Two Principal Points for Multivariate Location Mixtures of Spherically Symmetric Distributions

Yamamoto, Wakako; Shinozaki, Nobuo

doi:10.14490/jjss1995.30.53

Cited by 20 publications

(20 citation statements)

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“…In fact, principal points are applied to various problems, including product design and quality control problems, such as finding an optimal set of designs for masks (Flury, 1993), selective assembly for two parts in the manufacturing field (Mease et al, 2004;Mease and Nair, 2006;Matsuura, 2011), functional data analysis (Tarpey and Kinateder, 2003;Shimizu and Mizuta, 2007;Shimizu and Mizuta, 2008), and classification analysis of the placebo effect (Tarpey, et al, 2010). The properties of principal points have been widely studied in the literature (e.g., Tarpey and Flury, 1995;Zoppè, 1995;Yamamoto and Shinozaki, 2000;Gu and Mathew, 2001;Kurata, 2008;Bali and Boente, 2009;Matsuura and Kurata, 2011;Matsuura and Kurata, 2014), and the parametric estimation of principal points of some continuous distributions has also been discussed (e.g., Flury, 1993;Tarpey, 1997;Stampfer and Stadlober, 2002;Tarpey, 2007).…”

Section: Introductionmentioning

confidence: 99%

Comparison of model selection methods for the estimation of principal points for a multivariate binary distribution

Yamashita

Matsuura

Suzuki

2015

TQS

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Abstract:Recently, a parametric estimation method for principal points for a multivariate binary distribution using a log-linear model has been proposed, and Akaike information criterion (AIC) has been applied to model selection for log-linear model. This paper compares three model selection methods based on AIC, Bayesian information criterion (BIC), and the likelihood ratio test (LRT) for estimating principal points for a multivariate binary distribution. The performances of the model selection methods are shown through numerical simulation studies

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Section: Introductionmentioning

confidence: 99%

Comparison of model selection methods for the estimation of principal points for a multivariate binary distribution

Yamashita

Matsuura

Suzuki

2015

TQS

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“…Proposition 2 (Yamamoto and Shinozaki [41], Theorem 1). Let X be distributed as the location mixture given by (1.4).…”

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confidence: 97%

“…The mixture treated here is much more general than those of Yamamoto and Shinozaki [41] and Kurata [14]. We derive a principal subspace theorem for n-principal points.…”

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confidence: 99%

“…Hence conditions for its uniqueness, which can be viewed as the uniqueness of optimal partition of a distribution, have been discussed by many authors (e.g., [15,19,26,[38][39][40][41][42][43]). For further reference, see, for example, [2,7,8,11,12,16,21,22,24,25,27,[29][30][31][32][33][34][35][36][37].…”

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confidence: 99%

“…Hence it is of interest to know whether the principal subspace theorem remains valid even under such distributions. Yamamoto and Shinozaki [41] dealt with a location mixture of spherically symmetric distributions and derived a version of the principal subspace theorem for 2-principal points. To state their theorem, let f be the pdf of a p-dimensional spherically symmetric distribution with mean 0 and covariance matrix σ 2 I p .…”

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Principal points of a multivariate mixture distribution

Matsuura

Kurata

2011

Journal of Multivariate Analysis

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a b s t r a c tA 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 npoint-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.

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Two principal points for location mixtures

Yamamoto

Shinozaki

2000

Compstat

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Two Principal Points for Multivariate Location Mixtures of Spherically Symmetric Distributions

Cited by 20 publications

References 11 publications

Comparison of model selection methods for the estimation of principal points for a multivariate binary distribution

Comparison of model selection methods for the estimation of principal points for a multivariate binary distribution

Principal points of a multivariate mixture distribution

Two principal points for location mixtures

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