Abstract

Data in many fields of science are sampled from processes that can most naturally be described as functional. Examples include growth curves, temperature curves, curves of financial transaction data and patterns of pollution data. Functional data analysis (FDA) is concerned with the statistical analysis of such data.

An important tool in many empirical and theoretical problems related to FDA is the functional principal analysis (FPCA) which allows to represent or approximate curves in low dimension. It is certainly the most common approach to obtain dimension reduction for functional data. In fact, it achieves in some sense optimal dimension reduction if data are independent. However, it is all but uncommon that functional data are serially correlated.

A typical example is if the observations are segments from a continuous time process (e.g. days). Then, although cross-sectionally uncorrelated for a fixed observation, the classical FPC -score vectors have non-diagonal cross-correlations. This means that we cannot analyze them componentwise (like in the i.i.d. case), but we need to consider them as vector time series which are less easy to handle and to interpret. In particular, a functional principal component with small eigenvalue, hence negligible instantaneous impact on some observation, may have a major impact on the lagged values. Regular FPCs, thus, in a time series context, will not lead to an adequate dimension reduction technique, as they do in the i.i.d. case. This motivates the development of a time series version of functional PCA . The idea is to transform the (possibly infinite dimensional) functional time series, into a vector time series (of low dimension 3 or 4, say), where the individual component processes are mutually uncorrelated, and explain a bigger part of the dynamics and variability of the original process.

In this talk we will propose such a dynamic version of FPCA for general data structures (Hilbertian data) and study its properties. An empirical analysis and a real data example will be given.

The talk is based on joint work with Lukasz Kidziński (EPFL) and Marc Hallin (ULB).