Functional data analysis
MATH-420 / 5 credits
Teacher: Panaretos Victor
Language: English
Remark: Cours donné en alternance tous les deux ans
Summary
A rigorous introduction to the statistical analysis of random functions and random operators. Viewing random functions either as random Hilbert vectors or as stochastic processes, we explore the rich interplay between nonparametrics and multivariate statistics in infinite dimensions.
Content
A rigorous introduction to the statistical analysis of random functions and associated random operators. Random functions can be viewed as random vectors in a Hilbert space, or as sample paths of stochastic processes. The former is mathematically convenient, whereas the latter is often more suitable from an applied perspective. This course considers the statistical analysis of random functions through both lenses, showcasing the rich interplay between nonparametrics and multivariate statistics in infinite dimensions, and revealing some of the curses and blessings of infinite-dimensional phenomena. The title of the course is deliberately ambiguous: while we consider the analysis of functional data, we will also see the other side of the coin, namely the functional analysis of data, where infinite-dimensional methods are brought to bear on more classical statistical questions concerning probability distributions arising from traditional and/or high-dimensional data.
Topics:
- Topological and metric peculiarities of Hilbert spaces
- Bounded vs Compact Operators and Schatten classes
- Singular value decomposition and the spectral theorem
- Reproducing Kernel Hilbert Spaces and Mercer's theorem
- Random vectors in Banach/Hilbert spaces and Bochner moments
- Gaussian vectors, Feldman-Hajek dichotomy, conditional independence
- Mean square continuity and the Karhunen-Loève theorem
- Mean square vs pathwise regularity and the Kolmogorov-Centsov theorem
- Law of large numbers and central limit theorem in Hilbert/Banach spaces
- Moment estimation and the problem of measurement
- Functional principal component analysis
- Intrinsic and extrinsic functional graphical models
- Kernel embeddings of probability distributions
- The separation of measure phenomenon
Note. This is primarily a mathematically oriented course in probability, statistics, and functional analysis. The emphasis is on theoretical foundations, infinite-dimensional phenomena, and rigorous reasoning rather than on computation/implementation. Topics such as spline smoothing, numerical algorithms, and data analysis are not a central focus of the course.
Keywords
- Infinite-Dimensional Probability & Statistics
- Functional Analysis
- Stochastic Processes
- Operator Theory
- Gaussian Measures
- Reproducing Kernel Hilbert Spaces
Learning Prerequisites
Required courses
There are no formal prerequisites. However, prior exposure to functional analysis, measure theory, and multivariate statistics is highly beneficial. In particular, MATH-302 (Functional Analysis I), MATH-303 (Measure Theory), and MATH-444 (Multivariate Statistics) provide a valuable foundation for the course material. Students who have completed one or more of these courses will generally find the transition into the material considerably smoother. Students with no prior exposure to any of these topics should expect a steeper learning curve, particularly during the first weeks of the course.
Learning Outcomes
By the end of the course, the student must be able to:
- Define the principal mathematical objects arising in infinite-dimensional probability and statistics
- Describe the main infinite-dimensional phenomena that distinguish functional data from finite-dimensional multivariate data
- Identify the assumptions and mathematical structures underlying major results in functional data analysis.
- Analyze random functions and stochastic processes through both Hilbert-space and stochastic-process perspectives.
- Apply fundamental results from functional analysis, probability, and operator theory to statistical problems.
- Derive key properties of covariance operators, Gaussian measures, and spectral representations in infinite-dimensional settings.
- Interpret phenomena unique to functional probability and statistics through the lens of infinite-dimensional geometry
- Prove selected theoretical results in operator theory and infinite-dimensional probability/statistics
- Elaborate on the uses of functional tools for more classical data analysis
Teaching methods
Lectures ex cathedra, exercises in class
Assessment methods
Final exam
Resources
Bibliography
The course will follow the insstructor's own lecture notes. The following references can serve as additional sources:
- Hsing & Eubank, Theoretical Foundations of Functional Data Analysis, Wiley
- Kukush, Gaussian Measures in Hilbert Space, Wiley
- Kreyszig, Introductory Functional Analysis with Applications, Wiley
Moodle Link
In the programs
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Functional data analysis
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Functional data analysis
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Functional data analysis
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Functional data analysis
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Functional data analysis
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Functional data analysis
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Functional data analysis
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional