The course requires tools from various areas in mathematics such as measure theory and function spaces. We briefly introduce these concepts at the beginning of the course. As we will otherwise not be able to cover interesting theory, the idea is to discuss some of these underlying concepts in less depth. An introduction to mathematical statistics, measure theory, theory of stochastic processes and functional analysis will therefore be very helpful but it is not required.
In statistics, we estimate/reconstruct objects from data. Given a noisy image for example, the aim is to reconstruct the underlying true image.
A first course in statistics typically deals with reconstruction of finite dimensional parameters, such as the mean or the standard deviation of a distribution. For many interesting applications, however, we want to assume as little as possible about the true underlying objects. Taking a fixed number of parameters is then not appropriate. Instead this should be modelled by assuming a high-dimensional or even infinite dimensional parameter space. To reconstruct an image, for instance, we can think of it as a two-dimensional function and take as a parameter space a function class.
The mathematical theory of complex statistical models has been developed largely during the past years but remains a topic of active research with many challenging open problems. Moreover, recent advances in technology and data sampled made available data sampled at increasingly high frequency and complex form. Such data are often called functional and mathematically they can be taken to be random element in Hilbert Spaces.
The course deals with the theoretical foundation of modern non parametric statistical methods as well as of modern concepts in functional data analysis.
In the course we give a mathematical introduction to this field. Class notes will be made available after the lesson. We start with a short introduction of mathematical prerequisites. We then discuss general estimation methods and non parametric regression. The final third of the course will be devoted to teach the theoretical fundation of functional data analysis. To illustrate the mathematical theory we discuss applications in linguistics and image reconstruction. One lecture will be devoted to the statistical theory of neural networks.
The schedule for the course can be found on MyTimeTable.
Mode of instruction
Blackboard lectures and homeworks
Weekly homework assignments with math problems (1/3) and a final exam (2/3).
Homework counts as practical and there is no retake for it.
Depending on the number of students the final exam will be oral or written.
L. Wasserman, All of Nonparametric Statistics
F. Ferraty, P. Vieu, Nonparametric Functional Data Analysis
T. Hsing, R. Eubank, Theoretical Foundations of Functional Data Analysis, with an Introduction to Linear Operators
From the academic year 2022-2023 on every student has to register for courses with the new enrollment tool MyStudyMap. There are two registration periods per year: registration for the fall semester opens in July and registration for the spring semester opens in December. Please see this page for more information.
Please note that it is compulsory to both preregister and confirm your participation for every exam and retake. Not being registered for a course means that you are not allowed to participate in the final exam of the course. Confirming your exam participation is possible until ten days before the exam.
Extensive FAQ's on MyStudymap can be found here.
Docent: Valentina Masarotto, Snellius Building room 221, v.masarotto[at]math.leideuniv.nl
Teaching Assistant: Thijs Bos, j.m.bos[at]math.leidenuniv.nl