input layer [Elm90]. Recurrent neural networks have
been proposed to detect automobile emission problems that occur over time. One recent
dissertation has studied the temporal NNs [1198].
The various applications of time to association rules is becoming quite popular in
the research community. Intertransaction association rules were introduced in [TLHF99] .
Calendric association rules were proposed in [RMS98] as an extension to the earlier
proposed cyclic association rules [ORS98]. The authors pmpose a calendar algebra to
manipulate time intervals using a predefined calendric system.
SPADE was first introduced in 1 998 [Zak98]. The concept of subsequence gen
eralization was examined in [SA96b] The approach of applying windows to events in
WINEPI was proposed in [MTV95].
Z t , taken at 5-minute
time intervals: {50, 52, 55, 58, 60, 57, 66, 62, 60}. Plot both Z t+2 and Zt Does there
appear to be an autocorrelation? Calculate the correlation coefficient.
3. Plot the following time series values as well as the moving average found by
replacing a given value with the average of it and the ones preceding and following
it: {5, 1 5 , 7, 20, 1 3 , 5, 8, 1 0, 12, 1 1 , 9, 1 5 } . For the first and last values, you are to
use only the two values available to calculate the average.
4. Using the MM in Figure 9.2, determine the probability that the model is in the
Figure 9.2.
6.
9.8
(Research) Investigate and describe two techniques which have been used to predict
future stock prices.
BIBLIOGRAPHIC NOTES
The original string matching algorithms, KMP and BM, were proposed ver 20 years
ago [BM77] [KMP77] . A variation on KMP proposed by Aho and Corastck constructs
an FSM that can recognize multiple patterns [AC75] .
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YM98.
A c 2 , 275
IR, 1 1 6
4Thought, 274
Data D:istribution, 1 8 1
DBCLASD, 241
DBS CAN, 154
JDA Intellect, 2 8 1
Linear Discriminant Analysis
, 1 23
NeoVista Decision Series,
281
ZaiO l .
Osmar R. Zaiane, Building Virtual Web Views, 39(2) : 1 43- 1 63, 200 1 .
ACC, 1 20
Zak98.
GA Clustering, 1 47
Generate Rules, 1 1 5
Genetic Algorithm, 69
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ZRL96.
activation function, 64
bipolar, 65
Gradient Descent, 1 1 0
HITS, 206
Gaussian, 66
hyperbolic tangent, 66
linear, 65
hrnm, 25 1
K Nearest Neighbors , 92
K-Means, 141
sigmoid, 66
step, 65
threshold, 65
unipolar, 65
adaptive, 1 63
adaptive classifier combina
tion, 1 20
additive, 3 1
affinity, 145
affinity analysis, 8
Affinium Model, 274
agglomerative clustering, 128
aggregate proximity, 24 1
aggregate proximity relationsh
aggregation hierarchy , 29
AGNES, 1 62
AI, 1 2, 42
AI Trilogy, 274
algorithms
l R, 1 1 7
Agglomerative, 1 3 3
Apriori, 1 73
Apriori-Gen, 1 72
ARGen, 1 69
Averagelink:, 137
Backpropagation, 1 09
BIRCH, 1 5 1
Count Distribution, 1 80
CURE, 1 57
ip, 242
Nearest Neighbor, 1 42
OAT, 2 1 7
PAM, 1 44
Partition, 1 7 8
Partitional MST, 1 39
PRISM, 1 1 9
Propagation, 1 06
SPADE, 264
305
306
Index
I ndex
alleles, 67
bias, 47
alternative hypothesis, 54
Amadea, 275
bipolar, 65
ANN, 61
AnswerTree, 275
BIRCH, 150
approximation, 1 3 , 241
bitmap index, 34
Apriori, 169-170
bivariate regression, 55
Apriori-Gen, 170
BM, 258
AprioriAll, 262
ARIMA, 257
boosting, 101
ARMA, 257
border point, 1 5 3
artificial intelligence, 1 2, 42
box plot, 51
Boyer-Moore, 258
association, 8
Braincel, 276
BrainMaker, 276
broker, 201
257
autocorrelation, 254
autocorrelation coefficient, 254
autoregression, 256
average, 130
average link, 133, 1 37
B-tree, 223
b-tree, 34
C4.5, 100-101
confusion matrlx, 80
connected component, 134
ID3, 97-100
contains, 267
Issues, 77-80
KNN , 90-92
RainForest, 103
context layer, 25 1
regression, 80-86
SPRINT, 103
classification rule, 1 14
classification tree, 93
conviction, 1 8 8
Clementine, 277
cookie, 209
Clever, 205
click, 208
clickstrearn, 206
clique, 1 37
correlation rule, 1 87
clustering, 7, 125-163
correlogram, 254
C5, 100
agglomerative, 132-137
cosine, 58
C5.0, 1 0 1 , 286
CACTUS, 162
BEA, 145-146
covariance, 254
BIRCH, 150-152
covering, 1 1 6
candidate, 168
categorical, 157-159
crawler, 198
CRH, 242
Capri, 286
CARMA, 1 9 1
CURE, 154-157
CRISP-DM, 1 8, 20
DBSCAN, 152-154
cross, 67
CCPD, 191
divisive, 1 3 8
crossover, 67, 68
Cubist, 277
cell, 223
hierarchical, 1 3 1-138
CURE, 154
center, 90
K-Means, 140-141
customer-sequence, 261
CF tree, 151
CFC, 200
CHAID, 123
PAM, 142-145
CHAMELEON, 163
partitional, 1 38-149
characterization, 8 , 234
backlink, 205
back-percolation, 276
children, 67
chromosome, 67
CLARA, 144
backward traversal, 2 1 1
CLARANS, 144
BANG, 24 1
class, 76, 89
classification, 5, 75-124
Bayes, 86, 1 99
B ayes Rule, 52
C4.5, 100-101
Bayes Theorem, 52
CART, 102-103
BEA, 145
compression, 12
concept, 27
concept hierarchy, 27, 184
confidence, 166, 1 8 8 , 234, 26 1 , 268-27 1
confidence interval, 47
DAG, 28, 2 1 5
Darwin; 277
data bubbles, 162
Data Distribution, 1 80
data mart, 3 8
data mining, 3 , 9
Data Mining Query Language, 1 8
data model, 2 1
data parallelism, 1 7 8
data scrubbing, 3 8
data staging, 38
data warehouse, 35, 36
database, 1 2, 2 1
Database Management System, 2 1
database segmentation, 125
DataEngine, 278
DataMite, 278
DB, 1 2
DBCLASD, 240
307
308
I ndex
I ndex
DBMiner, 279
DT model, 60
DBMS, 17, 21
hidden node, 6 1
EIS, 28
DBSCAN, 1 5 2
DCS, 120
DDA, 1 80
Decider, 279
Decider-Online, 279
decision support systems, 28
EM, 49
encompassing circle, 242
Enterprise Miner, 280
entity, 2 1
entropy, 97, 98
DecisionTime, 279
Eps, 1 5 2
flattened, 3 3
FN, 79
forecasting, 256
forward references, 2 1 5
FP , 79
frequency distribution, 5 1
frequent, 261
frequent itemset, 1 68
FSM, 248, 258
fuzzy association rule, 1 92
fuzzy logic, 24
Bps-neighborhood, 1 5 2
DENCLUE, 1 63
equivalence classes, 7 6
dendrogram, 1 3 1 , 23 1
equivalent, 263
g-sequence, 2 1 8
density-reachable, 153
ER data model, 2 1
GA, 68
descriptive model, 5
ER diagram, 2 1
Gain, 98
diameter, 129, 1 5 1
ESS, 28
GainRatio, 1 0 1
DIANA, 1 62
Dice, 58
dice, 40
fuzzy set, 23
dissimilarity measure, 58
distance, 58, 227
distance measure, 58, 90, 1 29
Euclidean, 58
manhattan, 58
distance scan, 223
distiller, 198
distributed, 178
division, 82
divisive clustering, 128
DMA, 1 9 1
DMG, 20
DMQL, 1 8, 202
domain, 22
downward closed, 170
drill down, 29, 40, 229
DSS, 28
Hybrid Distribution, 1 9 1
Hybrid OLAP, 40
hyperbolic tangent, 66
hyperbolic tangent activation function, 66
HyperText Markup Language, 198
hypothesis testing, 54
gene, 67
dissimilarity, 58
HTML, 198
hub, 1 98 , 205
IDD, 1 9 1
evolutionary computing, 67
HOLAP, 40
HPA, 1 9 1
Gaussian, 66
dimensional modeling, 29
HMM, 249
HNC Risk Suite, 280
GDBSCAN, 244
HITS, 205
GainSmarts, 280
histogram, 5 1
gatherer, 20 1
dimensions, 1 5
high dimensionality, 15
Essence, 20 1
evaluation, 1 0
dimensionality reduction, 1 5
hierarchical classifier, 1 99
Euclidean, 58
dimension table, 3 1
dimension, 29
GIS, 221
extrinsic, 1 28
Google, 205
fact, 29
fact table, 3 1
fallout, 80
false negative, 79
false positive, 79
GSP, 265
farthest neighbor, 1 37
GST, 2 1 1
FEATUREMINE, 266
feedback, 63
hard focus, 1 99
feedforward, 63
Harvest, 20 1
harvest rate, 1 99
hash tree, 1 82
Firefly, 204
HD, 1 9 1
fires, 65
heapify, 156
firing rule, 65
fitness, 68
fitness function, 68
hidden layer, 6 1
introduction, 3-20
inverse document frequency, 26, 20 1
IR, 12, 26
309
310
I n dex
I ndex
ISO/IEC, 20
MINT, 218
nonlinear regression, 85
mismatch, 272
nonparametric model, 46
311
ISP, 209
itemset, 1 66
lift, 188
MLDB, 201
iterative, 1 63
likelihood, 49
MLE, 49
nonstationary, 256
linear, 65
MLP, 1 1 3
Jaccard, 5 8
MM, 248
now, 247
mode, 5 1
jackknife estimate, 47
MOLAP, 39
null hypothesis, 54
link analysis, 8
momentum, 1 1 2
IDA Intellect, 28 1
location, 222
monothetic, 1 28
JDBCMine, 282
logistic regression, 85
OAT, 2 1 7
JDM, 285
LOGIT, 283
MSapriori, 1 87
join index, 34
OC curve, 79
MSE, 47, 1 08
Ockham's razor, 5 1
machine learning, 43
Magnify, 283
Magnum Opus, 283
major table, 3 1
manhattan, 58
manhattan distance, 58
Mantas, 283
map overlay, 222
KDD object, 23
KDD process, 1 0
KDDMS, 1 8
KDnuggets, 274
key, 21
KlvlP, 258
knowledge and data discovery management
system, 1 8
knowledge discovery i n databases, 9
knowledge discovery in spatial databases, 221
KnowledgeSEEKER, 282
KnowledgeSTUDIO, 282
Knuth-Morris-Pratt algorithm, 258
Kohonen, 148
Kohonen self organizing map, 148
lag, 254
Multidimensional OLAP, 3 9
multilayer perceptron, 1 1 3
OLTP, 23
multimedia data, 1 5
mutation, 68
operational data, 35
operative characteristic curve, 79
MarketMiner, 284
OPUS, 191
Opus, 283
OPTICS, 163
MARS, 284
KNN, 90
My Yahoo, 203
KDD, 9
MST, 1 3 5
Multidimensional Database, 3 9
sequences, 215, 2 1 6
outlier detection, 1 30
output layer, 61
neighborhood, 152
output node, 61
overfitting, 14, 77
MBR, 223
overlap, 58
MDD, 39
neural network, 63
mean, 51
page, 208
PageRank, 205
perceptron, 1 1 2
PAM, 142
median, 5 1
propagation, 105
PAR, 1 9 1
medoid, 129, 1 30
RBF, 1 1 2
parallel, 178
SOFM, 148
parallelization, 1 80
large, 261
parametric model, 46
metric, 129
parents, 67
Partek, 285
NN, 6 1 , 63, 1 03
partial-completeness, 1 92
NN model, 63
Partition, 177
LDA, 123
minor table, 3 1
noise, 82
partitional, 1 3 8
partitional clustering, 128
learning, 106
Minotaur, 284
Noisy data, 15
learning parameter, 1 1 0
Partitioning, 177
MinPts, 1 5 2
nonhierarchical, 1 3 8
312
Index
I n d ex
radial function, 1 12
S-Plus, 287
radius, 1 29
Sampling, 173
SLIQ, 124
RainForest, 103
SAND, 222
smoothing, 253
pattern discovery, 2 1 1
range, 5 1
satisfy, 269
snapshot, 245
pattern matching, 44
snowflake schema, 34
PDM, 1 9 1
raster, 226
scatter diagram, 52
SOM, 148
Pearson's r, 254
Scenario, 286
SPADE, 262
perceptron, 1 12, 1 1 3
RBF, 1 1 2
schema, 2 1
RBF network, 1 1 2
SD(CLARANS), 239
performance, 79
trees, 103
SOFM, 148
soft focus, 199
performance measures, 78
Re:order, 286
search, 13
recall, 26
search engine, 1 98
SeeS, 286
personalization, 202
point estimation, 47
PolyAnalyst, 285
seed URLs, 1 98
segmentation, 8, 125
polythetic, 128
segments, 8
population, 67
selection, 10
linear, 80
precision, 26
logistic, 85
nonlinear, 85
regression coefficients, 8 1
spider, 198
regressor, 55
sequence, 260
splitting attributes, 94
relation, 22
splitting predicates, 94
relational algebra, 22
SPRINT, 103, 1 24
SQL, 1 7, 23
relational calculus, 22
relational model, 22
sequence discovery, 9
Relational OLAP, 39
sequential analysis, 9
relationship, 2 1
squashing function, 64
serial, 128
tServ, 274
standard deviation, 5 1
relevance, 233
sessionize, 208
star schema, 3 1 , 32
relevant, 26
set, 23
stationary, 256
profiling, 1 6
reproduction, 67
SGML, 2 1 9
response, 55
shock, 256
statistical inference, 42
SIGKDD, 20
statistical sigrilficance, 54
pruning, 1 00
rms, 47
sigmoid, 66
statistics, 1 2
prefix, 263
preprocessing, 10
prior probability, 52, 87
PRISM, 1 1 7
privacy, 16
curve, 79
RMSE, 47
step, 65
RNN, 251
robot, 198
similarity, 26, 57
STING, 23 1
ROC curve, 79
strength, 166
cosine, 58
ROI, 1 5, 36
Dice, 58
subepisode, 267
ROLAP, 39
Jaccard, 58
subsequence, 260
overlap, 5 8
subseries, 252
simultaneous, 1 28
suffix tree, 2 1 0
slice, 40
summarization, 8
313
314
I n dex
SuperQuery, 287
269, 27 1
SurfAid Analytics, 288
surprise, 1 89
variance, 5 1
targeting, 1 96
vector, 226
Margaret H. Dunham received the B .A. and the M . S . in mathematics from Miami Uni
versity in Oxford, Ohio. She earned the Ph.D. degree in computer science from Southern
Methodist University. Professor Dunham' s research interests encompass main memory
databases, data mining, temporal databases, and mobile computing. She is currently an
Associate Editor for IEEE Transactions on Knowledge and Data Engineering. She has
published numerous technical papers in such research areas as database concurrency con
trol and recovery, database machines, main memory databases, and mobile computing.
VWV, 201
WAP-tree, 214. 2 1 9
WaveCluster, 241
time line, 1 2
TN, 79
topological relationship, 227
TP, 79
training data, 76
WebAnalyst, 288
transaction, 261
WEBMINER, 220
WebML, 202, 2 1 9
WebSIFf, 220
Whatlf, 279
WizWhy, 289
trie, 209
WordNet, 202
true negative, 79
true positive, 79
unbiased, 47
XML, 1 98
unipolar, 65
315