Functional Principal Points and Functional Cluster Analysis

Abstract

In this chapter, we deal with functional principal points and functional cluster analysis. The k principal points [4] are defined as the set of k points which minimizes the sum of expected squared distances from every points in the distribution to the nearest points of the set, and are mathematically equivalent to centers of gravity by k-means clustering [3]. The concept of principal points can be extended for functional data analysis [16]. We call the extended principal points functional principal points.Random function [6] is defined in a probability space, and functional principal points of random functions have a close relation to functional cluster analysis. We derive functional principal points of polynomial random functions using orthonormal basis transformation. For functional data according to Gaussian random functions, we discuss the relation between the optimum clustering of the functional data and the functional principal points.We also evaluate the numbers of local solutions in functional k-means clustering of polynomial random functions using orthonormal basis transformation.KeywordsFunctional data analysisk-means clusteringGaussian random functionsOrthogonal basisOptimization