Assessing the pattern of covariance matrices via an augmentation multiple testing procedure

Greselin, Francesca; Ingrassia, Salvatore; Punzo, Antonio

doi:10.1007/s10260-010-0157-5

Cited by 18 publications

(8 citation statements)

References 35 publications

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“…, k exhibit some common structures, and several models have been proposed in this direction (see, e.g., Flury 1984Flury , 1986aFlury , 1987Boik 2002;Greselin et al 2011). The assessment of a common covariance structure, in addition to allow for parsimony, can provide more information about the group conditional distributions (Greselin et al 2011;Greselin and Punzo 2013) and it is of intrinsic interest in several fields such as biometry (refer to Sect. 4).…”

Section: Preliminariesmentioning

confidence: 99%

Unconstrained representation of orthogonal matrices with application to common principal components

Bagnato

Punzo

2020

Comput Stat

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Many statistical problems involve the estimation of a $$\left( d\times d\right) $$ d × d orthogonal matrix $$\varvec{Q}$$ Q . Such an estimation is often challenging due to the orthonormality constraints on $$\varvec{Q}$$ Q . To cope with this problem, we use the well-known PLU decomposition, which factorizes any invertible $$\left( d\times d\right) $$ d × d matrix as the product of a $$\left( d\times d\right) $$ d × d permutation matrix $$\varvec{P}$$ P , a $$\left( d\times d\right) $$ d × d unit lower triangular matrix $$\varvec{L}$$ L , and a $$\left( d\times d\right) $$ d × d upper triangular matrix $$\varvec{U}$$ U . Thanks to the QR decomposition, we find the formulation of $$\varvec{U}$$ U when the PLU decomposition is applied to $$\varvec{Q}$$ Q . We call the result as PLR decomposition; it produces a one-to-one correspondence between $$\varvec{Q}$$ Q and the $$d\left( d-1\right) /2$$ d d - 1 / 2 entries below the diagonal of $$\varvec{L}$$ L , which are advantageously unconstrained real values. Thus, once the decomposition is applied, regardless of the objective function under consideration, we can use any classical unconstrained optimization method to find the minimum (or maximum) of the objective function with respect to $$\varvec{L}$$ L . For illustrative purposes, we apply the PLR decomposition in common principle components analysis (CPCA) for the maximum likelihood estimation of the common orthogonal matrix when a multivariate leptokurtic-normal distribution is assumed in each group. Compared to the commonly used normal distribution, the leptokurtic-normal has an additional parameter governing the excess kurtosis; this makes the estimation of $$\varvec{Q}$$ Q in CPCA more robust against mild outliers. The usefulness of the PLR decomposition in leptokurtic-normal CPCA is illustrated by two biometric data analyses.

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Section: Preliminariesmentioning

confidence: 99%

Unconstrained representation of orthogonal matrices with application to common principal components

Bagnato

Punzo

2020

Comput Stat

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“…The idea is to treat λ g , D g and A g as independent sets of parameters and either constrain them to be the same for each cluster or allow them to vary among clusters, see Table 1. In a quite different context, similar ideas have been introduced in Greselin et al (2011).…”

Section: Gaussian Parsimonious Clustering Modelsmentioning

confidence: 99%

Eigenvalues and constraints in mixture modeling: geometric and computational issues

García-Escudero

Gordaliza

Greselin

et al. 2017

Adv Data Anal Classif

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This paper presents a review about the usage of eigenvalues restrictions for constrained parameter estimation in mixtures of elliptical distributions according to the likelihood approach. These restrictions serve a twofold purpose: to avoid convergence to degenerate solutions and to reduce the onset of non interesting (spurious) maximizers, related to complex likelihood surfaces. The paper shows how the constraints may play a key role in the theory of Euclidean data clustering. The aim here is to provide a reasoned review of the constraints and their applications, along the contributions of many authors, spanning the literature of the last thirty years.Keywords Mixture Model · EM algorithm · Eigenvalues · Model-based clustering IntroductionFinite mixture distributions play a central role in statistical modelling, as they combine much of the flexibility of non parametric models with nice analytical properties of parametric models, see e.g.

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“…was studied by Greselin and Punzo (2013). They assessed this case, named as EVV, as a part of their hierarchy of models to assess the similarity between covariance matrices (see, Greselin et al 2011). A Bartlett adjustment (Bartlett 1937;Lawley 1956) to the LRT for testing the hypothesis H Scale 0 against H Scale 1 , was considered by Najarzadeh (2017).…”

Section: Introductionmentioning

confidence: 99%

Testing equality of standardized generalized variances of k multivariate normal populations with arbitrary dimensions

Najarzadeh

2019

Stat Methods Appl

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For a p-variate normal distribution with covariance matrix , the standardized generalized variance (SGV) is defined as the positive pth root of | | and used as a measure of variability. Testing equality of the SGVs, for comparing the variability of multivariate normal distributions with different dimensions, is still regarded as matter of interest. The most classical test for this problem is the likelihood ratio test (LRT). In this article, testing equality of the SGVs of k multivariate normal distributions with possibly unequal dimensions is studied. To test this hypothesis, two approximations for the null distribution of the LRT statistic are proposed based on the well known Welch-Satterthwaite and Bartlett adjustment distribution approximation methods. Furthermore, the high-dimensional behavior of these approximated distributions is also investigated. Through a wide simulation study: first, the performance of the proposed tests with the classical LRT is compared in terms of type I error, power, and alpha adjusted equivalents; second, the robustness of the procedures with respect to departures from normality assumption is evaluated. Finally, the proposed methods are illustrated with two real data examples. KeywordsStandardized generalized variance • Likelihood ratio test • Welch-Satterthwaite chi-squared approximation • Bartlett adjustment • Taylor expansion • Alpha adjusted critical level • Alpha adjusted power B Dariush Najarzadeh

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Assessing the pattern of covariance matrices via an augmentation multiple testing procedure

Cited by 18 publications

References 35 publications

Unconstrained representation of orthogonal matrices with application to common principal components

Unconstrained representation of orthogonal matrices with application to common principal components

Eigenvalues and constraints in mixture modeling: geometric and computational issues

Testing equality of standardized generalized variances of k multivariate normal populations with arbitrary dimensions

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