Closed Likelihood Ratio Testing Procedures to Assess Similarity of Covariance Matrices

Greselin, Francesca; Punzo, Antonio

doi:10.1080/00031305.2013.791643

Cited by 24 publications

(20 citation statements)

References 20 publications

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“…However, the use of the PLR decomposition of an orthogonal matrix is not limited to CPCA, and other statistical models may benefit from its use. Indeed, the PLR decomposition may be used to simplify the ML estimation of the orthogonal matrix related, only to mention a few, to: CPCA based on further non-normal distributions for the groups, other multiple group models allowing for common covariance struc-tures (Flury 1986a;Greselin and Punzo 2013), parsimonious model-based clustering, classification and discriminant analysis (Banfield and Raftery 1993;Flury et al 1994;Celeux and Govaert 1995;Fraley and Raftery 2002;Andrews and McNicholas 2012;Bagnato et al 2014;Lin 2014;Vrbik and McNicholas 2014;Dang et al 2015;Punzo et al 2018;Dotto and Farcomeni 2019), and sophisticated multivariate distributions (Forbes and Wraith 2014;Punzo and Tortora 2019). We pursue to handle these possibilities in future works.…”

Section: Discussionmentioning

confidence: 99%

“…, 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%

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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: Discussionmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Unconstrained representation of orthogonal matrices with application to common principal components

Bagnato

Punzo

2020

Comput Stat

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“…By assuming a normal distribution for the covariates in each group, the ML estimates of the mean and the standard deviation are 11.718 and 2.090 in group 1, and 12.138 and 2.414 in group 2 (see Greselin et al, 2011, Greselin and Punzo, 2013, and Bagnato et al, 2014. Based on these estimates, and further introducing a transition probabilities matrix…”

Section: Sensitivity Study Based On the Blue Crabs Datamentioning

confidence: 99%

Model-based time-varying clustering of multivariate longitudinal data with covariates and outliers

Maruotti

Punzo

2017

Computational Statistics & Data Analysis

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A class of multivariate linear models under the longitudinal setting, in which unobserved heterogeneity may evolve over time, is introduced. A latent structure is considered to model heterogeneity, having a discrete support and following a first-order Markov chain. Heavy-tailed multivariate distributions are introduced to deal with outliers. Maximum likelihood estimation is performed to estimate parameters by using expectation-maximization and expectation-conditional-maximization algorithms. Notes on model identifiability and robustness are provided, along with all computational details needed to implement the proposal. Three applications on artificial and real data are illustrated. These focus on the potential effects of outliers on clustering and their identification.

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“…For each specimen, we consider P = 2 measurements (in millimeters), namely the rear width (RW) and the length along the midline of the carapace (CL). Mardia's test suggests that it is reasonable to assume that the two group-conditional distributions are bivariate normal (see Greselin et al, 2011, Greselin and Punzo, 2013, and Bagnato et al, 2014 for details). The ML estimates of the parameters μ 1 , μ 2 , Σ 1 , and Σ 2 are given in Greselin et al (2011, p. 158); based on these estimates, and further introducing a transition probabilities matrix…”

Section: Artificial Longitudinal Blue Crabs Datamentioning

confidence: 99%

Clustering Multivariate Longitudinal Observations: The Contaminated Gaussian Hidden Markov Model

Punzo¹,

Maruotti²

2016

Journal of Computational and Graphical Statistics

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The Gaussian hidden Markov model (HMM) is widely considered for the analysis of heterogeneous continuous multivariate longitudinal data. To robustify this approach with respect to possible elliptical heavy-tailed departures from normality, due to the presence of outliers, spurious points, or noise (collectively referred to as bad points herein), the contaminated Gaussian HMM is here introduced. The contaminated Gaussian distribution represents an elliptical generalization of the Gaussian distribution and allows for automatic detection of bad points in the same natural way as observations are typically assigned to the latent states in the HMM context. Once the model is fitted, each observation has a posterior probability of belonging to a particular state and, inside each state, of being a bad point or not. In addition to the parameters of the classical Gaussian HMM, for each state we have two more parameters, both with a specific and useful interpretation: one controls the proportion of bad points and one specifies their degree of atypicality. A sufficient condition for the identifiability of the model is given, an expectation-conditional maximization algorithm is outlined for parameter estimation and various operational issues are discussed. Using a large scale simulation study, but also an illustrative artificial dataset, we demonstrate the effectiveness of the proposed model in comparison with HMMs of different elliptical distributions, and we also evaluate the performance of some well-known information criteria in selecting the true number of latent states. The model is finally used to fit data on criminal activities in Italian provinces.

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Closed Likelihood Ratio Testing Procedures to Assess Similarity of Covariance Matrices

Cited by 24 publications

References 20 publications

Unconstrained representation of orthogonal matrices with application to common principal components

Unconstrained representation of orthogonal matrices with application to common principal components

Model-based time-varying clustering of multivariate longitudinal data with covariates and outliers

Clustering Multivariate Longitudinal Observations: The Contaminated Gaussian Hidden Markov Model

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