Functional Clustering and Functional Principal Points

  • Authors:

  • Nobuo Shimizu;Masahiro Mizuta

  • Affiliations:

  • The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569, Japan;Information Initiative Center, Hokkaido University, Kita 11 Nishi 5, Kita-ku, Sapporo 060-0811, Japan

  • Venue:
  • KES '07 Knowledge-Based Intelligent Information and Engineering Systems and the XVII Italian Workshop on Neural Networks on Proceedings of the 11th International Conference
  • Year:
  • 2007

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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.