The multivariate leptokurtic‐normal distribution and its application in model‐based clustering

Bagnato, Luca; Punzo, Antonio; Zoia, Maria Grazia

doi:10.1002/cjs.11308

Cited by 40 publications

(17 citation statements)

References 86 publications

(105 reference statements)

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“…79). Following this idea, we consider the multivariate leptokurtic-normal distribution (Bagnato et al 2017) in CPCA. Compared to the normal distribution, the leptokurtic-normal has an additional parameter β governing the excess kurtosis and, advantageously with respect to other heavy-tailed elliptical distributions, its parameters correspond to quantities of direct interest (mean, covariance matrix, and excess kurtosis).…”

Section: The Modelmentioning

confidence: 99%

“…Compared to the normal distribution, the leptokurtic-normal has an additional parameter β governing the excess kurtosis and, advantageously with respect to other heavy-tailed elliptical distributions, its parameters correspond to quantities of direct interest (mean, covariance matrix, and excess kurtosis). Such a distribution was successfully applied in the modelling of biometric and financial data (Bagnato et al 2017;Maruotti et al 2019).…”

Section: The Modelmentioning

confidence: 99%

“…The maximization of l LN-CPC ( ) with respect to does not admit a closed-form solution (see Bagnato et al 2017, for the case k = 1). Furthermore, the maximization problem is constrained due to Q, j , and β j , j = 1, .…”

Section: Computational Detailsmentioning

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: The Modelmentioning

confidence: 99%

Section: The Modelmentioning

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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“…For MN mixtures (MNMs), one of the possible solutions used to deal with mild outliers is the "component-wise" approach: the component MN distributions are separately protected against mild outliers by embedding them in more general heavy-tailed, usually symmetric, multivariate distributions. Examples are Mt mixtures (MtMs; Peel, 1998 and, MPE mixtures (MPEMs; Zhang andLiang, 2010 andDang et al, 2015), MLN mixtures (Bagnato et al, 2017), and MSt mixtures (Forbes and Wraith, 2014). These methods robustify the estimation of the component means and covariance matrices with respect to mixtures of MN distributions, but they do not allow for automatic detection of bad points, although an a posteriori procedure (i.e., a procedure taking place once the model is fitted) to detect bad points with MStMs is illustrated by McLachlan and Peel (2000).…”

Section: The Mixtures Of Mscn Distributionsmentioning

confidence: 99%

Multiple scaled contaminated normal distribution and its application in clustering

Punzo

Tortora

2019

Statistical Modelling

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The multivariate contaminated normal (MCN) distribution represents a simple heavytailed generalization of the multivariate normal (MN) distribution to model elliptical contoured scatters in the presence of mild outliers, referred to as "bad" points. The MCN can also automatically detect bad points. The price of these advantages is two additional parameters, both with specific and useful interpretations: proportion of good observations and degree of contamination. However, points may be bad in some dimensions but good in others. The use of an overall proportion of good observations and of an overall degree of contamination is limiting. To overcome this limitation, we propose a multiple scaled contaminated normal (MSCN) distribution with a proportion of good observations and a degree of contamination for each dimension. Once the model is fitted, each observation has a posterior probability of being good with respect to each dimension. Thanks to this probability, we have a method for simultaneous directional robust estimation of the parameters of the MN distribution based on down-weighting and for the automatic directional detection of bad points by means of maximum a posteriori probabilities. The term "directional" is added to specify that the method works separately for each dimension. Mixtures of MSCN distributions are also proposed as an application of the proposed model for robust clustering. An extension of the EM algorithm is used for parameter estimation based on the maximum likelihood approach. Real and simulated data are used to show the usefulness of our mixture with respect to well-established mixtures of symmetric distributions with heavy tails.

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“…For general elliptically-contoured mixture components see e.g. McLachlan and Peel [9], Zhang and Liang [10] or Bagnato et al [11]. Finally, a structure is assumed for the cluster-specific covariance matrices.…”

Section: Introductionmentioning

confidence: 99%

Robust inference for parsimonious model-based clustering

Dotto

Farcomeni

2018

Journal of Statistical Computation and Simulation

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We introduce a robust clustering procedure for parsimonious model-based clustering. The classical mclust framework is robustified through impartial trimming and eigenvalue-ratio constraints (the tclust framework, which is robust but not affine invariant). An advantage of our resulting mtclust approach is that eigenvalue-ratio constraints are not needed for certain model formulations, leading to affine invariant robust parsimonious clustering. We illustrate the approach via simulations and a benchmark real data example. R code for the proposed method is available at https://github.com/afarcome/mtclust.

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The multivariate leptokurtic‐normal distribution and its application in model‐based clustering

Cited by 40 publications

References 86 publications

Unconstrained representation of orthogonal matrices with application to common principal components

Unconstrained representation of orthogonal matrices with application to common principal components

Multiple scaled contaminated normal distribution and its application in clustering

Robust inference for parsimonious model-based clustering

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