Important Notes On Dyanamic
Important Notes On Dyanamic
Important Notes On Dyanamic
COURSES
BROCHURE
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Analysis (Course A)
Analysis (Course A)
PREREQUISITES
Calculus and mathematical Analysis in one and several real variables. Ordinary Differential Equations.
COURSE OBJECTIVES
The course introduces the participants to the theory of infinite-dimensional vector spaces and of linear operators
between them, with a special focus on the concepts of normed vector spaces, completeness, compactness, and
the different topologies which characterize the infinite dimensional vector spaces. Applications concern various
spaces of functions and operators between them (in particular, integral and differential operators). The
course presents basic tools of modern mathematical analysis which are of fundamental importance in many
branches of pure and applied mathematics, in particular in probability theory, statistics, numerical analysis, partial
differential equations and dynamical systems.
COURSE AIMS
The student will acquire knowledge of many basic tools which are of common use in the analysis of infintite
dimensional vector spaces. In particular he will learn the theory of Banach and Hilbert spaces and their dual spaces,
of linear, bounded, and compact operators, and he will know the theory of distributions (generalized functions). He
will be able to apply this knowledge to solve simple problems and exercises related to the theory (in particular, to
solve simple integral or differential equations) and he will be able to rigorously prove main results of the theory.
COURSE DELIVERY
The assessment consists in a written test followed by an oral examination, after completion of the course.
The written test consists in open questions and exercises on the topics treated in class and has a duration of 180
minutes. The mark will be expressed in thirtieth; the single points (30 in total) will be distributed to the questions and
exercises on the basis of their importance and length; the final score will be given by summing up the partial scores
of each question and exercise.
The oral examination is scheduled after the written test and can be given only after having passed the written test
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with a mark of 18 or better. The oral examination consists of questions on the written test and on the topics treated
in class and listed in the examination programme (which is available to the participants on the web-site of the
course).
Both written test and oral examination will result in a final mark expressed in thirtieth; the minimal mark allowed for
successful assessment is 18. Otherwise, the student's performance is considered insufficient and the student has to
repeat the examination (both written test and oral examination).
Both written test and oral examination must be achieved in the same examination period.
SYLLABUS
Banach spaces.
Linear operators.
Hilbert spaces, projections, orthonormal basis.
Generalized Fourier series.
Dual spaces: linear functionals, weak convergence.
Compactness in finite dimensional spaces.
Compact operators and applications to integral equations.
Fundamentals of spectral theories
Distributions (generalized functions)
Fourier transform
Laplace transform
Bryan P. Rynne and Martin A. Youngson, Linear Functional Analysis, Second Edition, Springer, 2008.
Dudley, R. M., Real Analysis and Probability, Cambridge University Press.
Royden, H.L. Real Analysis. MacMillan.
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Analysis (Course B)
Analysis (Basics)
PREREQUISITES
A good knowledge of basic calculus and real analysis.
COURSE OBJECTIVES
This course is a 9-credit course aimed at introducing and developing many of the mathematical tools necessary in
many fields of Probablity, statistics and applied mathematics. It introduces in particular the theory of infinite-
dimensional vector spaces with a special focus on the concepts of normed vector spaces, completeness,
compactness, and other characteristic properties of infinite dimensional vector spaces. Concrete applications to
spaces of functions will be provided. Application of this theory to the study of ordinary differential equation and
Fourier analysis will be also given.
COURSE AIMS
- using the basic tools and results to pose, formalize and solve a complex mathematical problem of applied
interest.
- being able to think about possible and useful generalizations of the various results studied during the
lectures.
- being able to communicate such findings using appropriate and clear mathematical notation and language
COURSE DELIVERY
The course is articulated in 72 hours of formal in‐class lecture time, and in at least as many hours of at‐home work
solving practical exercises.
The course grade is determined solely on the basis of a written examination. The examination (2 hours and 45
minutes) test the student's ability to do the following:
Present briefly the main ideas, concepts and results developed in the course, also explaining intuitively the
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meaning and scope of the definitions and the arguments behind the validity of the result. Students will be required
to know the definitions, the statements of the theorems, the idea behing the proofs and their applications.
Use effectively the concepts and the result to answer questions pertaining to functional analysis.
The above is accomplished by asking the student to answer 5‐6 questions. Each of the questions has an essay part,
and some of the questions also have a more practical ("exercise ") part.
SUPPORT ACTIVITIES
The course includes exercises classes; extra exercises are suggested as homework.
SYLLABUS
The course is divided in 4 parts:
Fourier Analysis.
KHURI, A.I. Advanced calculus with applications in statistics. Wiley Series in Probability and Statistics.
- RYNNE, B. P. and Martin A. YOUNGSON, M.A., Linear Functional Analysis, Second Edition, Springer 2008
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Bayesian nonparametric statistics (not offered in 2017/2018)
PREREQUISITES
STOCHASTIC MODELLING FOR STATISTICAL APPLICATIONS
COURSE OBJECTIVES
The course aims at providing a modern overview of Bayesian nonparametric statistical methods.
COURSE AIMS
Students will learn how to model statistical problems with Bayesian nonparametric tools, study theoretical
properties of the involved objects and devise appropriate computational algorithms for their implementation.
COURSE DELIVERY
The course consists mainly of class lectures, with some additional computer lab sessions using R.
Oral examination and optional paper presentation or discussion of an essay elaborated by the student.
SYLLABUS
This course covers the fundamentals of Bayesian nonparametric inference and focuses on the key
probabilistic concepts and stochastic modelling tools at the basis of the most recent advances in the field:
• foundations of Bayesian nonparametric inference: exchangeability and de Finetti's representation
theorem
• the Dirichlet process
• models beyond the Dirichlet process
• mixture models for density estimation and clustering
• random partitions
• dependent priors for partially exchangeable data
• elements of Bayesian asymptotics
16 hours of the course will be taught by Visiting Professor Ramses Mena on Random partitions and dependent
processes.
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SUGGESTED TEXTBOOKS AND READINGS
GHOSAL and VAN DER VAART (2016). Theory of nonparametric Bayesian inference. Cambridge University
Press.
HJORT, HOLMES, MUELLER and WALKER (eds.) (2010). Bayesian Nonparametrics. Cambridge University
Press.
GHOSH, RAMAMOORTHI. (2003). Bayesian Nonparametrics. Springer.
NOTE
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Bayesian statistics
Bayesian statistics
PREREQUISITES
STOCHASTIC MODELLING FOR STATISTICAL APPLICATIONS
COURSE OBJECTIVES
The course aims at providing a modern overview of Bayesian statistical methods, covering the fundamentals of both
the parametric and the nonparametric approach. The course will focus on the key probabilistic concepts, stochastic
modelling tools and most widely used computational strategies at the basis of the most recent advances in the field.
A short module of the course will be taught by Visiting Professor Jim Griffin on a topic TBA (see International visiting
professors)
COURSE AIMS
Students will learn how to model statistical problems with Bayesian parametric and nonparametric tools, study the
theoretical properties of the involved objects and devise appropriate computational algorithms for their
implementation.
COURSE DELIVERY
The course consists of roughlyt 80% of class lectures, and 20% of computer lab sessions.
Oral examination on the material covered in class, plus optional paper presentation or discussion of an essay
elaborated by the student.
SYLLABUS
- Motivation and foundations of Bayesian inference: exchangeability and de Finetti's representation theorems
- Conjugacy, posteriors and parametric families of conjugate models
- Markov chain Monte Carlo methods for parametric inference
- The Bayesian nonparametric approach
- The Dirichlet process: definition, properties and constructions
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- Hierarchical priors derived from the Dirichlet process
- Models beyond the Dirichlet process
- Markov chain Monte Carlo methods for nonparametric inference.
If time allows, the course will also cover a brief introduction of the following topics:
- Elements of Bayesian asymptotics
- Dependent priors for partially exchangeable data
NOTE
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Complex networks
Complex networks
PREREQUISITES
A strong working knowledge of probability and linear algebra (at the level of a bachelor degree in a scientific
discipline) will certainly be helpful, as is some mathematical maturity. The ability to write code is important, because
programming skills are required for the coursework project.
COURSE OBJECTIVES
This module introduces the fundamental concepts, principles and methods in the interdisciplinary field of network
science, with a particular focus on analysis techniques, modeling, and applications for the World Wide Web and
online social media. Topics covered include graphic structures of networks, mathematical models of networks,
common networks topologies, structure of large scale graphs, community structures, epidemic spreading,
PageRank and other centrality measures, dynamic processes in networks, graphs visualization.
COURSE AIMS
COURSE DELIVERY
A Moodle webpage is created for the course. All course materials, such as lecture notes and online resources will
be shared. By using the Moodle, students will also be able to discuss ideas and questions with the lecturer and other
students.
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Coursework I (20%): essay writing (2000-3000 words).
Coursework II (20%): individual project on network data analysis (programming is usually required).
To pass the module students must achieve a pass mark of 60% when all elements are combined.
SYLLABUS
Network science
• Random networks
• Small-world networks
• Scale-free networks
• Evolving networks
• Degree correlations
• Communities
• Spreading phenomena
• Network visualizations
D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning About a Highly Connected World, Cambridge
University Press, 2010.
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M. E. J. Newman. Networks: An Introduction, Oxford University Press, 2010.
NOTE
This course is borrowed from Complex Networks, delivered at the Computer Science Department.
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Computational methods for statistics
COURSE OBJECTIVES
This course aims at introducing the students with computational statistics methods. The program includes some
computationally intensive methods in statistics, such as Monte Carlo methods, bootstrap, and permutation tests. An
important part of the course will be devoted to practicals: all the methods discussed during the course will be will
be implemented in the R software.
COURSE AIMS
After this course the students will be familiar with pseudo-random number generators and with the statistical
software R. They will know how to sample an independent and identically distributed sequence or (pseudo)
random number with a given distribution, and will be able to implement a Monte Carlo integration algorithm in R.
Moreover, students will learn some of the most common statistical methods based on sampling strategies (e.g.,
Bootstrap, Jackknife, Bayesian estimation).
COURSE DELIVERY
Half of the lectures will be devoted to the theoretical aspects of simulation and Monte Carlo Integration and the
remaining half to their practical implementation in the R software considering both the related numerical and
computational issues. Exercises will be assigned during lectures and lab sessions.
The exam consists of two parts: the first part is a written exam on theory; the second part is a practical session with
R.
SYLLABUS
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Bootstrap and Jackknife.
Permutation Tests for Equal Distributions.
Rizzo, M.L. (2015) "Statistical Computing with R (Second Edtion)" -- Chapman & Hall/CRC The R Series.
Ross. S.M. (2006) "Simulation 4th edition" -- Academic Press.
Jones, O., Maillardet, R. and Robinson A. (2009). "Introduction to scientific programming and simulation usig R" --
Chapman and Hall/CRC;
NOTE
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Database and algorithms
PREREQUISITES
Knowledge on programming.
PROPEDEUTIC FOR
Complex networks, Introduction to Data Mining
COURSE OBJECTIVES
The objectives are formalized for each of the two parts of the course.
Databases
This course will teach the fundamentals of relational theory, SQL language and its relationships with relational
algebra, design of data in relational databases by means of the conceptual and logical design of databases. The
course will introduce the students to the basic notions of NoSQL databases, important for the new generation of
databases and the management of big data. In the laboratory the students will be able to work with a practical
database management system.
Algorithms
In this course students will learn several fundamental principles of algorithm design and how to implement some
fundamental data structures (e.g., graphs, arrays, trees, hash tables). This course aims at providing a solid
methodological background for the analysis of algorithms in terms of their correctness, complexity (in time and in
space), and tractability.
COURSE AIMS
Databases
After the course students will be able to design data for relational databases, formulate a query in SQL or relational
algebra, interact with a real database management system and will have the basic notions of NoSQL databases.
Algorithms
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After the course students will be able to approach a problem through the design, analysis and implementation of
appropriate algorithms and data structures.
COURSE DELIVERY
This course consists in two parts: the former is on Databases and the latter is on Algorithms.
Databases
The course will consist of 32 hours of frontal lessons and 16 hours of practical assignments at the computer or at
assigned exercises. Personal training on the assigned exercises on both the theory and practice modules is
fundamental to successfully pass the final exam.
Algorithms
The course will consist of 32 hours of frontal lessons and 16 hours of practical assignments at the computer.
Personal training on the assigned exercises on both the theory and practice modules is fundamental to successfully
pass the final exam.
The final exam will consist of a written test and a following oral discussion. In the written test the candidate will be
asked to solve some data design problems, write in SQL a data retrieval request, present and discuss the practical
assignments implemented during the course. Usually the examination on the two parts of the course (Databases and
Algorithms) are held in the same day (one in the morning and the other one in the afternoon), but can be overcome
separately (but in the same year).
At the end, both the tests on the two parts of the course (Databases and Algorithms) must be satisfactory to allow
the student to overcome the overall examination.
During the following, oral exam (planned some days after the written part) the student will be asked to discuss the
presented solution in the written part.
Students are required to pass the written test before to be admitted the oral part.
SUPPORT ACTIVITIES
The laboratory will consists in assignments that will be solved by means of practical activities at the computer that
will support the theorical notions learnt during the course.
SYLLABUS
Databases
Databases; database management systems; data models; database languages; the relational model and its
languages; integrity constraints; relational algebra; SQL;
Database design methodologies and models; the database design process; the Entity-Relationship model;
conceptual design; requirement collection and analysis; general data modelling criteria; design strategies; qualities
of a conceptual schema; a general methodology for database design; CASE tools for database design; logical
design; translation towards the relational model;
Normalization theory for database design; redundancies and anomalies; functional dependencies; Boyce- Codd
normal form; qualities of decompositions; third normal form; normalization and the design process
Algorithms
Problems and algorithms: solvability, correctness, complexity. Termination and non-termination. Unsolvable
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problems: undecidability of the halting problem. Correctness of algorithms. Mathematical induction principles:
simple induction, complete induction, structural induction.
Analysis of algorithms. Complexity (in time and in space) of algorithms. Complexity of problems. Tractable vs.
intractable problems and algorithms.
Data structures, Abstract Data Types (ADT), structure invariants. Sequences, arrays, linked lists, stacks, queues,
trees, dictionaries, hash tables. Graph representation and primitives.
Databases
Algorithms
Suggested book:
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Decision and Uncertainty
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Decision theory (deactivated)
NOTE
In the a.y. 2017/18 the course's name and code will change to MAT0071 Decisions and Uncertainty.
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Econometrics
Econometrics
COURSE OBJECTIVES
The main purpose of this course is to give a general and comprehensive overview of the different econometric
methodogies and approaches, focusing on what is relevant for doing and understanding empirical work on large
data-bases. The number of econometric techniques that can be used is numerous and their validity often depends
crucially upon the validity of the underlying assumption. This course attempts to guide students through this array of
estimation and testing procedures by also offering several computer-lab sessions where students will face real
world empirical cases.
COURSE AIMS
Knowledge and understanding: this course will provide students with a deep and up-to-date knowledge of modern
econometric theories and related estimation and testing techniques.
- Applying knowledge and understanding: students will learn how to apply econometrics techniques to actual
economic problems. To this aim students will be introduced to a professional econometric software which will be
used for the computations presented in this course.
- Making judgements: the students will learn how to assess the validity of the assumptions of a wide range of
econometric models with the purpose of realizing potential drawbacks or dangers in their application to relevant
empirical economic questions.
- Communication skills: students will learn how to effectively organize ideas both in written and oral form, possibly
with the help of presentation of scientific papers during the course.
- Learning skills: this course will enable students to understand the recent developments in econometrics and will be
a suitable basis for further research work in the area.
COURSE DELIVERY
The course consists of 46 lecture hours. Strong interaction between teachers and students is warmly encouraged.
Part of the course will be given at the Computer Lab.
75 m. (max.) written exam with closed books at the end of the course
SYLLABUS
- The Classical Linear Regression Model and Its Violations (chap. 2-3-4)
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- Endogeneity, Instrumental Variables and GMM (chap. 5)
- MaximumLikelihood Estimation and Specification Tests (chap. 6)
- Models with Limited Dependent Variables (chap. 7)
The course is mostly based on Verbeek's A Guide to Modern Econometrics (4th edition, 2012). For most topics
lecture notes with further references will be also circulated .
NOTE
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Game theory
Game theory
NOTE
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Information theory
Information theory
COURSE OBJECTIVES
COURSE AIMS
At the end of the course the student will have the capacity to apply information theory tools and approaches to both
theoretical and practical problems related to information management, coding, representation, protection and
information metrics.
COURSE DELIVERY
The course will be based on theretical lessons followed by in class exercises and computer based experiments.
Personal training on assigned exercises is important for the success in this class.
SYLLABUS
The first part of the course is devoted to the classical information theory. In particular, the addressed topics are:
definition of information and source types, the concept of entropy, source coding, Shannon's first theorem (source
coding), uniquely decodable codes, optimality of Huffman coding, models of noisy channels, definition of the
channel capacity according to Shannon's theorem (channel coding).
The second part of the course is devoted to the study of source coding and channel coding algorithms used in many
applications, communication systems and networks. The selected topics include arithmetic coding, the Lempel-Ziv-
Welch algorithms and state of the art standards for image and video compression. As far as channel coding is
regarded the course will introduce linear block codes, cyclic codes, convolutional codes and fountain codes.
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Thomas M. Cover, Joy A. Thomas, "Elements of Information Theory, 2nd Edition", ISBN: 978-0-471-24195-9
NOTE
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Introduction to data mining
PREREQUISITES
Databases and Algorithms, Programming
COURSE OBJECTIVES
The objectives of the course will be introduce students to the field of Data Mining and Machine Learning, that merge
together competencies of statistics and computer science.
The course will teach the differences between tasks and models and will introduce the students to some of the
popular models in Machine Learning such as binary classification and related tasks, transformation of a binary
classification model into a multiple class model, concept learning by means of logical formulas, tree models and
their purposes, rule models, subgroup discovery, linear models (least squares, regression), perceptron, Support
Vector Machines, Kernel methods.
The course will introduce the algorithms for the training of the models.
The laboratory part of the course will introduce the students to a practical open software suite that includes the
algorithms of learning of the models seen during the course (and much more).
COURSE AIMS
The results of the learning outcomes will be mastering some the main concepts in Data Mining and Machine Learning
and using them in the context of a practical open software suite for data analysis and machine learning.
COURSE DELIVERY
The final exam will be oral in which the students will be asked to show that they master the theorical lessons
(knowledge of the models and of their purposes) and use of the practical software suite (Weka) for data analysis in
some use cases.
SUPPORT ACTIVITIES
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Machine learning experiments in Laboratory with a software suite for Data Mining.
The laboratory will be a practical support to the learning of the theorical lessons by means of practical data analysis
assignments on public data-sets.
SYLLABUS
Tasks and models; Binary classification and related tasks; Beyond binary classification (transformation of a binary
classification model into a multiple class model; Concept learning by means of logical formulas; Version Space;
learning hypothesis by means of Horn clauses; Tree models (decision trees, regression trees, features trees,
ranking trees); rule models (list of rules and sets of rules); subgroup discovery; linear models (least squares,
regression); perceptron; Support Vector Machines; Kernel methods;
Peter Flach, Machine Learning - The Art and Science of Algorithms that Make Sense of Data, Cambridge University
Press, 2012.
NOTE
This course is borrowed from Machine Learning and Intelligent Data Analysis and will be delivered at the Computer
Science Department.
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Multivariate statistical analysis
PREREQUISITES
Probability Theory
PROPEDEUTIC FOR
Statistical Machine Learning
COURSE OBJECTIVES
The course aims at introducing multivariate analysis in statistical modeling. All the methods will be implemented on
real datasets in the R language.
COURSE AIMS
The student will learn the basic techniques for analyzing multi-dimensional data (including visualization), study
multivariate distributions and their properties, discuss various methods for dimension reduction.
COURSE DELIVERY
The course is composed of 48 hours of class lectures. Examples and exercises will be dealt with at class through the
R language.
Problem Sets:
There will be 2 problem sets assigned throughout the course. They will be posted in due time on
https://sites.google.com/a/carloalberto.org/pdeblasi/teaching
together with an indication of the deadline.
Problem sets must be submitted and there are no late submissions. They are an essential part of the course,
providing students with a guide on how well they are grasping the material on a "real time" basis. They request the
solution of exercises, solution which might require the use of a statistical software. Students are encouraged to work
in groups on the problem sets. However, students should understand the material on their own, and hand in their
own problem sets.
Exam:
There will be a final exam, check out for dates on
http://www.master-sds.unito.it
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The final examination consists of a written test, either a short or a long test according to the problem sets.
Specifically,
(1) First 2 exam dates: the course grade is determined by the problem sets and the final exam. The final exam
consists of a short written test (1h) on the part of the program not covered by problem sets followed by an oral
examination. The final grade will be a combination of the problem sets grades (70%), and the final exam grade
(45%). For students who have failed to submit the solutions of the problem sets, case (2) below applies.
(2) From the 3rd exam date on: the final exam consists of a long written test (3h) on the whole program and the final
grade will be determined solely by it (100%).
SYLLABUS
- Introduction
- summary statistics for multivariate data
- multivariate data visualization
- multivariate Gaussian distributions
- Principal Component Analysis (PCA):
- geometric and algebraic basics of PCA
- calculation and choice of components
- plotting PCs, interpretation
- Factor Analysis (FA):
- model definition and assumptions
- estimation of loadings and communalities
- choice of the number of factors
- factor rotation
- Canonical Correlation Analysis:
- computation and interpretation
- relationship with multiple regression
- Discriminant Analysis and Classification:
- classification rules
- linear and quadratic discrimination
- error rates
- Cluster Analysis:
- measure of similarity
- hierarchical clustering
- K-means clustering
- model based clustering
- R.A. Johnson and D.W. Wichern (2007). Applied Multivariate Statistical Analysis. Prentice-Hall, 6th Ed.
Suggested readings:
- Hastie, Tibshirani, Friedman (2009). The Elements of Statistical Learning, 2nd ed., Springer
- Afifi A., May S., Clark V.A. (2012). Practical Multivariate Analysis, 5th ed., Chapman & Hall/CRC
- Everitt B. (2005). An R and S-PLUS Companion to Multivariate Analysis. Springer
- Rencher A. C., Christensen W. F. (2012). Methods of multivariate analysis, 3rd ed., Wiley
- Rencher A.C. (1992). Interpretation of canonical discriminant functions, canonical variates and principal
components. The American Statistician 46, 217-225.
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Course webpage: http://www.master-sds.unito.it/do/corsi.pl/Show?_id=u23n
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Partial differential equations
COURSE OBJECTIVES
To give a general overview of various methods to solve (free, boundary value, initial value) problems associated
with PDEs commonly met in modelling and applications.
COURSE AIMS
It is expeced that the students are able, at the end of the lectures, to solve basic exercises of the types examined
during the lectures.
COURSE DELIVERY
SUPPORT ACTIVITIES
SYLLABUS
The method of characteristics. First order linear PDEs in two variables, First order linear PDEs in n variables,
Semilinear first order PDEs in n variables.
Second order linear, semilinear and quasilinear PDEs. Elliptic, parabolic and hyperbolic PDEs.
Methods based on (generalized) Fourier series for boundary value problems and initial value problems associated
with linear PDEs.
The Fourier-Laplace method for free problems, boundary value problems and initial value problems associated
with linear PDEs. Laplacian, heat and wave operators.
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S. Salsa, Equazioni a derivate parziali. Springer (2010)
NOTE
Lectures will take place in classroom Vercelli (13-14/9) and Novara (19-21/9). Please, notice that the first lecture on
tue 13/9 will take place from 11:00 to 13:00.
The final assessment test will consist in the solution of some exercises of the types examined during the lectures. It
has a self-evaluation character, it is fully optional and provides no credits.
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Probability theory
Probability theory
PREREQUISITES
An undergrauate level class in Probability and good knowledge of real analysis. Good abilities in elementary
probabilistic problem solving are also necessary for the success in this class.
PROPEDEUTIC FOR
Stochastic Processes, Statistics for Stochastic Processes and EDS-Stochastic Dfferential Equations use concepts
and tools introduced in this course.
COURSE OBJECTIVES
Topics taught in this class are essential tools required to a statistician and a probabilist. They are fundamental for
any modern mathematician. Students re-think to subjects of their undergraduate studies with a different level of
abstraction.This new approach allows them to control some advanced methods of probability theory, useful for
applications as wel as for research.
COURSE AIMS
Students attain a detailed knowledge of the foundations of the theory of probability and related topics in measure
theory. They attain good ability in probabilistic problem solving becoming able to deal both with theoretical and
applied problems related with conditional expectation, convergence features, characteristic functions and
martingales.
They become able to prove new results related with the studied theory, furthermore they become used to learn
using different textbooks.
COURSE DELIVERY
There will be 72 hours of lessons, including 16 hours of in class exercises. Personal training on assigned exercises is
important for the success in this class.
The final exam includes both a written and an oral tests. The two tests are scheduled on different dates. The written
test is valid until the following oral exam. The written test requires the solution of two exercises and the proof of a
theorem (selected from those discussed during classes). It is mandatory to pass this test to be admitted to the oral
test. The use of textbooks and personal notes during the written test is not allowed. The oral examination includes a
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discussion on the written test as well as two question, taken at random by the student. The list of the possible
questions for the oral examination will be provided in advance.
SUPPORT ACTIVITIES
The course include exercises classes; extra exercises are suggested as homework.
SYLLABUS
Overview of elementary probability. Construction of probability measures on R and random variables. Integrals
over probability measures. Independent random variables. Distributions on Rn. Sums of random variables. 0-1 Laws.
Convergence of sequences of random variables. Weak convergence and characteristic functions. Laws of large
numbers and central limit theorem. Conditional expectations. Discrete time martingales, optional stopping , Doob
decomposition and martingale inequalities. Convergence properties of discrete time martingales. Introduction to
continuous time martingales.
Textbooks:
NOTE
Note that the exams' rules have changed since last academic year. These set of rules will be applied from the
session of January 2018.
The old set of rules (academic year 2016/2017 - see on the corresponding tab on the top of the page) will be valid
until the session of September 2017 included.
Attention: students must register on the exams web-page to be admitted to the written/oral exams (in time). Not
registered students will not be admitted being impossible to register their final mark.
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Programming for data science
PREREQUISITES
ELEMENTS OF STATISTICS Basic knowledge in Calculus as provided by the first year Mathematics course. ELEMENTS
OF COMPUTER SCIENCE No specific computer science knowledge is required.
COURSE OBJECTIVES
Aim of the course is to introduce methods, techniques and related computer science instruments for the analysis of
experimental data.
It provides the basic knowledge to use computer science applications as Spreadsheet (e.g. Excel, Calc,...) and
programming languages for statistical computing and graphics (e.g. R programming language)
COURSE AIMS
KNOWLEDGE AND UNDERSTANDING – Completing the course students will be able to:
1) use suitable descriptive and inferential statistics techniques to describe and understand the phenomena being
studied;
2) manage suitable computer science instruments such as worksheet or dedicated software programs for
statistical data analysis.
APPLYING KNOWLEDGE AND UNDERSTANDING – Students will perform the statistical analyses required by the
problem under study by selecting the most computationally and graphically suitable computer science support.
MAKING JUDGEMENTS – Students will decide which statistical techniques to use according to the available data sets
to describe and understand the phenomena under consideration.
COMMUNICATION – The student will be able to justify the choices for the analysis to be performed and to give a
synthetic description of the techniques employed and of the results obtained.
COURSE DELIVERY
The course consists of 10 hours of lectures, and 14 hours of laboratories . Laboratories include exclusively practical
activities.
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The slides presented during lectures are available to students as online materials.
Attendance to lessons is not mandatory, but highly recommended due to the necessity of learning and employing
specific computer science instruments.
The exam consists of a written test and requires a practice exercise on R programming languages
WRITTEN EXAMINATION:
- ten multiple choice questions on course topics (4 options, with the possibility of 0-4 correct options);
SYLLABUS
The teaching material used for lessons and a series of practical exercises are available on the web site of the
course.
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Simulation
Simulation
PREREQUISITES
The basis of Probability Theory and Elements of Statistics are assumed to be known by the students. The knowledge
of a general purpose programming language is required in order to implement the simulators required as part of the
homework exercises and of the final project.
COURSE OBJECTIVES
Simulation is one of the most common techniques used for the evaluation of the performance and if the reliability of
Discrete Event Dynamic Systems (DEDS) often modelled with Stochastic Processes. Discrete Event Simulation
consists on the execution of a program which results in the production of a realization of a stochastic process driven
by Monte-Carlo methods. Learning how to construct a simulator is the main objective of this course, together with
the development of the techniques needed for the statistical analysis of the simulation output. To deeply
understand the difficulty of writing an efficient simulator equipped with the output analysis components, students will
be required to write a few simple simulators "from scratch" without using available tools and libraries.
COURSE AIMS
At the end of the course the students will be able to perform the simulation of non-trivial Discrete Event Sysrtems.
The exercises and the final project will provide the students with the capability of writing the simulators using a
general purpose programming language of their choice. Having developed the simulators "fromn scratch" will allow
the students to understand the potentials and the limits of the Discrete Event Simultauion technique, thus providing
them with the capability of using professional simulators with competence
COURSE DELIVERY
The course will be based on theretical lessons as well as on the soltion of class exercises. Computer
implementations will be required as homework assignments. Personal training on assigned exercises is important
for the success in this class.
The final examination will consist in the discussion of a project developed individually by the students used as the
basis for asking questions on the theoretical aspects of the exercise. Students will not be required to be able to
reproduce the derivations used to obtain the results discussed during the course, but will have to know the
definitions and the applications of the theory.
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The final grade will be out of thirty.
SUPPORT ACTIVITIES
Exercises will be assigned as homework. The course will include sessions devoted to the discussion of the solutions
of selected homework, as well as to the solution of additional exercises.
SYLLABUS
Introduction
Operational Analysis
- Balance equations
Simulation
- Validation
-Lawrence M. Leemis and Stephen K. Park, "Discrete-Event Simulation: a first course", Pearson Prentice Hall, 2006.
-George S. Fishman, "Principles of Discrete Event Digital Simulation", John Wiley & Sons, 1978.
-Hisashi Kobayashi, "Modeling and Analysis: An Introduction to System Performance Evaluation Methodology",
Addison-Wesley, 1978.
-Kishor S. Trivedi, "Probability and Statistics with Reliability, Queueing and Computer Science Applications",
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Prentice Hall, 1982.
NOTE
This course is borrowed from Simulation and Modelling and will be delivered at the Computer Science Department.
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Simulation models for economics
NOTE
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Statistical inference
Statistical inference
PREREQUISITES
Mathematical, probabilistic and statistical tools acquired in the three-year undergraduate program. A detailed list of
the required backgroud will be provided during the first lecture.
COURSE OBJECTIVES
Ability to apply statistical concepts and statistical techniques with respect to the point estimation, hyphotesis testing
and confidence sets.
COURSE AIMS
Making judgements
Students will be able to discern the different aspects of statistical modeling.
Communication skills
Students will properly use statistical and probabilistic language arising from the classical statistics.
Learning skills
The skills acquired will give students the opportunity of improving and deepening their knowledge of the different
aspects of statistical modeling.
COURSE DELIVERY
Main lectures are devoted to the theorerical aspects of statistical inference. Exercises will be assigned during these
lectures. Lecture devoted to exercises are included in the course.
The exam consists of two parts: the first part is a formal discussion of one of the main topics of statistical infence; the
second part consists of two exercises, typically with more than two questions.
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SYLLABUS
Properties of random samples: random samples and their distributions; functions of random samples; Hoeffding's
and Bernstein's inequality; Efron-Stein inequality; generating random samples; the likelihood function and the formal
likelihood principle; exponential families of distributions.
Estimators and principle of data reduction: sufficient statistics; minimal sufficient statistics; Fisher factorization and
Lehmann-Scheffé theorem; finite-sample properties of estimators; Cramer-Rao lower bound and Rao-Blackwell
theorem; large-sample properties of estimators.
Hypothesis testing: probabilistic structure of hypothesis tests; Neyman-Pearson lemma; likelihood ratio test; the
Karlin-Rubin test; asymptotics for likelihood ration test; other large-sample hypothesis tests; hypothesis testing
under the Gaussian model; oneway analysis of variance
Regression models; simple and multiple linear regression; least squares estimators and maximum likelihood
estimators; Gauss-Markov theorem; hypothesis testing for regression models; generalized linear regression; the
logistic regression model; the poisson regression models.
Bickel, P.J. and Doksum, K.A. (2015). Mathematical Statistics: basic ideas and selected topics. Chapman and Hall/CRC
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Statistical machine learning
COURSE OBJECTIVES
The course introduces methods and models to extract important patterns and trends from big amount of data, and
presents basic concepts of machine learning and data mining from a statistical perspective. All the methods will be
introduced from a theoretical point of view and implemented on real datasets in the R language.
COURSE AIMS
Advances knowledge of parametric and nonparametric models for prediction and classification
Ability to convert various problems and data into statistical models to perform several type of
prediction/classification.
Making judgements
Students will be able to discern the different aspects of statistical learning in modern settings.
Communication skills
Students will properly use statistical language to comunicate the results of their findings.
Learning skills
The skills acquired will give students the opportunity of improving and deepening their knowledge of
statistical modeling.
COURSE DELIVERY
Half of the lectures are devoted to the theorerical aspects of statistical machine learning and the remaining half to
their practical implemetation in the R software considering both the related numerical and computational issues.
Exercises will be assigned during lectures and lab sessions.
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The exam consists of three parts: the first part is a written exam on theory; the second part is a practical session with
R; the last part is an oral discussion.
SYLLABUS
Introduction
Regression
Classification:
Miscellanea:
AZZALINI, SCARPA. Data analysis and data mining . Oxford University Press
HASTIE, TIBSHIRANI AND FRIEDMAN. The elements of statistical learning: data mining, inference and
prediction. Springer-Verlag.
NOTE
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Statistics for stochastic processes
PREREQUISITES
Good knowledge of probability theory and the basics of stochastic processes. In more details you will need - laws
of large numbers and central limit theorems - measure theory - conditional expectations - L^p spaces with respect
to a probability measure - Hilbert spaces (some introductory material on this topic is present in the text books)
COURSE OBJECTIVES
The goal of lectures is to introduce statistical inference for time series taking into account both the
theoretical/mathematical aspects and their practical application to data analysis.
Time series are considered, aiming to characterize properties, asymptotic behavior, estimations and forecasting,
spectral analysis as well as decomposition in trend and seasonal components. Such concepts are applied to the
analysis of simulated data or existing databases in order to infer and validate a model supporting the data.
COURSE AIMS
At the end of the course, students will have understood how to model time series with focus on forecasting and
estimation of the moments, of the spectrum and of the parameters of time series models.
Moreover they will know which are the main steps of the analysis of a dataset, and which tools are available to this
aim:
- descriptive statistics, moment and spectrum estimation
- formulation of models, parameter estimation, model selection, model verification
- forecasting
COURSE DELIVERY
We will mainly deliver frontal lectures, but a computer lab is also included. During the lectures we will alternate a
formal presentation of some topics, including proofs and technical details, with a more informal part where we will
introduce some concepts that are useful for the analysis of data sets. In the computer lab we will use R to simulate
and analyse datasets from ARMA processes or existing databases. We refer to some particular packages useful to
deal with simulations, decompositions and forecasting.
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Who wants to be examined on the syllabus of
a.a.<2015/16: send an e-mail to Elvira Di Nardo, one week before the practical session, to organize the methods
a.a.=2015/16: a practical session on the analysis of a dataset in the computer lab is followed by writing a short essay
on one of the arguments introduced by Prof. Sirovich. The final evaluation with a regular oral examination will be
after the correction of this essay and the analysis in the computer lab a couple of days later.
a.a.=2016/17: a practical session on the analysis of a dataset in the computer lab is followed by writing a short essay
on one of the arguments introduced by Prof. Rinott. The final evaluation with a regular oral examination will be after
the correction of this essay and the analysis in the computer lab a couple of days later.
SUPPORT ACTIVITIES
Computer lab.
SYLLABUS
Time series: ARMA processes, covariance and spectrum. Estimation and elimination of trend, seasonal components
and periodicities. Linear filtering, causality and smoothing. Recursive methods for computing the best linear
predictors: Durbin-Levinson algorithm, innovations algorithms. Spectral representation of simple processes.
Herglotz Theorem. Spectral density, the relation to characteristic functions and their inversion in probability.
Computing the spectral density for ARMA models. Applying the spectral density to obtain causal invertible models.
Stochastic integrals: definition, existence, examples, properties, relation to spectral distributions. Spectral
representation of stationary processes by stochastic integrals and applications to prediction in ARMA. Estimation of
the mean, the covariance, the partial autocorrelation. Estimation of the parameters and model selection. Diagnostic
tools. Asymptotic theory: m-dependent variables and the associated CLT. Computer lab: simulation and statistical
analysis of time series with R.
Lectures will not adhere to the material of any single text, but the students can find material on the topics we teach
on different books. References for each topic will be made available during the course.
- Brockwell and Davis, Introduction to Time Series and Forecasting, Second Edition. Springer texts in statistics. 2002
- Brockwell and Davis, Time Series, theory and methods, Springer (collana SSS), New York, 1991
- Priestley, Spectral Analysis and Time Series, Academic Press, Vol I, 1981
- Shumway and Stoffer, Time series Analysis and Its Applications, Springer, 2011.
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Stochastic differential equations
COURSE OBJECTIVES
The course aims to put the student in a position to understand the mathematical formulation of various models of
applied sciences and financial mathematics which involve stochastic differential equations. The course uses
probabilistic concepts and tools that are developed in the course ``Probability Theory'' and elements of
Functional Analysis (see ``Analysis''); these concepts are briefly mentioned in the first lectures. The proofs of the
main results of the course are carried out completely. They show important links between Analysis and Probability.
To improve the skills of reading and study the teacher proposes the reading of some scientific articles. Together
with the course ``Stochastic Processes'' it suggests an approach to the research in stochastic environments. The
course also provides basic concepts on parabolic equations of Kolmogorov type.
COURSE AIMS
Knowledge of the Ito stochastic integral and the related stochastic differential equations. Knowledge of the
relations between stochastic differential equations and Kolmogorov equations. Ability to apply stochastic
differential equations to solve problems in applied sciences.
COURSE DELIVERY
Oral examination. Questions on the program (theorems, remarks and examples). Concerning the proofs we require
to know in details 3 important proofs. Such required proofs are given in the folder ``Teaching material'' below. This
folder also contains more information on the examination.
SYLLABUS
- Reminder of basic notions on measure theory and probability theory. Multidimensional Gaussian distributions.
- Brownian motion (its construction by means of Haar functions; regularity properties of trajectories); the Wiener
measure.
- The Ito stochastic integral (basic properties; comparison between the stochastic integral and the Riemann-
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Stieltjes integral)
- Markov property of solutions of stochastic differential equations; connections between stochastic differential
equations and parabolic Kolmogorov equations
- Possible applications of stochastic differential equations to Mathematical Finance and Population Dynamics
- Lectures notes
- I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, Second Edition,
1991.
- Arnold, L., Stochastic Differential Equations, Theory and Applications, New York. John Wiley & Sons. 1974
NOTE
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Stochastic modelling for statistical applications
PREREQUISITES
PROBABILITY THEORY (MAT0034)
PROPEDEUTIC FOR
BAYESIAN STATISTICS (MAT0070)
COURSE OBJECTIVES
The course introduces to the theory of Markov chains, in discrete and continuous time, and Lévy processes. These
are nowadays considered essential probabilistic instruments which should be part of a modern statistician's toolbox.
As an illustrative application, some time will be devoted to introduce the basics of Markov chain Monte Carlo
methods, with a few examples of the most widely used strategies.
COURSE AIMS
The student will possess a quite detailed knowledge of Markov chain theory in discrete and continuous time,
knowing how to formulate a model relative to the required task or application and how to analyse its properties,
and will have acquired sufficient familiarity with Levy processes and Markov chain Monte Carlo methods to be able
to autonomously comprehend a scientific paper on those topics.
COURSE DELIVERY
The final assessment consists in an oral examination on the material covered during the course.
The possibility of presenting a scientific paper whose content is coherent with the course's syllabus will be
discussed at the beginning of the course.
SYLLABUS
- Introduction: stochastic processes; finite dimensional distributions; existence theorem; classes of stochastics
processes based on path properties.
- Markov chains: transition matrices, Chapman-Kolmogorov equations, strong Markov property, classification of
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states, invariant measures, reversibility, convergence to equilibrium.
- Elements of Markov chain Monte Carlo methods: Monte Carlo principle; Markov chain Monte Carlo principle;
Metropolis-Hastings algorithm; Gibbs sampler; slice sampler.
- Continuous time Markov chains: transition functions and Chapman-Kolmogorov equations; transition rates and
infinitesimal generators; backward and forward Kolmogorov equations; embedded chains and holding times;
uniformisation; stationarity; reversibility; scaling limits and diffusion approximations.
- Levy processes: definition; infinite divisibility; Levy-Khintchine formula; Levy-Ito decomposition; Poisson random
measures.
The material introduced will be throughly discussed and illustrated with numerous examples.
An 8 hours module, included in the courseload, will be taught by visiting professor Dario Spanò on "Introduction to
stochastic modelling in Population Genetics".
Main references:
NORRIS, J.R. Markov chains. Cambridge Series in Statistical and Probabilistic Mathematics.
BREMAUD, P. Markov Chains. Springer.
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Stochastic processes
Stochastic processes
PREREQUISITES
Good knowledge of Probability and Analysis
COURSE OBJECTIVES
The course is aimed at giving the students the skills to use diffusion processes to represent different realities of
practical interest. The student should use the different techniques for carrying out the analysis of the models. The
student will demonstrate both the ability of self-study of advanced topics, connected to the content of the course,
and the ability to collaborate. Students should also use the software Mathematica to perform some assigned
simulations.
COURSE AIMS
At the end of the course, students will know several important methods to study stochastic models of applied
interest. They will know some of the important classes of stochastic processes and will be able to study their main
functional and features.
COURSE DELIVERY
During the course homeworks are assigned. Solution of these exercises is part of the final exam. Teamwork is
allowed for this part of the work. Exam is oral. Students that do not make homeworks will solve exercises
immediately before the oral exam.
The evaluation of homeworks is valid only for the Summer exam session. From September session students are
required to solve exercises immediately before the oral exam.
SYLLABUS
Brownian Motion: Markov property, existence of the Brownian motion; maximum and first passage time distribution;
arcosine law; iterated logarithm law; Reflected Brownian motion; Heat equation and Brownian motion;
multidimensional Brownian motion.
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Stationary Processes: mean square distance; autoregressive processes; ergodic theory and stationary processes;
Gaussian processes
Diffusion Processes: differential equations associated with some functionals of the process; backward and forward
equations; stationary measures; boundary classification for regular diffusion processes; conditioned diffusion
processes; spectral representation of the transition density for a diffusion; diffusion processes and stochastic
differential equations; jump-diffusion processes; first passage time problems for diffusion processes.
An 8 hours module, included in the courseload, will be taught by Visiting Professor Vassili Kolokoltsov on Brownian
motion.
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