Principal points of a multivariate mixture distribution

Shun Matsuura, Hiroshi Kurata

Research output: Contribution to journalArticlepeer-review

18 Citations (Scopus)


A set of n-principal points of a distribution is defined as a set of n points that optimally represent the distribution in terms of mean squared distance. It provides an optimal n-point-approximation of the distribution. However, it is in general difficult to find a set of principal points of a multivariate distribution. Tarpey et al. [T. Tarpey, L. Li, B. Flury, Principal points and self-consistent points of elliptical distributions, Ann. Statist. 23 (1995) 103-112] established a theorem which states that any set of n-principal points of an elliptically symmetric distribution is in the linear subspace spanned by some principal eigenvectors of the covariance matrix. This theorem, called a "principal subspace theorem", is a strong tool for the calculation of principal points. In practice, we often come across distributions consisting of several subgroups. Hence it is of interest to know whether the principal subspace theorem remains valid even under such complex distributions. In this paper, we define a multivariate location mixture model. A theorem is established that clarifies a linear subspace in which n-principal points exist.

Original languageEnglish
Pages (from-to)213-224
Number of pages12
JournalJournal of Multivariate Analysis
Issue number2
Publication statusPublished - 2011 Feb
Externally publishedYes


  • Location mixture
  • Mean squared distance
  • Primary
  • Principal points
  • Secondary
  • Self-consistency
  • Spherically symmetric distribution

ASJC Scopus subject areas

  • Statistics and Probability
  • Numerical Analysis
  • Statistics, Probability and Uncertainty


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