Visualizing Association Rules: Introduction To The R-Extension Package Arulesviz

Visualizing Association Rules: Introduction to the
R-extension Package arulesViz
Michael Hahsler Sudheer Chelluboina

Southern Methodist University Southern Methodist University
Abstract
Association rule mining is a popular data mining method available in R as the extension
package arules. However, mining association rules often results in a very large number
of found rules, leaving the analyst with the task to go through all the rules and discover
interesting ones. Sifting manually through large sets of rules is time consuming and
strenuous. Visualization has a long history of making large data sets better accessible
using techniques like selecting and zooming. In this paper we present the R-extension
package arulesViz which implements several known and novel visualization techniques to
explore association rules. With examples we show how these visualization techniques can
be used to analyze a data set.
Keywords: data mining, association rules, visualization.
1. Introduction
Many organizations generate a large amount of transaction data on a daily basis. For exam-
ple, a department store like “Macy’s” stores customer shopping information at a large scale
using check-out data. Association rule mining is one of the major techniques to detect and
extract useful information from large scale transaction data. Mining association rules was fist
introduced by Agrawal, Imielinski, and Swami (1993) and can formally be defined as:
Let I = {i1 , i2 , . . . , in } be a set of n binary attributes called items. Let D = {t1 , t2 , . . . , tm }
be a set of transactions called the database. Each transaction in D has an unique transaction
ID and contains a subset of the items in I. A rule is defined as an implication of the form
X ⇒ Y where X, Y ⊆ I and X ∩ Y = ∅. The sets of items (for short itemsets) X and Y
are called antecedent (left-hand-side or LHS) and consequent (right-hand-side or RHS) of the
rule. Often rules are restricted to only a single item in the consequent.
Association rules are rules which surpass a user-specified minimum support and minimum
confidence threshold. The support supp(X) of an itemset X is defined as the proportion of
transactions in the data set which contain the itemset and the confidence of a rule is defined
conf(X ⇒ Y ) = supp(X ∪ Y )/supp(X). Therefore, an association rule X ⇒ Y will satisfy:
supp(X ∪ Y ) ≥ σ
and
conf(X ⇒ Y ) ≥ δ
where σ and δ are the minimum support and minimum confidence, respectively.
2 Visualizing Association Rules
Another popular measure for association rules used throughout this paper is lift (Brin, Mot-
wani, Ullman, and Tsur 1997). The lift of a rule is defined as
lift(X ⇒ Y ) = supp(X ∪ Y )/(supp(X)supp(Y ))
and can be interpreted as the deviation of the support of the whole rule from the support
expected under independence given the supports of both sides of the rule. Greater lift values
( 1) indicate stronger associations. Measures like support, confidence and lift are generally
called interest measures because they help with focusing on potentially more interesting rules.
For a more detailed treatment of association rules we refer the reader to the in introduction
paper for package arules (Hahsler, Buchta, Grün, and Hornik 2010; Hahsler, Grün, and Hornik
2005) and the literature referred to there.
Association rules are typically generated in a two-step process. First, minimum support is
used to generate the set of all frequent itemsets for the data set. Frequent itemsets are itemsets
which satisfy the minimum support constraint. Then, in a second step, each frequent itemsets
is used to generate all possible rules from it and all rules which do not satisfy the minimum
confidence constraint are removed. Analyzing this process, it is easy to see that in the worst
case we will generate 2n − n − 1 frequent itemsets with more than two items from a database
with n distinct items. Since each frequent itemset will in the worst case generate at least two
rules, we will end up with a set of rules in the order of O(2n ). Typically, increasing minimum
support is used to keep the number of association rules found at a manageable size. However,
this also removes potentially interesting rules with less support. Therefore, the need to deal
with large sets of association rules is unavoidable when applying association rule mining in a
real setting.
Visualization is successfully used to communicate both abstract and concrete ideas in many ar-
eas like education, engineering and science (Prangsmal, van Boxtel, Kanselaar, and Kirschner
2009). According to Chen, Unwin, and Hardle (2008), the application of visualization falls
into two phases. First, the exploration phase where the analysts will use graphics that are
mostly incompatible for presentation purposes but make it easy to find interesting and impor-
tant features of the data. The amount of interaction needed during exploration is very high
and includes filtering, zooming and rearranging data. After key findings are discovered in the
data, these findings must be presented in a way suitable for presentation for a larger audi-
ence. In this second phase it is important that the analyst can manipulate the presentation
to clearly highlight the findings.
Many researchers introduced visualization techniques like scatter plots, matrix visualizations,
graphs, mosaic plots and parallel coordinates plots to analyze association rules (see Bruzzese
and Davino (2008) and Jentner and Keim (2017) for a recent overview). This paper discusses
existing techniques and demonstrates how their implementation in arulesViz can be used via
a simple unified interface. We extend most plots using techniques of color shading and re-
ordering to improve their interpretability. Finally, this paper also introduces a completely new
method called “grouped matrix-based visualization” which is based on a novel way of cluster-
ing rules. Clustering rules is usually based on distances defined on the items included in the
rules or on shared transactions covered by the rules. However, here we cluster rules especially
for visualization using similarities between sets of values of a selected interest measure.
The rest of the paper is organized as follows. In Section 2 we give a very short example of
how to prepare data using package arules and then introduce the unified interface provided
Michael Hahsler, Sudheer Chelluboina 3
by arulesViz for association rule visualization. In Sections 3 to 8 we describe the different

visualization techniques and give examples. Most of the techniques are enhanced using color
shading and reordering. Grouped matrix-based visualization in Section 5 is a novel visual-
ization technique. In Section 9 compares the presented visualization techniques. Section 11
concludes the paper.
2. Data preparation and unified interface of arulesViz

Before we start, we set the number of displayed significant digits to two to make the output
easier to read, and we set the seed for the random number generator for predictability.
> options(digits = 2)
> set.seed(1234)
To use arulesViz we fist have to load the package. The package automatically loads other
needed packages like arules (Hahsler et al. 2010) for handling and mining association rules.
For the examples in this paper we load the “Groceries” data set which is included in arules.
> library("arulesViz")
> data("Groceries")
Groceries contains sales data from a local grocery store with 9835 transactions and 169 items
(product groups). The summary shows some basic statistics of the data set. For example,
that the data set is rather sparse with a density just above 2.6%, that “whole milk” is the
most popular item and that the average transaction contains less than 5 items.
> summary(Groceries)
transactions as itemMatrix in sparse format with

9835 rows (elements/itemsets/transactions) and
169 columns (items) and a density of 0.026
most frequent items:

whole milk other vegetables rolls/buns soda
2513 1903 1809 1715
yogurt (Other)
1372 34055
element (itemset/transaction) length distribution:

sizes
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
2159 1643 1299 1005 855 645 545 438 350 246 182 117 78 77 55
16 17 18 19 20 21 22 23 24 26 27 28 29 32
46 29 14 14 9 11 4 6 1 1 1 1 3 1
Min. 1st Qu. Median Mean 3rd Qu. Max.

1 2 3 4 6 32
includes extended item information - examples:

labels level2 level1
1 frankfurter sausage meat and sausage
2 sausage sausage meat and sausage
3 liver loaf sausage meat and sausage
Next we mine association rules using the Apriori algorithm implemented in arules.
> rules <- apriori(Groceries, parameter=list(support=0.001, confidence=0.5))
Apriori
Parameter specification:
confidence minval smax arem aval originalSupport maxtime support minlen
0.5 0.1 1 none FALSE TRUE 5 0.001 1
maxlen target ext
10 rules FALSE
Algorithmic control:
filter tree heap memopt load sort verbose
0.1 TRUE TRUE FALSE TRUE 2 TRUE
Absolute minimum support count: 9
set item appearances ...[0 item(s)] done [0.00s].

set transactions ...[169 item(s), 9835 transaction(s)] done [0.00s].
sorting and recoding items ... [157 item(s)] done [0.00s].
creating transaction tree ... done [0.00s].
checking subsets of size 1 2 3 4 5 6 done [0.02s].
writing ... [5668 rule(s)] done [0.00s].
creating S4 object ... done [0.00s].
> rules
set of 5668 rules
The result is a set of 5668 association rules. The top three rules with respect to the lift
measure, a popular measure of rule strength, are:
> inspect(head(rules, n = 3, by ="lift"))
lhs rhs support confidence

[1] {Instant food products,soda} => {hamburger meat} 0.0012 0.63
[2] {soda,popcorn} => {salty snack} 0.0012 0.63
[3] {flour,baking powder} => {sugar} 0.0010 0.56

lift count
[1] 19 12
[2] 17 12
[3] 16 10
However, it is clear that going through all the 5668 rules manually is not a viable option.
We therefore will introduce different visualization techniques implemented in arulesViz. All
implemented visualization techniques share the following interface:
> args(getS3method("plot", "rules"))
function (x, method = NULL, measure = "support", shading = "lift",

interactive = NULL, engine = "default", data = NULL, control = NULL,
...)
NULL
where x is the set of rules to be visualized, method is the visualization method, and measure
and shading contain the interest measures used by the plot. Further arguments are described
in the manual page.
Using engine, different plotting engines can be specified to render the plot. the default engine
typically uses grid, many plots can also be rendered using the engine "htmlwidget" resulting
in an interactive HTML widget.
In the following sections we will introduce the different visualization methods implemented
in arulesViz and demonstrate how easy it is to use them.
3. Scatter plot
A straight-forward visualization of association rules is to use a scatter plot with two interest
measures on the axes. Such a presentation can be found already in an early paper by Bayardo,
Jr. and Agrawal (1999) when they discuss sc-optimal rules.
The default method for plot() for association rules in arulesViz is a scatter plot using support
and confidence on the axes. In addition a third measure (default: lift) is used as the color
(gray level) of the points. A color key is provided to the right of the plot.
> plot(rules)
This plot for the rules mined in the previous section is shown in Figure 1. We can see that rules
with high lift have typically a relatively low support. Bayardo, Jr. and Agrawal (1999) argue
that the most interesting rules (sc-optimal rules) reside on the support/confidence border,
which can be clearly seen in this plot. We will show later how the interactive features of this
plot can be used to explore these rules.
Any measure stored in the quality slot of the set of rules can be used for the axes (vector of
length 2 for parameter measure) or for color shading (shading). The following measures are
available for our set of rules.
Scatter plot for 5668 rules
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support
Figure 1: Default scatter plot.
> head(quality(rules))
support confidence lift count

1 0.0011 0.73 2.9 11
2 0.0012 0.52 2.8 12
3 0.0013 0.59 2.3 13
4 0.0013 0.57 2.2 13
5 0.0013 0.52 2.0 13
6 0.0037 0.64 2.5 36
These are the default measures generated by Apriori. To add other measures we refer the
reader to the function interestMeasure() included in arules. For example we can customize
the plot by switching lift and confidence:
> plot(rules, measure = c("support", "lift"), shading = "confidence")
Figure 2 shows this plot with lift on the y-axis. Here it is easy to identify all rules with high
lift.
Unwin, Hofmann, and Bernt (2001) introduced a special version of a scatter plot called Two-
key plot. Here support and confidence are used for the x and y-axes and the color of the
points is used to indicate “order,” i.e., the number of items contained in the rule. Two-key
plots can be produced using the unified interface by:
> plot(rules, method = "two-key plot")

Scatter plot for 5668 rules
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Figure 2: Scatter plot with lift on the y-axis.
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support
Figure 3: Two-key plot.

The resulting Two-key plot is shown in Figure 3. From the plot it is clear that order and
support have a very strong inverse relationship, which is a known fact for association rules
(Seno and Karypis 2005).
In addition to using order for shading, we also give the plot a different title (main). Other
control options including pch (best with filled symbols: 20–25), cex, xlim and ylim are
available and work in the usual way expected by R-users.
For exploration, the scatter plot method offers interactive features for selecting and zooming.
Interaction is activated using interactive=TRUE.
> sel <- plot(rules, measure=c("support", "lift"), shading = "confidence",

+ interactive = TRUE)
Interactive features include:
Inspecting individual rules by selecting them and clicking the inspect button.
Inspecting sets of rules by selecting a rectangular region of the plot and clicking the
inspect button.
Zooming into a selected region (zoom in/zoom out buttons).
Filtering rules using the measure used for shading by clicking the filter button and
selecting a cut-off point in the color key. All rules with a measure lower than the cut-off
point will be filtered.
Returning the last selection for further analysis (end button).
The result of an example interaction is shown in Figure 4. Using a box selection the rules
with the highest lift are selected. Using the inspect button, the rules are displayed in the
terminal below the plotting device.
4. Matrix-based visualizations
Matrix-based visualization techniques organize the antecedent and consequent itemsets on the
x and y-axes, respectively. A selected interest measure is displayed at the intersection of the
antecedent and consequent of a given rule. If no rule is available for a antecedent/consequent
combination the intersection area is left blank.
Formally, the visualized matrix is constructed as follows. We start with the set of association
rules
R = {ha1 , c1 , m1 i, . . . hai , ci , mi i, . . . han , cn , mn i}
where ai is the antecedent, ci is the consequent and mi is the selected interest measure for
the i-th rule for i = 1, . . . , n. In R we identify the set of K unique antecedents and L unique
consequent. We create a L × K matrix M with one column for each unique antecedent and
one row for each unique consequent. Finally, we populate the matrix by setting Mlk = mi
for i = 1, . . . , n and l and k corresponding to the position of ai and ci in the matrix. Note
that M will contain many empty cells since many potential association rules will not meet
the required minimum thresholds on support and confidence.
Figure 4: Interactive mode for scatter plot (inspecting rules with high lift).
Ong, leong Ong, Ng, and Lim (2002) presented a version of the matrix-based visualization
technique where a 2-dimensional matrix is used and the interest measure is represented by
color shading of squares at the intersection. An alternative visualization option is to use 3D
bars at the intersection (Wong, Whitney, and Thomas 1999; Ong et al. 2002).
For this type of visualization the number of rows/columns depends on the number of unique
itemsets in the consequent/antecedent in the set of rules. Since large sets of rules typically
have a large number of different itemsets as antecedents (often not much smaller than the
number of rules themselves), the size of the colored squares or the 3D bars gets very small
and hard to see. We reduce the number of rules here by filtering out all rules with a low
confidence score.
> subrules <- rules[quality(rules)$confidence > 0.8]

> subrules
set of 371 rules
> plot(subrules, method = "matrix", measure = "lift")
The resulting plot is shown in Figure 5. Since there is not much space for long labels in the
plot, we only show numbers as labels for rows and columns and the complete itemsets are
printed to the terminal for look-up. We omit the complete output here, since this plot and
the next few plots print several hundred labels to the screen. The output looks like:
Itemsets in Antecedent (lhs)

[1] "{liquor,red/blush wine}"
[2] "{curd,cereals}"
[3] "{yogurt,cereals}"
[4] "{butter,jam}"
[5] "{soups,bottled beer}"
(lines omitted)
[343] "{tropical fruit,root vegetables,rolls/buns,bottled water}"
[344] "{tropical fruit,root vegetables,yogurt,rolls/buns}"
Itemsets in Consequent (rhs)
[1] "{bottled beer}" "{whole milk}" "{other vegetables}"
[4] "{tropical fruit}" "{yogurt}" "{root vegetables}"
An alternative representation is to use 3D bars (method “matrix3D”) instead of colored rect-

angles.
> plot(subrules, method = "matrix3D", measure = "lift")
The 3D visualization is shown in Figure 6.
5. Grouped matrix-based visualization

Matrix with 371 rules

Consequent (RHS)
6
10
5
4 8
3
6
2
1 4
lift
1 613223140495867768594105118131144157170183196209222235248261274287300313326339352
Antecedent (LHS)
Figure 5: Matrix-based visualization with colored squares.
Matrix with 371 rules

12
10
8
Antecedent (LHS)
lift
400
300
4
200
100
0
2
1 2 3 4 5 6
Consequent (RHS)
Figure 6: Matrix-based visualization with 3D bars.

Matrix-based visualization is limited in the number of rules it can visualize effectively since
large sets of rules typically also have large sets of unique antecedents/consequents. Here we
introduce a new visualization techniques (Hahsler and Chelluboina 2011; Hahsler and Karpi-
enko 2016) that enhances matrix-based visualization using grouping of rules via clustering to
handle a larger number of rules. Grouped rules are presented as an aggregate in the matrix
and can be explored interactively by zooming into and out of groups.
A direct approach to cluster itemsets is to define a distance metric between two itemsets Xi
and Xj . A good choice is the Jaccard distance defined as
|Xi ∩ Xj |
dJaccard (Xi , Xj ) = 1 − .
|Xi ∪ Xj |
The distance simply is the number of items that Xi and Xj have in common divided by the
number of unique items in both sets. For a set of m rules we can calculate the m(m − 1)/2
distances and use them as the input for clustering. However, using clustering on the itemsets
directly has several problems. First of all, data sets typically mined for association rules
are high-dimensional, i.e., contain many different items. This high dimensionality is carried
over to the mined rules and leads to a situation referred is as the “course of dimensionality”
where, due to the exponentially increasing volume, distance functions lose their usefulness.
The situation is getting worse since minimum support used in association rule mining leads
in addition to relatively short rules resulting in extremely sparse data.
Several approaches to cluster association rules and itemsets to address the dimensionality
and sparseness problem were proposed in the literature. Toivonen, Klemettinen, Ronkainen,
Hatonen, and Mannila (1995), Gupta, Strehl, and Ghosh (1999) and Berrado and Runger
(2007) propose clustering association rules by looking at the number of transactions which
are covered by the rules. Using common covered transactions avoids the problems of clus-
tering sparse, high-dimensional binary vectors. However, it introduces a strong bias towards
clustering rules which are generated from the same frequent itemset. By definition of frequent
itemsets, two subsets of a frequent itemset will cover many common transactions. This bias
will lead to mostly just rediscovering the already known frequent itemset structure from the
set of association rules.
Here we pursue a completely different approach. We start with the matrix M defined in
Section 4 which contains the values of a selected interest measure of the rules in set R.
The columns/rows are the unique antecedents/consequents in R, respectively. Now grouping
antecedents becomes the problem of grouping columns in M. To group the column vectors
fast and efficient into k groups we use k-means clustering. The default interest measure used
is lift. The idea is that antecedents that are statistically dependent on the same consequents
are similar and thus can be grouped together. Compared to other clustering approaches for
itemsets, this method enables us to even group antecedents containing substitutes (e.g., butter
and margarine) which are rarely purchased together since they will have similar dependence
to the same consequents.
The same grouping method can be used for consequents. However, since the mined rules are
restricted to a single item in the consequent there is no problem with combinatorial explosion
and such a grouping is typically not necessary.
To visualize the grouped matrix we use a balloon plot with antecedent groups as columns
and consequents as rows (see Figure 7). The color of the balloons represent the aggregated
interest measure in the group with a certain consequent and the size of the balloon shows the
aggregated support. The default aggregation function is the median value in the group. The
number of antecedents and the most important (frequent) items in the group are displayed
as the labels for the columns. Furthermore, the columns and rows in the plot are reordered
such that the aggregated interest measure is decreasing from top down and from left to right,
placing the most interesting group in the top left corner.
The matrix visualization with grouped antecedents for the set of 5668 rules mined earlier can
be easily created by
> plot(rules, method = "grouped")
The resulting visualization is shown in Figure 7. The group of most interesting rules according
to lift (the default measure) are shown in the top-left corner of the plot. There are 3 rules
which contain “Instant food products” and up to 2 other items in the antecedent and the
consequent is “hamburger meat.”
To increase the number of groups we can change k which defaults to 20.
> plot(rules, method = "grouped", control = list(k = 50))
The resulting, more detailed plot is shown in Figure 8.

An interactive version of the grouped matrix visualization is also available.
> sel <- plot(rules, method = "grouped", interactive = TRUE)
Here it is possible to zoom into groups and to inspect the rules contained in a selected group.
6. Graph-based visualizations
Graph-based techniques (Klemettinen, Mannila, Ronkainen, Toivonen, and Verkamo 1994;
Rainsford and Roddick 2000; Buono and Costabile 2005; Ertek and Demiriz 2006) visualize
association rules using vertices and edges where vertices annodated with item labels represent
items, and itemsets or rules are reptesented as a second set of vertices. Items are connected
with itemsets/rules using arrows. For rules arrows pointing from items to rule vertices indicate
LHS items and an arrow from a rule to an item indicates the RHS. Interest measures are
typically added to the plot by using color or size of the vertices representing the itemsets/rules.
Graph-based visualization offers a very clear representation of rules but they tend to easily
become cluttered and thus are only viable for very small sets of rules. For the following plots
we select the 10 rules with the highest lift.
> subrules2 <- head(rules, n = 10, by = "lift")
arulesViz contains several graph-based visualizations rendered using either the igraph library
via package igraph (Csardi and Nepusz 2006) or the interface to the GraphViz software in
package Rgraphviz (Gentry, Long, Gentleman, Falcon, Hahne, Sarkar, and Hansen 2010). By
default igraph is used. The following plot represents items and rules as vertices connecting
them with directed edges (shown in Figure 9). This representation focuses on how the rules
are composed of individual items and shows which rules share items.
14
Items in LHS Group

3 rules: {Instant food products, soda, +1 items}
4 rules: {processed cheese, ham, +1 items}
56 rules: {popcorn, white wine, +35 items}
646 rules: {liver loaf, frozen potato products, +80 items}
332 rules: {semi−finished bread, dog food, +50 items}
1022 rules: {specialty chocolate, packaged fruit/vegetables, +100 items}
295 rules: {soups, beverages, +75 items}
30 rules: {hard cheese, hamburger meat, +11 items}
111 rules: {jam, salt, +31 items}
16 rules: {hard cheese, sugar, +6 items}
345 rules: {mayonnaise, whole milk, +57 items}
70 rules: {instant coffee, hard cheese, +23 items}
239 rules: {rice, herbs, +37 items}
914 rules: {flower (seeds), detergent, +68 items}
351 rules: {soups, butter milk, +50 items}
184 rules: {pasta, mustard, +46 items}
36 rules: {ham, frozen fish, +20 items}
486 rules: {dog food, curd, +50 items}
147 rules: {instant coffee, turkey, +56 items}
381 rules: {vinegar, UHT−milk, +80 items}
{curd}
{beef}
{sugar}
Grouped Matrix for 5668 Rules
{butter}
Visualizing Association Rules
RHS
{salty snack}
{white bread}
{bottled beer}
+ 15 supressed
{domestic eggs}
Figure 7: Grouped matrix-based visualization.

{cream cheese }
{hamburger meat}
Size: support
Color: lift
Figure 8: Grouped matrix with k = 50.
Items in LHS Group
1 rules: {popcorn, soda}
1 rules: {flour, baking powder}
3 rules: {Instant food products, soda, +1 items}
4 rules: {processed cheese, ham, +1 items}
3 rules: {curd, whipped/sour cream, +2 items}
39 rules: {specialty bar, mayonnaise, +34 items}
9 rules: {liquor, red/blush wine, +12 items}
31 rules: {candy, cream cheese , +24 items}
30 rules: {hard cheese, hamburger meat, +11 items}
5 rules: {pastry, cream cheese , +4 items}
16 rules: {hard cheese, sugar, +6 items}
61 rules: {misc. beverages, UHT−milk, +38 items}
8 rules: {misc. beverages, specialty chocolate, +6 items}
24 rules: {herbs, fruit/vegetable juice, +15 items}
70 rules: {instant coffee, hard cheese, +23 items}
84 rules: {grapes, turkey, +46 items}
22 rules: {soft cheese, margarine, +14 items}
47 rules: {frozen dessert, sliced cheese, +19 items}
463 rules: {cooking chocolate, specialty chocolate, +83 items}
46 rules: {semi−finished bread, frozen meals, +22 items}
91 rules: {rice, herbs, +23 items}
36 rules: {ham, frozen fish, +20 items}
40 rules: {rice, flour, +20 items}
128 rules: {dog food, grapes, +38 items}
94 rules: {herbs, oil, +29 items}
50 rules: {frozen meals, rice, +23 items}
123 rules: {rice, soft cheese, +25 items}
72 rules: {turkey, frozen fish, +23 items}
192 rules: {jam, salt, +48 items}
144 rules: {mayonnaise, specialty cheese, +42 items}
234 rules: {specialty bar, soups, +71 items}
96 rules: {beverages, candy, +40 items}
102 rules: {meat, butter milk, +34 items}
164 rules: {cereals, hygiene articles, +56 items}
82 rules: {butter milk, chicken, +26 items}
295 rules: {soft cheese, curd, +39 items}
161 rules: {dog food, frozen meals, +40 items}
225 rules: {candles, dishes, +66 items}
168 rules: {soups, mayonnaise, +40 items}
293 rules: {honey, cereals, +71 items}
Grouped Matrix for 5668 Rules
121 rules: {jam, specialty cheese, +37 items}

92 rules: {canned vegetables, frozen dessert, +37 items}
54 rules: {white wine, popcorn, +32 items}
270 rules: {vinegar, UHT−milk, +74 items}
44 rules: {tidbits, spread cheese, +30 items}
192 rules: {flower (seeds), pickled vegetables, +50 items}
214 rules: {detergent, pot plants, +44 items}
294 rules: {liver loaf, frozen potato products, +65 items}
234 rules: {house keeping products, ice cream, +55 items}
396 rules: {packaged fruit/vegetables, pudding powder, +76 items}
{curd}
{beef}
{sugar}
{butter}
RHS
{salty snack}
{white bread}
{bottled beer}
+ 15 supressed
{domestic eggs}
{cream cheese }
{hamburger meat}
Size: support
Color: lift
15 Michael Hahsler, Sudheer Chelluboina
> plot(subrules2, method = "graph")
An interactive visualization is available in arulesViz, however, the built-in graph based visu-
alizations are only useful for small set of rules. To explore large sets of rules with graphs,
advanced interactive features like zooming, filtering, grouping and coloring nodes are needed.
Such features are available in interactive visualization and exploration platforms for networks
and graphs like Gephi (Bastian, Heymann, and Jacomy 2009). From arulesViz graphs for
sets of association rules can be exported in the GraphML format or as a Graphviz dot-file to
be explored in tools like Gephi. For example the 1000 rules with the highest lift are exported
by:
> saveAsGraph(head(rules, n = 1000, by = "lift"), file = "rules.graphml")
Figure 10 shows a screenshot of exploring these rules interactively. Rules can be explored by
zooming, filtering and coloring vertices and edges.
7. Parallel coordinates plot

Parallel coordinates plots are designed to visualize multidimensional data where each di-
mension is displayed separately on the x-axis and the y-axis is shared. Each data point is
represented by a line connecting the values for each dimension. Parallel coordinates plots were
used previously to visualize discovered classification rules (Han, An, and Cercone 2000) and
association rules (Yang 2003). Yang (2003) displays the items on the y-axis as nominal values
and the x-axis represents the positions in a rule, i.e., first item, second item, etc. Instead of a
simple line an arrow is used where the head points to the consequent item. Arrows only span
enough positions on the x-axis to represent all the items in the rule, i.e., rules with less items
are shorter arrows.
> plot(subrules2, method = "paracoord")
Figure 11 shows a parallel coordinates plot for 10 rules. The width of the arrows represents
support and the intensity of the color represent confidence. It is obvious that for larger rule
sets visual analysis becomes difficult since with an increasing number of rules also the number
of crossovers between the lines increases Yang (2003).
The number of crossovers can be significantly reduced by reordering the items on the y-axis.
Reordering the items to minimize the number of crossovers is a combinatorial problem with
n! possible permutations. However, for visualization purposes a suboptimal but fast solution
is typically acceptable. We applies a variation of the well known 2-opt heuristic Bentley
(1990) for travelers salesman problem to the reordering problem. The objective function is
to minimize the number of crossovers. The simple heuristic uses the following steps:
1. Choose randomly two items and exchange them if it improves the objective function.
2. Repeat step 1 till no improvement is found for a predefined number of tries.
Reordering is achieved with reorder=TRUE.

Graph for 10 rules

size: support (0.001 − 0.002)
color: lift (11.279 − 18.996)
flour
baking powder
sugar
popcorn
soda
curd whipped/sour cream salty snack

hamburger meat
Instant food products

cream cheese
yogurt
butter
whole milk
other vegetables
ham
tropical fruit white bread
domestic eggs
processed cheese
Figure 9: Graph-based visualization with items and rules as vertices.

{white{shopping bags}
{sausage,rolls/buns,canned beer}
{fruit/vegetable juice,red/blush wine}
{coffee,misc. beverages}
{pastry} bread}
{citrus fruit,whole milk,whipped/sour cream,rolls/buns}
{butter,yogurt,hard cheese}
{hamburger meat,butter,yogurt}
{root vegetables,yogurt,frozen vegetables}
{citrus fruit,whole milk,rolls/buns,pastry}
{other vegetables,curd,frozen vegetables}
{butter,yogurt,pastry} {yogurt,cream cheese ,margarine}

{other vegetables,whole milk,butter,soda}
{rolls/buns,soda,canned beer}
{other vegetables,butter,sugar}
{ham,processed cheese}
{hamburger{soda}
{butter,hard cheese}
meat}
{root vegetables,other vegetables,yogurt,frozen vegetables}
{Instant food products,soda}

{sausage,citrus fruit,other vegetables,whole milk}
{other vegetables,instant coffee}

{tropical fruit,other vegetables,butter,fruit/vegetable juice}
{other vegetables,whole milk,butter,napkins}
{sausage,bottled water,bottled beer}
{curd}
{citrus fruit,whole milk,sugar}
{root vegetables,butter,napkins}
{root vegetables,whole milk,flour}
{processed cheese,domestic eggs}
{whole milk,Instant food products}
{citrus fruit,other vegetables,whole milk,cream cheese }
{curd,yogurt,sugar}
{other vegetables,misc. beverages,fruit/vegetable juice}
{citrus fruit,root vegetables,whole milk,fruit/vegetable juice}

{root vegetables,butter,fruit/vegetable juice}
{tropical fruit,other vegetables,yogurt,oil}

{pip fruit,root vegetables,whole milk,rolls/buns}
{other vegetables,rolls/buns,oil}
{yogurt,rolls/buns,bottled water,newspapers} {citrus fruit,other vegetables,soda}
{sausage,beef,whole milk}
{citrus fruit,pip fruit,rolls/buns} {pip fruit,other vegetables,frozen meals}

{soft cheese,cream cheese } {other vegetables,yogurt,sugar}
{whole milk,curd,yogurt,cream cheese }
{whole milk,frozen vegetables,napkins}
{pip fruit,other vegetables,curd,yogurt}

{other vegetables,whole milk,butter,yogurt}
{other vegetables,curd,bottled water}
{pip fruit,cat food}
{pip fruit,root vegetables,other vegetables,whole milk,yogurt}

{root vegetables,yogurt,sliced cheese}
{tropical fruit,other vegetables,whole milk,yogurt,bottled water}
{citrus fruit,herbs}
{other vegetables,whole milk,frozen meals}
{sausage,beef,rolls/buns}
{beef,citrus fruit,tropical fruit}
{beef,other vegetables,newspapers}
{sausage,beef}
{rolls/buns}
{tropical fruit,other vegetables,whole milk,white bread}
{other vegetables,yogurt,semi-finished bread} {grapes,other vegetables,whole milk,yogurt}

{sausage,beef,butter}
{spread cheese,newspapers}
{other vegetables,whole milk,yogurt,waffles}
{citrus fruit,pip fruit,other vegetables,whole milk} {other vegetables,soda,napkins} {curd,yogurt,cream cheese }
{pip fruit,misc. beverages}
{citrus fruit,other vegetables,whole milk,soda}
{pip fruit,whipped/sour cream,shopping bags}

{other vegetables,whipped/sour cream,soft cheese}
{beef,tropical fruit,other vegetables,whole milk,rolls/buns}
{root vegetables,whole milk,butter,yogurt}

{tropical fruit,whole milk,oil}
{beef,tropical fruit,whole milk}
{beef,root vegetables,whole milk,rolls/buns}
{chicken,tropical fruit,whole milk}

{tropical fruit,other vegetables,whole milk,frozen vegetables}
{root vegetables,yogurt,waffles}
{other vegetables,yogurt,whipped/sour cream,cream cheese } {grapes,other vegetables,pastry}
{whipped/sour cream}
{root vegetables,other vegetables,yogurt,domestic eggs}
{pip fruit,yogurt,sliced cheese}
{citrus fruit,other vegetables,soda,fruit/vegetable juice}
{beef,whole milk,butter,rolls/buns} {beef,other vegetables,yogurt,rolls/buns}

{other vegetables,butter,frozen vegetables}
{beef,pip fruit,whole milk}

{root vegetables,other vegetables,yogurt,whipped/sour cream}
{onions,other vegetables,whole milk,rolls/buns}

{root vegetables,other vegetables,yogurt,bottled water}
{citrus fruit,pip fruit,whole milk,yogurt}
{beef,dessert}
{whole milk,domestic eggs,chocolate}

{yogurt,soda,candy}
{onions,butter}
{yogurt,coffee,fruit/vegetable juice} {sausage,whole milk,yogurt,pastry}
{butter,yogurt,white bread}
{herbs,other vegetables,whole milk,rolls/buns}
{herbs,whole milk,curd}
{pip fruit,yogurt,brown bread}

{tropical fruit,whole milk,yogurt,margarine}
{other vegetables,whole milk,frozen vegetables,bottled water}

{sliced cheese,frozen vegetables}
{herbs,other vegetables,whole milk}

{pip fruit,whole milk,hygiene articles}
{citrus fruit,grapes,fruit/vegetable juice}
{other vegetables,whole milk,processed cheese}
{citrus fruit,other vegetables,whole milk,brown bread}
{other vegetables,yogurt,soft cheese}
{pork,other vegetables,whole milk,domestic eggs}

{citrus fruit,other vegetables,cat food} {tropical fruit,other vegetables,whole milk,yogurt,rolls/buns}
{UHT-milk,hygiene articles} {onions,butter,yogurt}
{ham,other vegetables,yogurt} {herbs,whole milk}
{ham,other vegetables,fruit/vegetable juice}
{sausage}
{butter,yogurt,brown bread}
{root vegetables,other vegetables,whole milk,oil} {citrus fruit,other vegetables,whole milk,oil}
{sausage,beef,other vegetables}
{pip fruit,whole milk,yogurt,white bread}
{chicken,tropical fruit,other vegetables,whole milk}
{herbs,other vegetables,bottled water}

{whole milk,white bread,napkins}
{pip fruit,other vegetables,whole milk,soda}
{chicken,other vegetables,whole milk,domestic eggs}
{yogurt,frozen vegetables,bottled water} {beef,other vegetables,whole milk,butter} {citrus fruit,tropical fruit,other vegetables,rolls/buns}
{frankfurter,pip fruit,domestic eggs}
{citrus fruit,tropical fruit,other vegetables,yogurt}

{root vegetables,other vegetables,whole milk,yogurt,oil}
{root vegetables,yogurt,whipped/sour cream,rolls/buns}
{pip fruit,herbs}
{whole milk,butter,curd,yogurt} {other vegetables,curd,yogurt,cream cheese }

{domestic eggs,semi-finished bread}
{herbs,whole milk,yogurt}
{domestic eggs}
{beef,other vegetables,butter}
{whole milk,bottled water,hygiene articles} {beef,tropical fruit,whole milk,yogurt}

{ham,pip fruit,yogurt}
{whole milk,butter,oil} {beef,whole milk,waffles}

{yogurt,dog food} {whole milk,curd,fruit/vegetable juice}
{frankfurter,sausage,yogurt}
{whole milk,yogurt,whipped/sour cream,domestic eggs} {citrus fruit,other vegetables,hard cheese}
{root vegetables,other vegetables,whole milk,butter,yogurt} {beef,whole milk,whipped/sour cream}
{root vegetables,other vegetables,yogurt,oil}

{root vegetables,brown bread,fruit/vegetable juice}
{sausage,beef,yogurt}
{pork,semi-finished bread}
{ham,pip fruit,other vegetables,whole milk} {cream cheese ,oil}
{other vegetables,butter,long life bakery product}
{chicken,other vegetables,fruit/vegetable juice}
{other vegetables,fruit/vegetable juice,napkins}
{citrus fruit,tropical fruit,other vegetables,fruit/vegetable juice}
{citrus fruit,other vegetables,whole milk,yogurt}

{yogurt,pastry,coffee} {tropical fruit,roll products } {onions,whole milk,whipped/sour cream}
{citrus fruit,other vegetables,whole milk,fruit/vegetable juice}

{whole milk,cream cheese ,sugar}
{root vegetables,other vegetables,whole milk,yogurt,rolls/buns}
{tropical fruit,whole milk,yogurt,brown bread}
{yogurt,rolls/buns,candy}
{citrus fruit,other vegetables,whole milk,yogurt,whipped/sour cream}
{beef,other vegetables,whole milk,soda} {citrus fruit,tropical fruit,other vegetables,whole milk}

{pip fruit,whole milk,yogurt,frozen meals}
{citrus fruit,grapes,other vegetables}
{beef,pasta}
{ham,other vegetables,whole milk,yogurt} {onions,other vegetables,frozen vegetables} {other vegetables,yogurt,mayonnaise}

{beef,sliced cheese}
{beef,butter}
{whole milk,white bread,bottled beer}

{other vegetables,butter,hard cheese}
{herbs,other vegetables,yogurt}
{other vegetables,yogurt,whipped/sour cream,bottled water}
{citrus fruit,root vegetables,other vegetables,yogurt}
{citrus fruit,rice}
{onions,hard cheese}
{other vegetables,whole milk,domestic eggs,fruit/vegetable juice}
{citrus fruit,tropical fruit,hygiene articles}

{root vegetables,whole milk,yogurt,domestic eggs} {root vegetables,whole milk,yogurt,soda}
{other vegetables,yogurt,whipped/sour cream,margarine} {other vegetables,whole milk,yogurt,oil}
{citrus fruit,tropical fruit,pip fruit,whole milk}

{other vegetables,yogurt,whipped/sour cream,rolls/buns}
{pip fruit,other vegetables,sliced cheese} {citrus fruit,whole milk,cream cheese }
{beef,whole milk,yogurt,rolls/buns}
{rice,fruit/vegetable juice}
{tropical fruit,pip fruit,other vegetables,whole milk}
{yogurt,whipped/sour cream,sliced cheese}
{onions,other vegetables,whole milk,whipped/sour cream}

{semi-finished bread,margarine}
{whole milk,domestic eggs,oil}

{other vegetables,whole milk,yogurt,rice}
{beef,yogurt,cream cheese }
{citrus fruit,misc. beverages}
{pip fruit,yogurt,napkins} {root vegetables,other vegetables,yogurt,margarine}

{pip fruit,root vegetables,rolls/buns} {butter,rice}
{other vegetables,butter,whipped/sour cream,fruit/vegetable juice}

{pip fruit,frozen fish}
{other vegetables,soft cheese,domestic eggs}
{other vegetables,butter,yogurt,domestic eggs}

{tropical fruit,other vegetables,whole milk,brown bread}
{pork,cat food} {citrus fruit,other vegetables,frozen vegetables}
{citrus fruit,pip fruit,root vegetables,other vegetables} {other vegetables,whole milk,sliced cheese} {sausage,chicken,yogurt} {other vegetables,whole milk,yogurt,shopping bags}
{sausage,beef,tropical fruit}
{other vegetables,jam} {citrus fruit,other vegetables,soft cheese}
{turkey,yogurt}
{other vegetables,whipped/sour cream,sliced cheese} {tropical fruit,other vegetables,yogurt,brown bread}
{beef,waffles}
{tropical fruit,yogurt,frozen meals}
{pip fruit,other vegetables,bottled beer} {tropical fruit,other vegetables,whole milk,domestic eggs}
{other vegetables,whole milk,whipped/sour cream,bottled water} {whole milk,butter,yogurt,sliced cheese}

{whole milk,curd,yogurt,domestic eggs}
{yogurt,whipped/sour cream,bottled water}
{citrus fruit,tropical fruit,other vegetables,whole milk,yogurt} {other vegetables,whole milk,butter,bottled water}
{whole milk,yogurt,semi-finished bread}

{whole milk,yogurt,whipped/sour cream,bottled water} {frankfurter,root vegetables,whole milk,yogurt}
{pip fruit,other vegetables,brown bread} {frankfurter,beef,rolls/buns}
{other vegetables,sugar,fruit/vegetable juice} {beef,citrus fruit,whole milk}
{citrus fruit,tropical fruit,frozen vegetables}

{whole milk,butter,rice}
{tropical fruit,whole milk,yogurt,domestic eggs}
{other vegetables,whole milk,yogurt,bottled water}
{turkey,root vegetables,other vegetables}

{whole milk,yogurt,candy}
{liquor,red/blush wine}
{butter,whipped/sour cream,napkins}
{root vegetables,other vegetables,butter,whipped/sour cream}

{other vegetables,yogurt,sliced cheese} {beef,citrus fruit,other vegetables,whole milk}
{tropical fruit}
{butter,oil}
{sausage,chicken,citrus fruit}
{citrus fruit,root vegetables,other vegetables,whipped/sour cream}
{root vegetables,yogurt,newspapers}
{other vegetables,yogurt,rolls/buns,newspapers}
{pork,citrus fruit,other vegetables}
{pork,other vegetables,fruit/vegetable juice}
{ham,tropical fruit,other vegetables,yogurt}
{other vegetables,whole milk,yogurt,sugar}
{beef,citrus fruit,other vegetables}

{sausage,yogurt,sliced cheese}
{other vegetables,whole milk,yogurt,white bread}
{root vegetables,yogurt,oil}
{other vegetables,whole milk,bottled water,fruit/vegetable juice}

{pip fruit,yogurt,frozen meals}
{sausage,whole milk,newspapers}
{tropical fruit,other vegetables,hard cheese}
{beef,herbs}
{frozen vegetables,semi-finished bread}
{whole milk,whipped/sour cream,soft cheese}
{pork,yogurt,frozen vegetables}
{citrus fruit,whole milk,oil}

{yogurt,domestic eggs,white bread}
{frankfurter,tropical fruit,whole milk,yogurt}

{frankfurter,chicken,whole milk}
{other vegetables,whole milk,frozen dessert} {tropical fruit,pip fruit,other vegetables,whole milk,yogurt}

{pip fruit,grapes}
{citrus fruit,root vegetables,other vegetables,fruit/vegetable juice} {beef,rolls/buns,pastry}
{citrus fruit,herbs,other vegetables,whole milk}
{beef,other vegetables,whole milk,rolls/buns}
{beef,other vegetables,soda}
{root vegetables,whole milk,yogurt,white bread}
{root vegetables}
{beef,hamburger meat,whole milk}
{root vegetables,other vegetables,butter,yogurt} {beef,tropical fruit,other vegetables,whole milk}

{citrus fruit,root vegetables,other vegetables,rolls/buns} {beef,citrus fruit,whipped/sour cream}
{soda,liquor}
{other vegetables,butter,sliced cheese}
{chicken,citrus fruit,other vegetables}
{other vegetables,butter,bottled water}
{root vegetables,yogurt,bottled water}

{herbs,domestic eggs}
{tropical fruit,whole milk,yogurt,frozen meals}
{pip fruit,root vegetables,other vegetables,whipped/sour cream} {whole milk,butter,yogurt,domestic eggs}
{whole milk,yogurt,rolls/buns,pastry}
{root vegetables,other vegetables,whole milk,yogurt,bottled water} {pip fruit,whole milk,yogurt,soda}

{citrus fruit,whole milk,frozen vegetables}
{other vegetables,whole milk,canned vegetables}
{bottled beer}
{whole milk,long life bakery product,newspapers} {herbs,curd}
{grapes,other vegetables,whole milk}
{tropical fruit,herbs} {other vegetables,rolls/buns,flour}
{pip fruit,salt}
{pip fruit,other vegetables,whole milk,brown bread} {citrus fruit,pip fruit,frozen vegetables}
{sausage,yogurt,long life bakery product}
{beef,tropical fruit,rolls/buns}
{whole milk,butter,hard cheese}
{beef,other vegetables,whole milk,whipped/sour cream} {citrus fruit,root vegetables,other vegetables,whole milk,yogurt}
{other vegetables,butter,curd,yogurt} {beef,tropical fruit,other vegetables}
{pork,citrus fruit,whole milk}

{root vegetables,yogurt,domestic eggs} {tropical fruit,other vegetables,whole milk,yogurt,oil}
{pip fruit}
{chicken,tropical fruit,other vegetables}
{tropical fruit,other vegetables,soft cheese}
{beef,butter,rolls/buns}
{sausage,whole milk,hygiene articles}
{whole milk,butter,sliced cheese}
{pip fruit,root vegetables,other vegetables,yogurt}
{root vegetables,butter,yogurt}
{root vegetables,whole milk,soft cheese}

{pip fruit,other vegetables,yogurt,whipped/sour cream}
{other vegetables,whole milk,oil}
{butter,yogurt,sliced cheese} {turkey,tropical fruit}
{beef}
{pip fruit,other vegetables,frozen fish}
{other vegetables,whole milk,whipped/sour cream,napkins}
{ham,pip fruit,other vegetables}
{other vegetables,whole milk,whipped/sour cream,sliced cheese}
{other vegetables,whole milk,yogurt,sliced cheese}

{other vegetables,whole milk,whipped/sour cream,hard cheese} {whole milk,butter,soft cheese}
{sausage,herbs}
{berries,bottled beer}
{whole milk,frozen meals,soda}
{other vegetables,domestic eggs,fruit/vegetable juice}
{beef,onions,other vegetables}
{yogurt,domestic eggs,margarine}
{citrus fruit,other vegetables,napkins}
{turkey,root vegetables}
{root vegetables,whole milk,yogurt,bottled water} {other vegetables,hard cheese,domestic eggs}
{frankfurter,whole milk,frozen meals} {curd,hard cheese}
{herbs,other vegetables,shopping bags} {tropical fruit,root vegetables,curd}

{curd,yogurt,bottled water}
{tropical fruit,whole milk,yogurt,sliced cheese}
{frankfurter,tropical fruit,domestic eggs}

{other vegetables,whole milk,rice}
{tropical fruit,other vegetables,yogurt,domestic eggs}
{onions,other vegetables,napkins}
{butter,whipped/sour cream,cream cheese }
{yogurt,whipped/sour cream,white bread} {tropical fruit,whipped/sour cream,shopping bags}
{pip fruit,root vegetables,whole milk,yogurt}
{whole milk,sliced cheese,fruit/vegetable juice}
{other vegetables,whole milk,butter,white bread}
{beef,root vegetables,other vegetables,whole milk,rolls/buns} {pip fruit,other vegetables,whole milk,bottled water}

{citrus fruit,other vegetables,whole milk,newspapers} {whole milk,curd,frozen meals}
{pip fruit,yogurt,hygiene articles} {onions,whole milk,butter}
{frankfurter,other vegetables,fruit/vegetable juice}

{frankfurter,other vegetables,whole milk,yogurt}
{other vegetables,domestic eggs,white bread}
{turkey,other vegetables,whole milk}
{tropical fruit,other vegetables,whole milk,cream cheese } {pip fruit,other vegetables,whole milk,cream cheese }
{hamburger meat,yogurt,whipped/sour cream}
{frankfurter,other vegetables,frozen meals}
{ham,pip fruit,other vegetables,yogurt} {yogurt,processed cheese} {tropical fruit,other vegetables,oil}
{whole milk,yogurt,whipped/sour cream,margarine}
{tropical fruit,root vegetables,whole milk,whipped/sour cream}

{root vegetables,curd,margarine}
{tropical fruit,other vegetables,frozen fish}

{tropical fruit,yogurt,whipped/sour cream,rolls/buns}
{tropical fruit,other vegetables,whipped/sour cream,rolls/buns}

{citrus fruit,root vegetables,other vegetables,whole milk}
{frankfurter,sausage,tropical fruit} {other vegetables,whole milk,frozen vegetables,fruit/vegetable juice}
{tropical fruit,herbs,whole milk} {beef,whole milk,cream cheese }
{citrus fruit,curd,whipped/sour cream}
{other vegetables,whole milk,yogurt,soft cheese} {other vegetables,rice} {tropical fruit,root vegetables,whole milk,pastry}
{other vegetables,butter,yogurt,white bread}
{citrus fruit,pip fruit,yogurt} {onions,other vegetables,whole milk,butter}
{other vegetables,frozen vegetables,bottled water}
{herbs,whole milk,rolls/buns}
{root vegetables,whole milk,yogurt,oil}
{root vegetables,whole milk,yogurt,sliced cheese} {citrus fruit,other vegetables,whole milk,curd}

{frankfurter,whole milk,sliced cheese} {root vegetables,whipped/sour cream,cream cheese }
{pip fruit,processed cheese}
{tropical fruit,curd,whipped/sour cream}
{tropical fruit,other vegetables,whole milk,butter}
{root vegetables,frozen vegetables,bottled water}

{citrus fruit,tropical fruit,chocolate}
{tropical fruit,other vegetables,whole milk,sliced cheese}
{whole milk,hard cheese,domestic eggs}

{beef,tropical fruit,other vegetables,rolls/buns} {other vegetables,whole milk,yogurt,hard cheese}
{whole milk,yogurt,frozen fish} {other vegetables,pastry,coffee}
{chicken,pip fruit,other vegetables}
{whipped/sour cream,frozen vegetables,fruit/vegetable juice}
{root vegetables,herbs,shopping bags}
{whole milk,butter,frozen meals}

{tropical fruit,curd,margarine} {whole milk,butter,whipped/sour cream,rolls/buns} {other vegetables,whole milk,whipped/sour cream,white bread}
{herbs,frozen vegetables}
{citrus fruit,root vegetables,whole milk,curd} {frankfurter,tropical fruit,root vegetables,whole milk}
{whole milk,yogurt,whipped/sour cream,rolls/buns}

{citrus fruit,herbs,whole milk}
{citrus fruit,other vegetables,yogurt,fruit/vegetable juice}
{ham,pip fruit,whole milk}
{tropical fruit,butter,yogurt,whipped/sour cream} {tropical fruit,other vegetables,margarine}
{tropical fruit,whole milk,yogurt,oil}
{citrus fruit,soda,chocolate}
{tropical fruit,whole milk,coffee}

{tropical fruit,whole milk,butter,yogurt,domestic eggs}
{pip fruit,ice cream} {tropical fruit,pip fruit,other vegetables,curd}

{tropical fruit,whole milk,whipped/sour cream,rolls/buns}
{beef,whole milk,butter}
{herbs,yogurt} {beef,other vegetables,curd}

{tropical fruit,whole milk,sliced cheese}
{tropical fruit,other vegetables,sliced cheese}

{curd,whipped/sour cream,margarine}
{pip fruit,whole milk,frozen meals}

{other vegetables,yogurt,frozen meals}
{tropical fruit,curd,frozen vegetables}
{whole milk,yogurt,whipped/sour cream,fruit/vegetable juice} {pip fruit,whipped/sour cream,cream cheese }
{other vegetables,curd,dessert}
{grapes,other vegetables,yogurt}
{citrus fruit,root vegetables,newspapers}
{beef,other vegetables,napkins}
{sausage,whipped/sour cream,brown bread}
{tropical fruit,yogurt,oil}
{pork,yogurt,fruit/vegetable juice}
{pip fruit,grapes,other vegetables}

{tropical fruit,root vegetables,oil}
{ham,curd}
{citrus fruit,pip fruit,chocolate} {herbs,bottled beer}
{grapes,other vegetables,fruit/vegetable juice}
{herbs,whole milk,butter}
{other vegetables,whole milk,whipped/sour cream,fruit/vegetable juice} {mustard,shopping bags}

{beef,other vegetables,sugar}
{root vegetables,whole milk,yogurt,frozen vegetables}
{other vegetables,whole milk,whipped/sour cream,frozen vegetables}
{pip fruit,curd,pastry}
{beef,pip fruit,other vegetables} {pip fruit,root vegetables,cream cheese }

{tropical fruit,grapes,yogurt}
{root vegetables,whipped/sour cream,newspapers}
{tropical fruit,whipped/sour cream,domestic eggs}
{whole milk,cream cheese ,margarine}
{yogurt,rice}
{other vegetables,pastry,fruit/vegetable juice}
{root vegetables,domestic eggs,pastry}

{frankfurter,chicken}
{yogurt,whipped/sour cream,hard cheese}
{tropical fruit,onions,other vegetables,whole milk}
{frankfurter,yogurt,whipped/sour cream}
{sausage,dessert,whipped/sour cream}
{tropical fruit,whipped/sour cream,salty snack} {citrus fruit,tropical fruit,whole milk,fruit/vegetable juice}
{other vegetables,whole milk,butter,yogurt,domestic eggs} {citrus fruit,herbs,other vegetables}
{other vegetables,curd,fruit/vegetable juice}
{butter,curd,yogurt} {herbs,bottled water}

{tropical fruit,rice}
{onions,other vegetables,soda}
{tropical fruit,dessert,whipped/sour cream}
{tropical fruit,other vegetables,whole milk,oil}
{pip fruit,other vegetables,whole milk,fruit/vegetable juice} {tropical fruit,whole milk,hard cheese} {dessert,butter milk} {curd,frozen meals}
{root vegetables,butter,cream cheese } {tropical fruit,root vegetables,other vegetables,whole milk,rolls/buns}
{butter}
{tropical fruit,root vegetables,other vegetables,margarine}
{tropical fruit,butter,yogurt,domestic eggs} {root vegetables,other vegetables,whole milk,white bread}

{citrus fruit,frozen vegetables,fruit/vegetable juice} {citrus fruit,other vegetables,oil} {other vegetables,butter milk,pastry}
{other vegetables,butter,yogurt,whipped/sour cream}

{pork,butter milk}
{beef,tropical fruit,whole milk,rolls/buns} {tropical fruit,pip fruit,whole milk,rolls/buns}

{herbs,shopping bags}
{ham,tropical fruit,other vegetables}
{pip fruit,other vegetables,yogurt,brown bread}
{other vegetables,whipped/sour cream,white bread}
{frankfurter,tropical fruit,frozen meals}
{butter,curd,pastry}
{citrus fruit,brown bread,fruit/vegetable juice} {pip fruit,whole milk,coffee}

{citrus fruit,root vegetables,cream cheese }
{other vegetables,curd,cream cheese }

{turkey,tropical fruit,other vegetables} {tropical fruit,whole milk,butter,yogurt}
{grapes,fruit/vegetable juice}
{whole milk,yogurt,rice}
{other vegetables,whole milk,pasta}
{onions,whole milk,napkins} {chicken,citrus fruit,domestic eggs}
{whole milk,curd,hygiene articles}

{pip fruit,whole milk,whipped/sour cream,domestic eggs}
{tropical fruit,whole milk,curd,domestic eggs} {tropical fruit,dog food}
{frankfurter,chicken,other vegetables}
{tropical fruit,root vegetables,whole milk,curd}
{tropical fruit,whole milk,yogurt,cream cheese }
{curd,soft cheese}
{tropical fruit,other vegetables,whole milk,rolls/buns}
{tropical fruit,pastry,coffee}
{pip fruit,whipped/sour cream,domestic eggs}
{beef,whole milk,newspapers}
{ham,tropical fruit,other vegetables,whole milk}
{pork,pip fruit,soda} {citrus fruit,oil,fruit/vegetable juice}
{other vegetables,whole milk,whipped/sour cream,margarine}

{frozen vegetables,rice}
{tropical fruit,root vegetables,onions,whole milk}

{pip fruit,whole milk,yogurt,brown bread}
{tropical fruit,herbs,other vegetables,whole milk}
{beef,other vegetables,shopping bags} {pip fruit,root vegetables,white bread}

{pip fruit,butter,pastry}
{citrus fruit,tropical fruit,other vegetables,whipped/sour cream} {tropical fruit,other vegetables,whole milk,margarine}
{citrus fruit,pip fruit,hygiene articles} {herbs,rolls/buns}
{pip fruit,rolls/buns,fruit/vegetable juice} {other vegetables,whole milk,soft cheese}
{other vegetables,whipped/sour cream,frozen meals}
{sausage,root vegetables,whipped/sour cream} {tropical fruit,pip fruit,curd}
{citrus fruit,root vegetables,soft cheese}

{other vegetables,oil,fruit/vegetable juice}
{sausage,whipped/sour cream,bottled water}

{citrus fruit,whipped/sour cream,frozen vegetables}
{butter milk,pastry}
{tropical fruit,root vegetables,whole milk,bottled water}
{butter,whipped/sour cream,sugar}
{other vegetables,whole milk,butter,rolls/buns}
{beef,tropical fruit,root vegetables,whole milk,rolls/buns}
{tropical fruit,other vegetables,rolls/buns,bottled water}

{tropical fruit,root vegetables,yogurt,fruit/vegetable juice}
{tropical fruit,oil,fruit/vegetable juice} {whole milk,rice}
{whole milk,frozen vegetables,bottled beer}
{other vegetables,curd,yogurt,whipped/sour cream} {onions,other vegetables,butter}

{pork,onions,other vegetables} {herbs,other vegetables}
{pip fruit,herbs,whole milk} {tropical fruit,soda,candy}
{butter,whipped/sour cream,fruit/vegetable juice}

{tropical fruit,other vegetables,whipped/sour cream,fruit/vegetable juice}
{pip fruit,whole milk,frozen fish}
{beef,butter,yogurt} {tropical fruit,whole milk,whipped/sour cream,domestic eggs}
{whole milk,whipped/sour cream,flour}
{tropical fruit,root vegetables,other vegetables,whole milk,oil}
{citrus fruit,whole milk,yogurt,fruit/vegetable juice} {tropical fruit,whole milk,candy}
{whipped/sour cream,domestic eggs,pastry}
{tropical fruit,other vegetables,whipped/sour cream,bottled water}

{tropical fruit,grapes,whole milk,yogurt}
{tropical fruit,root vegetables,yogurt,oil}

{beef,citrus fruit,tropical fruit,other vegetables}
{ham,tropical fruit,pip fruit}
{herbs,whole milk,bottled water}
{meat,root vegetables,yogurt}
{whole milk,frozen vegetables,salty snack}
{whole milk,oil,fruit/vegetable juice} {curd,sliced cheese}
{other vegetables,yogurt,rice} {tropical fruit,pip fruit,root vegetables,whole milk}
{whole milk,oil,shopping bags}

{whole milk,butter milk,cream cheese }
{pork,citrus fruit,other vegetables,whole milk} {herbs,other vegetables,rolls/buns} {tropical fruit,pip fruit,whole milk,brown bread}
{pip fruit,whipped/sour cream,fruit/vegetable juice} {tropical fruit,other vegetables,waffles}
{root vegetables,whole milk,rice}
{citrus fruit,whole milk,whipped/sour cream,cream cheese }

{citrus fruit,other vegetables,yogurt,whipped/sour cream}
{tropical fruit,grapes,root vegetables} {beef,citrus fruit,tropical fruit,root vegetables}
{tropical fruit,pip fruit,frozen fish}

{turkey,curd}
{tropical fruit,herbs,other vegetables} {citrus fruit,bottled water,napkins}
{whole milk,butter,cream cheese } {whole milk,curd,sliced cheese}
{citrus fruit,onions,whole milk}
{citrus fruit,root vegetables,whole milk,domestic eggs} {soft cheese,bottled water}
{onions,butter milk} {whole milk,dessert,butter milk}
{beef,tropical fruit,whole milk,whipped/sour cream}
{sausage,pip fruit,sliced cheese} {tropical fruit,whole milk,butter,curd}

{tropical fruit,whipped/sour cream,white bread} {tropical fruit,whipped/sour cream,pastry}
{citrus fruit,tropical fruit,herbs}

{tropical fruit,yogurt,whipped/sour cream,fruit/vegetable juice}
{tropical fruit,butter,curd}
{curd,whipped/sour cream,sugar}
{turkey,tropical fruit,root vegetables}
{citrus fruit,other vegetables,whole milk,frozen vegetables}

{sausage,citrus fruit,whipped/sour cream} {tropical fruit,whipped/sour cream,napkins}
{tropical fruit,other vegetables,whole milk,yogurt,domestic eggs}

{tropical fruit,whipped/sour cream,soft cheese}
{sausage,root vegetables,curd}
{root vegetables,pastry,fruit/vegetable juice} {herbs,whole milk,fruit/vegetable juice}
{tropical fruit,whole milk,curd,whipped/sour cream}
{meat,tropical fruit,whole milk}
{yogurt}
{curd,whipped/sour cream,domestic eggs}
{whole milk,whipped/sour cream,hard cheese}

{tropical fruit,whipped/sour cream,margarine}
{root vegetables,whole milk,cream cheese }
{other vegetables}
{tropical fruit,other vegetables,whipped/sour cream,margarine} {tropical fruit,pip fruit,whole milk,curd}
{chicken,root vegetables,pastry}
{tropical fruit,whipped/sour cream,hard cheese}
{tropical fruit,butter,sliced cheese} {cream cheese ,rolls/buns,soda}

{tropical fruit,butter,yogurt,white bread}
{ham,tropical fruit,pip fruit,whole milk}
{tropical fruit,root vegetables,whipped/sour cream,rolls/buns}
{tropical fruit,other vegetables,whole milk,butter,domestic eggs}
{pork,tropical fruit,fruit/vegetable juice}

{onions,waffles}
{tropical fruit,pip fruit,whole milk,whipped/sour cream}
{onions,sliced cheese}
{tropical fruit,pastry,margarine}
{onions,whole milk,curd}
{other vegetables,whole milk,pastry,fruit/vegetable juice}
{root vegetables,other vegetables,whole milk,rice}
{root vegetables,rolls/buns,candy} {tropical fruit,root vegetables,other vegetables,oil}
{citrus fruit,pip fruit,whipped/sour cream}

{coffee,specialty bar}
{tropical fruit,butter,domestic eggs}

{root vegetables,frozen vegetables,margarine}
{tropical fruit,root vegetables,whole milk,butter}
{tropical fruit,curd,bottled water}
{other vegetables,whole milk,curd,pastry}
{bottled water}
{butter,whipped/sour cream,margarine}
{beef,root vegetables,sugar}
{frankfurter,pip fruit,root vegetables,whole milk} {frankfurter,root vegetables,whole milk,rolls/buns}

{tropical fruit,grapes,whole milk} {tropical fruit,other vegetables,whole milk,bottled beer}
{tropical fruit,whole milk,soda,newspapers} {other vegetables,curd,whipped/sour cream,cream cheese }
{tropical fruit,whipped/sour cream,cream cheese }
{tropical fruit,butter,white bread}

{ham,tropical fruit,pip fruit,yogurt}
{root vegetables,whole milk,butter,white bread}
{root vegetables,curd,cream cheese }
{other vegetables,whole milk,specialty cheese}
{citrus fruit,whipped/sour cream,cream cheese }
{tropical fruit,soft cheese} {hamburger meat,other vegetables,fruit/vegetable juice}
{root vegetables,onions,napkins} {citrus fruit,root vegetables,other vegetables,whole milk,whipped/sour cream}

{whipped/sour cream,cream cheese ,margarine}
{tropical fruit,other vegetables,butter,curd}

{turkey,root vegetables,whole milk}
{sausage,root vegetables,whole milk,pastry}
{tropical fruit,root vegetables,whole milk,sliced cheese}
{pork,whole milk,butter milk}
{pip fruit,root vegetables,whole milk,fruit/vegetable juice}
{whole milk,butter milk,pastry}
{root vegetables,whole milk,yogurt,fruit/vegetable juice}
{pip fruit,root vegetables,whole milk,brown bread}
{tropical fruit,whipped/sour cream,sliced cheese}

{root vegetables,whole milk,whipped/sour cream,white bread}
{frankfurter,tropical fruit,whipped/sour cream}
{ham,whole milk,frozen vegetables}

{tropical fruit,root vegetables,whole milk,oil}
{pork,grapes,whole milk} {beef,tropical fruit,butter}
{citrus fruit,tropical fruit,root vegetables,whole milk}
{tropical fruit,root vegetables,other vegetables,domestic eggs} {whole milk,dessert,whipped/sour cream}
{herbs,fruit/vegetable juice}
{ham,tropical fruit,whole milk}
{other vegetables,flour,margarine}
{tropical fruit,root vegetables,cream cheese } {other vegetables,whole milk,whipped/sour cream,long life bakery product}
{whole milk,yogurt,whipped/sour cream,frozen vegetables} {chicken,citrus fruit,whipped/sour cream} {pastry,bottled water,fruit/vegetable juice}
{pip fruit,root vegetables,whole milk,soda}
{tropical fruit,root vegetables,long life bakery product}

{whole milk,curd,whipped/sour cream,cream cheese }
{tropical fruit,root vegetables,onions}
{yogurt,rolls/buns,soda,newspapers}
{tropical fruit,whipped/sour cream,fruit/vegetable juice} {berries,cream cheese } {whole milk,cream cheese ,soda}
{root vegetables,whole milk,frozen vegetables,bottled water}
{tropical fruit,pip fruit,whole milk,frozen meals}
{tropical fruit,yogurt,salty snack}

{tropical fruit,other vegetables,yogurt,white bread}
{pip fruit,root vegetables,brown bread}
{whole milk,cream cheese ,pastry}

{root vegetables,onions,frozen vegetables} {hamburger meat,butter,whipped/sour cream}
{citrus fruit,tropical fruit,root vegetables,whole milk,yogurt}
{root vegetables,whole milk,whipped/sour cream,cream cheese }

{tropical fruit,pip fruit,brown bread} {tropical fruit,yogurt,semi-finished bread} {sausage,whole milk,butter milk}
{tropical fruit,whole milk,rolls/buns,newspapers}

{sausage,tropical fruit,long life bakery product}
{citrus fruit,tropical fruit,yogurt,fruit/vegetable juice} {root vegetables,onions,whole milk,whipped/sour cream}
{tropical fruit,other vegetables,rolls/buns,newspapers} {root vegetables,cream cheese ,rolls/buns} {tropical fruit,root vegetables,whole milk,rolls/buns}
{sliced cheese,frozen meals}
{other vegetables,fruit/vegetable juice,salty snack}
{tropical fruit,other vegetables,semi-finished bread}
{pip fruit,butter milk,fruit/vegetable juice}

{citrus fruit,whipped/sour cream,soda}
{whole milk,curd,whipped/sour cream,rolls/buns}
{root vegetables,whipped/sour cream,white bread} {beef,tropical fruit,whipped/sour cream}
{tropical fruit,other vegetables,curd,whipped/sour cream} {tropical fruit,butter,margarine}

{tropical fruit,onions,yogurt} {tropical fruit,whole milk,butter,domestic eggs}
{butter,whipped/sour cream,soda}
{root vegetables,yogurt,rice}
{tropical fruit,whole milk,butter,whipped/sour cream}
{root vegetables,other vegetables,whole milk,hard cheese}

{butter,whipped/sour cream,long life bakery product}
{whole milk,butter,whipped/sour cream,soda}
{citrus fruit,root vegetables,whole milk,newspapers} {root vegetables,other vegetables,curd,cream cheese }
{citrus fruit,root vegetables,pastry} {tropical fruit,whipped/sour cream,bottled water}

{butter milk,waffles}
{whole milk,rolls/buns,soda,newspapers} {tropical fruit,whole milk,bottled water,soda}

{tropical fruit,whole milk,whipped/sour cream,fruit/vegetable juice}
{root vegetables,soft cheese,domestic eggs} {tropical fruit,root vegetables,whole milk,margarine}
{sausage,soda,newspapers}
{other vegetables,butter milk,fruit/vegetable juice}
{tropical fruit,other vegetables,butter,white bread}
{tropical fruit,whole milk,soft cheese}
{root vegetables,butter,white bread}

{root vegetables,whole milk,yogurt,rice} {tropical fruit,root vegetables,other vegetables,whole milk,bottled water} {whole milk,soft cheese,domestic eggs}
{tropical fruit,whole milk,butter,sliced cheese} {whole milk,butter milk,fruit/vegetable juice}
{tropical fruit,pastry,bottled water}
{ham,root vegetables,yogurt}
{tropical fruit,rolls/buns,waffles} {tropical fruit,other vegetables,whole milk,soft cheese}
{tropical fruit,root vegetables,butter,whipped/sour cream}
{citrus fruit}
{whole milk,curd,cream cheese }
{curd,whipped/sour cream,cream cheese }

{whole milk,butter,yogurt,white bread}
{cream cheese }
{hamburger meat,tropical fruit,whipped/sour cream}
{grapes,onions}
{tropical fruit,other vegetables,butter milk}
{root vegetables,yogurt,salty snack}
{citrus fruit,whole milk,curd}

{tropical fruit,other vegetables,butter,domestic eggs}
{tropical fruit,whole milk,rolls/buns,pastry}

{citrus fruit,whole milk,cat food} {tropical fruit,onions,butter}
{sausage,tropical fruit,curd}
{other vegetables,butter,white bread} {rolls/buns,bottled water,soda,newspapers}
{sausage,citrus fruit,whole milk,whipped/sour cream}
{hard cheese,oil}
{butter,curd,domestic eggs}
{tropical fruit,whipped/sour cream,rolls/buns} {whipped/sour cream,pastry,fruit/vegetable juice}

{citrus fruit,root vegetables,onions} {tropical fruit,root vegetables,other vegetables,butter}
{other vegetables,whipped/sour cream,frozen vegetables}
{citrus fruit,whole milk,whipped/sour cream,domestic eggs} {sausage,root vegetables,pastry}
{ham,whole milk,curd} {whole milk,whipped/sour cream,cream cheese }
{citrus fruit,root vegetables,herbs} {citrus fruit,root vegetables,whole milk,yogurt,whipped/sour cream}
{tropical fruit,whole milk,whipped/sour cream,margarine}
{citrus fruit,tropical fruit,root vegetables}

{tropical fruit,whole milk,curd}
{pip fruit,root vegetables,butter milk}

{tropical fruit,root vegetables,margarine} {root vegetables,whipped/sour cream,sliced cheese}
{chicken,tropical fruit,root vegetables}
{tropical fruit,margarine,bottled water} {pip fruit,root vegetables,curd}

{chicken,citrus fruit,rolls/buns}
{whole milk,curd,soft cheese}
{root vegetables,sugar,fruit/vegetable juice} {root vegetables,whipped/sour cream,hard cheese}
{beef,root vegetables,cream cheese }

{tropical fruit,root vegetables,whole milk,yogurt,oil}
{ham,tropical fruit,pip fruit,other vegetables}
{other vegetables,whole milk,curd,cream cheese }
{berries,butter milk}
{citrus fruit,tropical fruit,root vegetables,yogurt}
{sausage,pip fruit,curd}
{meat,margarine} {citrus fruit,tropical fruit,root vegetables,rolls/buns} {curd,whipped/sour cream,rolls/buns} {sausage,whipped/sour cream,cream cheese }
{other vegetables,butter,cream cheese }
{citrus fruit,tropical fruit,root vegetables,whipped/sour cream}

{rolls/buns,pastry,margarine}
{tropical fruit,butter,whipped/sour cream,fruit/vegetable juice} {tropical fruit,root vegetables,other vegetables,whole milk,butter}
{citrus fruit,tropical fruit,margarine}
{fruit/vegetable juice}
{root vegetables,rolls/buns,flour}
{tropical fruit,curd,pastry}
{butter,curd,whipped/sour cream}
{chicken,yogurt,domestic eggs}
{yogurt,oil,coffee}
{pip fruit,whipped/sour cream,brown bread} {tropical fruit,root vegetables,sliced cheese}

{whipped/sour cream,frozen vegetables,soda}
{tropical fruit,whole milk,whipped/sour cream,bottled water}
{tropical fruit,root vegetables,other vegetables,whole milk,whipped/sour cream}

{tropical fruit,rolls/buns,bottled water,soda} {other vegetables,UHT-milk,soda}
{citrus fruit,tropical fruit,whole milk,curd} {frankfurter,bottled water,soda}

{root vegetables,whole milk,whipped/sour cream,hard cheese}
{root vegetables,pastry,shopping bags}
{root vegetables,other vegetables,mayonnaise}
{whole milk,whipped/sour cream,rolls/buns,pastry}

{other vegetables,whole milk,cream cheese ,domestic eggs} {soft cheese,fruit/vegetable juice}
{root vegetables,oil,fruit/vegetable juice}

{other vegetables,whipped/sour cream,chocolate}
{pip fruit,whole milk,sliced cheese}
{sausage,tropical fruit,newspapers}
{soda,popcorn}
{sausage,soda,bottled beer}
{tropical fruit,root vegetables,whole milk,fruit/vegetable juice}

{root vegetables,whole milk,frozen fish}
{tropical fruit,other vegetables,whole milk,bottled water}
{root vegetables,other vegetables,soda,fruit/vegetable juice}

{sausage,curd,whipped/sour cream}
{citrus fruit,tropical fruit,grapes}

{tropical fruit,root vegetables,fruit/vegetable juice} {whole milk,soft cheese,fruit/vegetable juice}
{root vegetables,other vegetables,rice}
{salty snack} {tropical fruit,grapes,fruit/vegetable juice}

{root vegetables,other vegetables,whipped/sour cream,frozen vegetables}
{sugar}
{pip fruit,root vegetables,whole milk,white bread}
{frozen vegetables,specialty chocolate}
{flour,baking powder} {tropical fruit,other vegetables,misc. beverages}
Figure 10: Visualization of 1000 rules with Gephi (Fruchterman Reingold layout, vertex and
label size is proportional to the in-degree, i.e., the number of rules the consequent participates
in).
Parallel coordinates plot for 10 rules
salty snack
sugar
cream cheese
butter
popcorn
soda
baking powder
flour
ham
curd
processed cheese
white bread
domestic eggs
whole milk
other vegetables
tropical fruit
whipped/sour cream
yogurt
hamburger meat
5 4 3 2 1 rhs
Position
Figure 11: Parallel coordinate plot.
Parallel coordinates plot for 10 rules
salty snack
sugar
popcorn
white bread
baking powder
soda
whipped/sour cream
flour
ham
processed cheese
hamburger meat
whole milk
domestic eggs
yogurt
other vegetables
tropical fruit
cream cheese
butter
curd
5 4 3 2 1 rhs
Position
Figure 12: Parallel coordinate plot (reordered).

> plot(subrules2, method = "paracoord", control = list(reorder = TRUE))
Figure 12 shows the parallel coordinates plot with reordered items to reduce crossovers.
8. Double Decker plots

A double decker plot is a variant of a mosaic plot. A mosaic plot displays a contingency table
using tiles on a rectangle created by recursive vertical and horizontal splits. The size of each
tile is proportional to the value in the contingency table. Double decker plots use only a single
horizontal split.
Hofmann, Siebes, and Wilhelm (2000) introduced double decker plots to visualize a single
association rule. Here the displayed contingency table is computed for a rule by counting
the occurrence frequency for each subset of items in the antecedent and consequent from
the original data set. The items in the antecedent are used for the vertical splits and the
consequent item is used for horizontal highlighting.
We randomly choose a single rule and visualize it with a double decker plot.
> oneRule <- sample(rules, 1)

> inspect(oneRule)
lhs rhs support confidence lift count

[1] {pip fruit,
root vegetables,
cream cheese } => {other vegetables} 0.0011 0.61 3.2 11
> plot(oneRule, method = "doubledecker", data = Groceries)
Figure 13 shows the resulting plot. The area of blocks gives the support and the height of
the “yes” blocks is proportional to the confidence for the rules consisting of the antecedent
items marked as “yes.” Items that show a significant jump in confidence when changed from
“no” to “yes” are interesting. This is captured by the interest measure difference of confidence
defined by Hofmann and Wilhelm (2001).
9. Comparison of techniques
In this section, we compare the visualization techniques available in arulesViz based on the
size of the rule set which can be analyzed, the number of interest measures which are shown
simultaneously, if the technique offers interaction and reordering and how intuitive each vi-
sualization technique is. Note that most of these categories are only evaluated qualitatively
here, and the results presented in Table 1 are only meant to guide the user towards the most
suitable techniques for a given application.
Scatterplot (including two-key plots) and grouped matrix plot are capable to analyze large
rule sets. These techniques are interactive to allow the analyst to zoom and select interesting
rules. Matrix-based can accommodate rule sets of medium size. Reordering can be used to
improve the presentation. To analyze small rule sets the matrix-based method with 3D bars,
Doubledecker plot for 1 rule
other vegetables
no
yes
no yesno yes
noyes
no
yes
cream cheese
no yes no yesroot vegetables
no yes pip fruit
Figure 13: Double decker plot for a the rule pip fruit,root vegetables,cream cheese => other
vegetables.
Technique Method Rule set Measures Interactive Reordering Ease of use

Scatterplot "scatterplot" large 3 X ++
Two-Key plot "scatterplot" large 2 + order X ++
Matrix-based "matrix" medium 1 X 0
Matrix-b. (2 measures) "matrix" medium 2 X ––
Matrix-b. (3D bar) "matrix3D" small 1 X +
Grouped matrix "grouped" large 2 X X 0
Graph-based "graph" small 2 ++
Graph-b. (external) "graph" large 2 X X +
Parallel coordinates "paracoord" small 1 X –
Double decker "doubledecker" single rule (2) –
Table 1: Comparison of visualization methods for association rules available in arulesViz.
graph-based methods and parallel coordinates plots are suitable. Graphs for large rule sets
can be analyzed using external tools like Gephi. Finally, double decker plots only visualize a
single rule.
The techniques discussed in this paper can also be categorized based on the number of interest
measures simultaneously visualized. Most methods can represent two measures and scatter
plots are even able to visualize three measures for each rule in one plot.
Scatter plot and graph based techniques are the most intuitive while matrix-based visualiza-
tion with two interest measures, parallel coordinates and double decker require time to learn
how to interpret them correctly.
10. Further Reading

Details on the grouped matrix plot are reported in Hahsler and Karpienko (2016). Interactive
visualization using htmlWidgets with examples is describe in Hahsler (2017).
11. Conclusion
Association rule mining algorithms typically generate a large number of association rules
which poses a major problem for understanding and analyzing rules. In this paper we pre-
sented several visualization techniques implemented in arulesViz which can be used to explore
and present sets of association rules. In addition we present a new interactive visualization
method called grouped matrix-based visualization which can used to effectively explore large
rule sets.
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Affiliation:
Michael Hahsler
Engineering Management, Information, and Systems
Lyle School of Engineering
Southern Methodist University
P.O. Box 750123
Dallas, TX 75275-0123
E-mail: mhahsler@lyle.smu.edu
URL: http://lyle.smu.edu/~mhahsler

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Visualizing Association Rules: Introduction to the

R-extension Package arulesViz

Michael Hahsler Sudheer Chelluboina

Keywords: data mining, association rules, visualization.

lift(X ⇒ Y ) = supp(X ∪ Y )/(supp(X)supp(Y ))

by arulesViz for association rule visualization. In Sections 3 to 8 we describe the different

2. Data preparation and unified interface of arulesViz

transactions as itemMatrix in sparse format with

most frequent items:

element (itemset/transaction) length distribution:

Min. 1st Qu. Median Mean 3rd Qu. Max.

includes extended item information - examples:

> rules <- apriori(Groceries, parameter=list(support=0.001, confidence=0.5))

Absolute minimum support count: 9

set item appearances ...[0 item(s)] done [0.00s].

set of 5668 rules

> inspect(head(rules, n = 3, by ="lift"))

lhs rhs support confidence

[3] {flour,baking powder} => {sugar} 0.0010 0.56

> args(getS3method("plot", "rules"))

function (x, method = NULL, measure = "support", shading = "lift",

Scatter plot for 5668 rules

Figure 1: Default scatter plot.

support confidence lift count

> plot(rules, measure = c("support", "lift"), shading = "confidence")

> plot(rules, method = "two-key plot")

Scatter plot for 5668 rules

Figure 2: Scatter plot with lift on the y-axis.

0.005 0.01 0.015 0.02

Figure 3: Two-key plot.

> sel <- plot(rules, measure=c("support", "lift"), shading = "confidence",

Interactive features include:

 Zooming into a selected region (zoom in/zoom out buttons).

 Returning the last selection for further analysis (end button).

> subrules <- rules[quality(rules)$confidence > 0.8]

set of 371 rules

> plot(subrules, method = "matrix", measure = "lift")

Itemsets in Antecedent (lhs)

An alternative representation is to use 3D bars (method “matrix3D”) instead of colored rect-

> plot(subrules, method = "matrix3D", measure = "lift")

The 3D visualization is shown in Figure 6.

5. Grouped matrix-based visualization

Matrix with 371 rules

Figure 5: Matrix-based visualization with colored squares.

Matrix with 371 rules

Figure 6: Matrix-based visualization with 3D bars.

> plot(rules, method = "grouped")

> plot(rules, method = "grouped", control = list(k = 50))

The resulting, more detailed plot is shown in Figure 8.

> sel <- plot(rules, method = "grouped", interactive = TRUE)

> subrules2 <- head(rules, n = 10, by = "lift")

Items in LHS Group

Figure 7: Grouped matrix-based visualization.

121 rules: {jam, specialty cheese, +37 items}

> plot(subrules2, method = "graph")

> saveAsGraph(head(rules, n = 1000, by = "lift"), file = "rules.graphml")

7. Parallel coordinates plot

Zooming into a selected region (zoom in/zoom out buttons).

Returning the last selection for further analysis (end button).