Functional Principal Components Analysis, Implementation and Applications

This master thesis discusses selected topics of Functional Data Analysis (FDA). FDA deals with the random variables (and process) with realizations in the (smooth) functional space. The first part of this thesis introduces the basic assumptions, notation and ideas of FDA, here we will mainly focus on the functional basis approach. The second chapter deals with the one of the most popular FDA technique – Functional Principal Components Analysis (FPCA). FPCA is the functional analogue of the well known dimension reduction technique in the multivariate statistical analysis – search for the (pairwise orthogonal) linear transformations of the random vector with the maximal variance. In the second part of this thesis we discuss the k-sample problem in the framework of the FPCA. As the starting point we use the Common Principal Components Modelling in the multivariate statistical analysis. Apart from these theoretical considerations, the second main result of this thesis is the implementation of discussed FDA techniques in the statistical computing environment XploRe. Here we focus on the statistical macros (quantlets), graphical and plotting tools of functional data analysis.