Functional Data Analysis

In many different fields of applied statistics the object of interest is depending on some continuous parameter, i.e. continuous time. Typical examples in biostatistics are growth curves or temperature measurements. Although for technical reasons, we are able to measure temperature just in discrete intervals — it is clear that temperature is a continuous process. Temperature during one year is a function with argument “time”. By collecting one-year-temperature functions for several years or for different weather stations we obtain bunch (sample) of functions —

functional data set

. The questions arising by the statistical analysis of functional data are basically identical to the standard statistical analysis of univariate or multivariate objects. From the theoretical point, design of a stochastic model for functional data and statistical analysis of the functional data set can be taken often one-to-one from the conventional multivariate analysis. In fact the first method how to deal with the functional data is to discretize them and perform a standard multivariate analysis on the resulting random vectors. The aim of this chapter is to introduce the functional data analysis (FDA), discuss the practical usage and implementation of the FDA methods.

This chapter is organized as follows: Section 16.1 defines the basic mathematical and statistical framework for the FDA, Section 16.2 introduces the most popular implementation of functional data analysis — the functional basis expansion. In Section 16.4 we present the basic theory of the functional principal components, smoothed functional principal components and a practical application on the temperature data set of the Canadian Weather-stations.