On the Spectral Decomposition in Normal Discriminant Analysis

Bagnato, Luca; Greselin, Francesca; Punzo, Antonio

doi:10.1080/03610918.2012.735318

“…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%

Unconstrained representation of orthogonal matrices with application to common principal components

Bagnato

¹

,

Punzo

²

2020

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…There are different possibilities for further work, some of which are worth to be mentioned. First of all, our approach could be extended to the model-based clustering setting in the fashion of Bagnato et al (2014) andPunzo et al (2016). Secondly, our proposed methods could be generalized to the elliptically symmetric distributions.…”

Section: Discussionmentioning

confidence: 99%

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

Najarzadeh

¹

2019

View full text Add to dashboard Cite

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

show abstract

“…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

¹

,

Punzo

²

2017

Computational Statistics & Data Analysis

Self Cite

View full text Add to dashboard Cite

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.

show abstract

On the Spectral Decomposition in Normal Discriminant Analysis

Cited by 12 publications

References 15 publications

Unconstrained representation of orthogonal matrices with application to common principal components

Unconstrained representation of orthogonal matrices with application to common principal components

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

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

Contact Info

Product

Resources

About