Applications of Projection Pursuit in Functional Data Analysis: Goodness-of- fit, Forecasting, and Change-point Detection

Abstract

Dimension reduction methods for functional data have been avidly studied in recent years. However, existing methods are primarily based on summarizing the data by their projections into principal component subspaces, namely the functional principal component analysis (fPCA). While fPCA could be effective sometimes, in this thesis we show with both real and synthetic data examples some pitfalls of this approach, especially when the components of interest of the functional data are orthogonal to the leading principal components.

In multivariate data analysis, a possible alternative, the projection pursuit technique, was proposed by Kruskal (1972) and Friedman and Tukey (1974). In this thesis, we extend the idea of projection pursuit to functional data analysis. We develop several new computational tools needed to implement the high-dimensional projection pursuit. We apply this functional projection pursuit technique to three problems: (i) normality test for functional data; (ii) forecasting the functional time series; and (iii) change point detection for functional data. For each problem, a simulation study and several data analyses are provided to show the advantages of our proposed method to existing methods in the literature that mostly based on principal component analysis.