### Abstract

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 R^{p} 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.

Original language | English |
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Pages (from-to) | 853-870 |

Number of pages | 18 |

Journal | Statistical Papers |

Volume | 55 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2014 |

### Fingerprint

### Keywords

- Allometric extension model
- Elliptically symmetric distribution
- Multivariate mixture distribution
- Principal components
- Principal points
- Principal subspace theorem

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Statistical Papers*,

*55*(3), 853-870. https://doi.org/10.1007/s00362-013-0532-z

**Principal points for an allometric extension model.** / Matsuura, Shun; Kurata, Hiroshi.

Research output: Contribution to journal › Article

*Statistical Papers*, vol. 55, no. 3, pp. 853-870. https://doi.org/10.1007/s00362-013-0532-z

}

TY - JOUR

T1 - Principal points for an allometric extension model

AU - Matsuura, Shun

AU - Kurata, Hiroshi

PY - 2014

Y1 - 2014

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

UR - http://www.scopus.com/inward/record.url?scp=84903907940&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84903907940&partnerID=8YFLogxK

U2 - 10.1007/s00362-013-0532-z

DO - 10.1007/s00362-013-0532-z

M3 - Article

VL - 55

SP - 853

EP - 870

JO - Statistical Papers

JF - Statistical Papers

SN - 0932-5026

IS - 3

ER -