On parsimonious models for modeling matrix data

Sarkar, Shuchismita; Zhu, Xuwen; Melnykov, Volodymyr; Ingrassia, Salvatore

doi:10.1016/j.csda.2019.106822

Cited by 42 publications

(24 citation statements)

References 27 publications

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“…It is reasonable to assume that similar overparameterization concerns can affect also the MN-CWM when the dimensions of the matrices are quite high. To further increase the parsimony of our model, constrained parameterizations of the covariance matrices can be employed, by following the approaches of Sarkar et al (2020), Subedi et al (2013), Punzo and Ingrassia (2015), and Mazza et al (2018). Furthermore, to accommodate skewness or mild outliers in the data, skewed or heavy tailed matrix-variate distributions could also be considered for the mixing components of the model (e.g., Melnykov and Zhu 2018;Gallaugher and McNicholas 2018;Tomarchio et al 2020).…”

Section: Discussionmentioning

confidence: 99%

Matrix Normal Cluster-Weighted Models

2021

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Finite mixtures of regressions with fixed covariates are a commonly used model-based clustering methodology to deal with regression data. However, they assume assignment independence, i.e., the allocation of data points to the clusters is made independently of the distribution of the covariates. To take into account the latter aspect, finite mixtures of regressions with random covariates, also known as cluster-weighted models (CWMs), have been proposed in the univariate and multivariate literature. In this paper, the CWM is extended to matrix data, e.g., those data where a set of variables are simultaneously observed at different time points or locations. Specifically, the cluster-specific marginal distribution of the covariates and the cluster-specific conditional distribution of the responses given the covariates are assumed to be matrix normal. Maximum likelihood parameter estimates are derived using an expectation-conditional maximization algorithm. Parameter recovery, classification assessment, and the capability of the Bayesian information criterion to detect the underlying groups are investigated using simulated data. Finally, two real data applications concerning educational indicators and the Italian non-life insurance market are presented.

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Section: Discussionmentioning

confidence: 99%

Matrix Normal Cluster-Weighted Models

2021

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“…Specifically, constrained parameterizations of the covariance matrices can be employed, both for the distribution of the responses and the distribution of the covariates. This can be done by following two different routes: (i) the eigen decomposition approach in the fashion of Sarkar et al (2020) or (ii) the bilinear factor analyzers method in accordance to Gallaugher & McNicholas (2019b). Both proposals can drastically reduce the number of estimated parameters, allowing for more parsimonious models.…”

Section: Discussionmentioning

confidence: 99%

“…In all these cases we have p variables measured in r different occasions on N observations, so that the data can be arranged in a three-way array structure having the following three dimensions: variables (rows), occasions (columns) and observations (layers). Examples of contributions to the matrix-variate mixture models literature are Viroli (2011a,b), Gallaugher & McNicholas (2018, Melnykov & Zhu (2018, Sarkar et al (2020), Tomarchio et al (2020), Tomarchio, Gallaugher, Punzo & McNicholas (2021).…”

Section: Introductionmentioning

confidence: 99%

Model-based clustering via skewed matrix-variate cluster-weighted models

Gallaugher¹,

Tomarchio²,

McNicholas³

et al. 2021

Preprint

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Cluster-weighted models (CWMs) extend finite mixtures of regressions (FMRs) in order to allow the distribution of covariates to contribute to the clustering process. In a matrix-variate framework, the matrix-variate normal CWM has been recently introduced. However, problems may be encountered when data exhibit skewness or other deviations from normality in the responses, covariates or both. Thus, we introduce a family of 24 matrix-variate CWMs which are obtained by allowing both the responses and covariates to be modelled by using one of four existing skewed matrix-variate distributions or the matrix-variate normal distribution. Endowed with a greater flexibility, our matrix-variate CWMs are able to handle this kind of data in a more suitable manner. As a by-product, the four skewed matrix-variate FMRs are also introduced.Maximum likelihood parameter estimates are derived using an expectation-conditional maximization algorithm. Parameter recovery, classification assessment, and the capability of the Bayesian information criterion to detect the underlying groups are investigated using simulated data. Lastly, our matrix-variate CWMs, along with the matrix- * *

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“…By varying covariance matrix volumes, orientations, and shapes, 14 alternative models can be obtained. An extension of this idea to the matrix‐variate framework leads to 98 parsimonious models produced as combinations of 14 parameterizations of

bold-italicΣ

and seven representations of

bold-italicΨ

(Sarkar et al, ). The number of parameterizations of

bold-italicΨ

is lower than that of

Σ

due to an introduced constraint that makes the model identifiable.…”

Section: Methodsmentioning

confidence: 99%

“…Gaussian mixture models for matrix or three‐way data have been proposed and extensively studied by Viroli (, , ). Recently, mixtures capable of modelling nonnormal matrix data groups have been proposed by Dogru et al (), Gallaugher and McNicholas (), Melnykov and Zhu (, ), and Sarkar et al ().…”

Section: Introductionmentioning

confidence: 99%

On variable selection in matrix mixture modelling

Wang

Melnykov

2020

Stat

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SummaryFinite mixture models are widely used for cluster analysis, including clustering matrix data. Nowadays, high‐dimensional matrix observations arise in a variety of fields. It is known that irrelevant variables can severely affect the performance of clustering procedures. Therefore, it is important to develop algorithms capable of excluding irrelevant variables and focusing on informative attributes in order to achieve good clustering results. Several variable selection approaches have been proposed in the multivariate framework. We introduce and study a variable selection procedure that can be applied in the matrix‐variate context. The methodological developments are supported by several simulation studies and application to real‐life data sets, with good results.

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On parsimonious models for modeling matrix data

Cited by 42 publications

References 27 publications

Matrix Normal Cluster-Weighted Models

Matrix Normal Cluster-Weighted Models

Model-based clustering via skewed matrix-variate cluster-weighted models

On variable selection in matrix mixture modelling

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