Functional Data Analysis
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.
Frequency
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
Learning Prerequisites
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
Resources
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
Moodle Link
In the programs
- Number of places: 30
- Exam form: Written (session free)
- Subject examined: Functional Data Analysis
- Courses: 22 Hour(s)
- TP: 12 Hour(s)
- Type: mandatory