Functional Clustering and Functional Principal Points

Abstract

In this paper, we deal with the relation between functional clustering and functional principal points. The kprincipal points are defined as the set of kpoints which minimizes the sum of expected squared distances from every points in the distribution to the nearest point 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 clustering. We call the extended principal points functional principal points. Random function[5] is defined in a probability space, and functional principal points of a random function have a close relation to functional data analysis. We derive functional principal points of polynomial random functions using orthonormal basis transformation. For functional data following Gaussian random functions, we discuss the relation between the optimum clustering of the functional data and the functional principal points.