Functional and Topological Data Analysis
A.Y. 2023/2024
Learning objectives
The aim of the course is to introduce the main mathematical and statistical techniques that can be applied to analyse data that have an high geometrical complexity. Functional Data Analysis is applied to data that can be represented as functions, like for example time series, stochastic processes, density functions, etc. The functional data are here interpreted as patterns, and problems of classification, clustering, source of variation of the patterns are studied. Topological Data Analysis instead is focused on the analysis of the topological or geometrical structure of the data, like the presence of clusters, cavities (or regions with a low density), peaks (or regions with a high density), etc. In this framework data are represented still as functions, possibly multidimensional, or as graphs or networks.
Expected learning outcomes
At the end of the course the student will be able to address problems in which the geometrical or functional structure of the data is a relevant issue. In particular the student will be able to choose the 'right technique for the right problem', and to simplify problems in which the data are extremely high dimensional. The course will be complemented with a coumputer lab part, during which practical examples of functional or topological data analysis will be shown on specific case studies, so that the student will develop also the related needed computational skills.
Lesson period: Second trimester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
Second trimester
Course syllabus
Part A: Functional data analysis
A1. Functional data representation
A2. Functional data registration
A3. Functional Principal Components Analysis
A4. Functional regression techniques
Part B: Topological Data Analysis
B1. Topological spaces and basics topology
B2. Complexes and filtrations on data
B3. Topological persistence and persistence diagrams
B4. Applications to the analysis of graphs and networks
A1. Functional data representation
A2. Functional data registration
A3. Functional Principal Components Analysis
A4. Functional regression techniques
Part B: Topological Data Analysis
B1. Topological spaces and basics topology
B2. Complexes and filtrations on data
B3. Topological persistence and persistence diagrams
B4. Applications to the analysis of graphs and networks
Prerequisites for admission
A prerequisite for attending this course is to have followed and mastered the contents of the "Statistical Theory and Mathematics" course, of the first year of Data Science for Economics, or equivalent courses.
Teaching methods
Lectures are based on frontal teaching with the support of slides and handouts that are progressively published on the reference course website (myAriel platform). Throughout the lectures, examples and case studies of functional and topological data analysis in R are proposed and discussed.
Teaching Resources
PRIMARY TEXTBOOKS:
- Ramsay, J.O., Hooker, G., Graves, S. 2009. ``Functional data analysis with R and MATLAB''. Springer
- T.K. Dey, Y. Wang, 2022, "Computational Topology for Data Analysis", Cambridge University Press. A free copy can be downloaded from the web page: https://www.cs.purdue.edu/homes/tamaldey/book/CTDAbook/CTDAbook.html
- Notes, slides and codes from the teacher
SUPPORTING TEXTBOOKS:
- Ramsay, J.O. and B. W. Silverman, 2005, "Functional Data Analysis", Springer: New York.
- Ferraty, F., Vieu, P. 2006. Nonparametric Functional Data Analysis. New Springer, New York
- Mimi Zhang, Andrew Parnell, Review of clustering methods for functional data, 2022, https://arxiv.org/abs/2210.00847
- Herbert Edelsbrunner and John L. Harer, Computational topology, AMS
- Ramsay, J.O., Hooker, G., Graves, S. 2009. ``Functional data analysis with R and MATLAB''. Springer
- T.K. Dey, Y. Wang, 2022, "Computational Topology for Data Analysis", Cambridge University Press. A free copy can be downloaded from the web page: https://www.cs.purdue.edu/homes/tamaldey/book/CTDAbook/CTDAbook.html
- Notes, slides and codes from the teacher
SUPPORTING TEXTBOOKS:
- Ramsay, J.O. and B. W. Silverman, 2005, "Functional Data Analysis", Springer: New York.
- Ferraty, F., Vieu, P. 2006. Nonparametric Functional Data Analysis. New Springer, New York
- Mimi Zhang, Andrew Parnell, Review of clustering methods for functional data, 2022, https://arxiv.org/abs/2210.00847
- Herbert Edelsbrunner and John L. Harer, Computational topology, AMS
Assessment methods and Criteria
The exam consists in writing a report of about 10-15 pages containing either a description of experimental results and/or of analysis of specific data (experimental project) or an in-depth analysis of a theoretical topic (theoretical project). Both the data for the experimental project, and the topic for the theoretical project, must be agreed in advance between each student and the teacher. The project will be presented and discussed with the teacher during an oral exam.
MAT/06 - PROBABILITY AND STATISTICS - University credits: 6
Lessons: 40 hours
Professor:
Micheletti Alessandra
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Professor(s)