TY - JOUR

T1 - Principal points for an allometric extension model

AU - Matsuura, Shun

AU - Kurata, Hiroshi

N1 - Funding Information:
Acknowledgments The authors would like to thank the anonymous reviewers for their constructive comments that have resulted in significant improvements in this paper. Matsuura’s portion of this work was supported by JSPS KAKENHI Grant Number 23700341. Kurata’s portion of this work was supported by JSPS KAKENHI Grant Number 20243016, 21500272.

PY - 2014/7

Y1 - 2014/7

N2 - A set of n-principal points of a p-dimensional distribution is an optimal n-point-approximation of the distribution in terms of a squared error loss. It is in general difficult to derive an explicit expression of principal points. Hence, we may have to search the whole space Rp for n-principal points. Many efforts have been devoted to establish results that specify a linear subspace in which principal points lie. However, the previous studies focused on elliptically symmetric distributions and location mixtures of spherically symmetric distributions, which may not be suitable to many practical situations. In this paper, we deal with a mixture of elliptically symmetric distributions that form an allometric extension model, which has been widely used in the context of principal component analysis. We give conditions under which principal points lie in the linear subspace spanned by the first several principal components.

AB - A set of n-principal points of a p-dimensional distribution is an optimal n-point-approximation of the distribution in terms of a squared error loss. It is in general difficult to derive an explicit expression of principal points. Hence, we may have to search the whole space Rp for n-principal points. Many efforts have been devoted to establish results that specify a linear subspace in which principal points lie. However, the previous studies focused on elliptically symmetric distributions and location mixtures of spherically symmetric distributions, which may not be suitable to many practical situations. In this paper, we deal with a mixture of elliptically symmetric distributions that form an allometric extension model, which has been widely used in the context of principal component analysis. We give conditions under which principal points lie in the linear subspace spanned by the first several principal components.

KW - Allometric extension model

KW - Elliptically symmetric distribution

KW - Multivariate mixture distribution

KW - Principal components

KW - Principal points

KW - Principal subspace theorem

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U2 - 10.1007/s00362-013-0532-z

DO - 10.1007/s00362-013-0532-z

M3 - Article

AN - SCOPUS:84903907940

SN - 0932-5026

VL - 55

SP - 853

EP - 870

JO - Statistical Papers

JF - Statistical Papers

IS - 3

ER -