Functional and topological data analysis

A.A. 2024/2025
6
Crediti massimi
40
Ore totali
SSD
MAT/06
Lingua
Inglese
Obiettivi formativi
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.
Risultati apprendimento attesi
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.
Corso singolo

Questo insegnamento può essere seguito come corso singolo.

Programma e organizzazione didattica

Edizione unica

Responsabile
Periodo
Secondo trimestre

Programma
Part A: Functional data analysis
A1. Functional data representation
A2. Functional data registration
A3. Functional Principal Components Analysis
A4. Functional regression techniques
A5. Depth measures
A6. Clustering of functional data

Part B: An introduction to Topological Data Analysis
B1. Topological spaces
B2. Topological persistence and persistence diagrams
B3. Alpha-spheres
Prerequisiti
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.
Metodi didattici
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.
Materiale di riferimento
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
Modalità di verifica dell’apprendimento e criteri di valutazione
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 - PROBABILITA' E STATISTICA MATEMATICA - CFU: 6
Lezioni: 40 ore
Docente/i
Ricevimento:
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