MATH-665 / 2 credits

Teacher: Panaretos Victor

Language: English

Remark: The lectures will run weekly on Thursdays (13:15-15:00) starting on October 3rd and running through the end of the semester.

Only this year

## Summary

A rigorous introduction to the statistical analysis of random functions and associated random operators. Viewing random functions either as random Hilbert vectors or as stochastic processes, we will see the interplay between nonparametrics and multivariate statistics in infinite dimensions.

## Content

Random functions can be viewed as random vectors in a Hilbert space, or as stochastic processes. The former is mathematically convenient, whereas the latter is somewhat more suitable from an applied perspective. This course will consider the statistical analysis of random functions through both lenses and present some of the "curses" and "blessings" of infinite dimensions.

Bochner integration
Reproducing kernel Hilbert Spaces
Basic operator Theory, Mercer's theorem
Random vectors and random functions
Mean square vs sample path regularity
Karhunen-Loève theorem
Weak Convergence, tightness, CLT
Gaussian measures and the Hajék-Feldman dichotomy
The problem of measurement
Functional Principal Components
Estimation, testing, regression, (perfect) discrimination
The positive definite continuation problem
Intrinsic and extrinsic functional graphical models

## Keywords

Hilbert space, non-parametric statistics, stochastic processes

## Required courses

Multivariate Statistics (MATH-444), Probability Theory (MATH-432)

## Recommended courses

Functional Analysis I (MATH-302)

## Learning Outcomes

By the end of the course, the student must be able to:

• Describe the main features of the theory and methodology for functional data;
• Operate basic (nonparametric) statistical analyses pertaining to random functions

## Bibliography

Hsing & Eubank, "Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators", Wiley
DaPrato and Zabczyk "Stochastic Equations in Infinite Dimensions" Cambridge

## In the programs

• Number of places: 30
• Exam form: Written (session free)
• Subject examined: Functional Data Analysis
• Lecture: 22 Hour(s)
• Practical work: 12 Hour(s)
• Type: mandatory

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