Functional Data Anaysis, doctoral course

Instructor: Victor Panaretos

TA: None

Lectures: Thursdays 13:15-15:00, MA12

Exercises: None, but self-study expected to fill in missing details, examples, etc.

This course aims to give 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.

A wish list of topics is as follows:

• 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
  

The main references will be:

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

We will partly follow the structure in the first reference, whereas the second reference is a very good source for some of the probabilistic aspects of function spaces, and Gaussian measures thereon. 

Some further references are:

• Kreyszig, Introductory Functional Analysis with Applications, Wiley
  
• Hunter and Nachtergaele, Applied Analysis, World Scientific
• Ramsey and Silverman, Functional Data Analysis, Springer
• Horvath and Kokoszka, Inference for Functional Data with Applications, Springer