Functional and Topological Data Analysis

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.

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.