Matrix-variate Normal Distribution Research Articles

Empirical Bayes Poisson matrix completion

An empirical Bayes method for the Poisson matrix denoising and completion problems is proposed, and a corresponding algorithm called EBPM (Empirical Bayes Poisson Matrix) is developed. This approach is motivated by the non-central singular value shrinkage prior, which was used for the estimation of the mean matrix parameter of a matrix-variate normal distribution. Numerical experiments show that the EBPM algorithm outperforms the common nuclear norm penalized method in both matrix denoising and completion. The EBPM algorithm is highly efficient and does not require heuristic parameter tuning, as opposed to the nuclear norm penalized method, in which the regularization parameter should be selected. The EBPM algorithm also performs better than others in real-data applications.

A New Class of Bayes Minimax Estimators of the Mean Matrix of a Matrix Variate Normal Distribution

Bayes minimax estimation is important because it provides a robust approach to statistical estimation that considers the worst-case scenario while incorporating prior knowledge. In this paper, Bayes minimax estimation of the mean matrix of a matrix variate normal distribution is considered under the quadratic loss function. A large class of (proper and generalized) Bayes minimax estimators of the mean matrix is presented. Two examples are given to illustrate the class of estimators, showing, among other things, that the class includes classes of estimators presented by Tsukuma.

Low and high dimensional wavelet thresholds for matrix-variate normal distribution

The matrix-variate normal distribution is a probability distribution that is a generalization of the multivariate normal distribution to matrix-valued random variables. In this paper, we introduce a wavelet shrinkage estimator based on Stein’s unbiased risk estimate (SURE) threshold for matrix-variate normal distribution. We find a new SURE threshold for soft thresholding wavelet shrinkage estimator under the reflected normal balanced loss function in low and high dimensional cases. Also, we obtain the restricted wavelet shrinkage estimator based on non-negative sub matrix of the mean matrix. Finally, we present a simulation study to test the validity of the wavelet shrinkage estimator and two real examples for low and high dimensional data sets.

Matrix variate receiver operating characteristic curve for binary classification

In recent years, the study of Receiver Operating Characteristic (ROC) curve analysis has gained significant attention as a means of accurately assessing test performance and determining optimal cutoff points. Traditionally, ROC models have been developed for bi-distributional univariate and multivariate data, such as Bi-normal, Bi-Exponential, Multivariate ROC models, and so forth. However, in current practical scenarios, the prevalence of high-dimensional matrix variate data poses a challenge for accurate test evaluation. To address this issue, this paper presents a novel ROC model that incorporates matrix variate normal distribution to effectively explain the accuracy of a test in the context of matrix data. Further, the accuracy measure, Area under the Curve (AUC) is derived, which helps in explaining the variability of the curve and provides the sensitivity at a particular value of specificity and vice versa. The proposed methodology is supported by a real data set and simulation studies.

Matrix variate generalized asymmetric Laplace distributions

The generalized asymmetric Laplace (GAL) distributions, also known as the variance/mean-gamma models, constitute a popular flexible class of distributions that can account for peakedness, skewness, and heavier-than-normal tails, often observed in financial or other empirical data. We consider extensions of the GAL distribution to the matrix variate case, which arise as covariance mixtures of matrix variate normal distributions. Two different mixing mechanisms connected with the nature of the random scaling matrix are considered, leading to what we term matrix variate GAL distributions of Type I and II. While Type I matrix variate GAL distribution has been studied before, there is no comprehensive account of Type II in the literature, except for their rather brief treatment as a special case of matrix variate generalized hyperbolic distributions. With this work we fill this gap, and present an account for basic distributional properties of Type II matrix variate GAL distributions. In particular, we derive their probability density function and the characteristic function, as well as provide stochastic representations related to matrix variate gamma distribution. We also show that this distribution is closed under linear transformations, and study the relevant marginal distributions. In addition, we also briefly account for Type I and discuss the intriguing connections with Type II. We hope that this work will be useful in the areas where matrix variate distributions provide an appropriate probabilistic tool for three-way or, more generally, panel data sets, which can arise across different applications.

Identifying depression disorder using multi-view high-order brain function network derived from electroencephalography signal.

Brain function networks (BFN) are widely used in the diagnosis of electroencephalography (EEG)-based major depressive disorder (MDD). Typically, a BFN is constructed by calculating the functional connectivity (FC) between each pair of channels. However, it ignores high-order relationships (e.g., relationships among multiple channels), making it a low-order network. To address this issue, a novel classification framework, based on matrix variate normal distribution (MVND), is proposed in this study. The framework can simultaneously generate high-and low-order BFN and has a distinct mathematical interpretation. Specifically, the entire time series is first divided into multiple epochs. For each epoch, a BFN is constructed by calculating the phase lag index (PLI) between different EEG channels. The BFNs are then used as samples, maximizing the likelihood of MVND to simultaneously estimate its low-order BFN (Lo-BFN) and high-order BFN (Ho-BFN). In addition, to solve the problem of the excessively high dimensionality of Ho-BFN, Kronecker product decomposition is used for dimensionality reduction while retaining the original high-order information. The experimental results verified the effectiveness of Ho-BFN for MDD diagnosis in 24 patients and 24 normal controls. We further investigated the selected discriminative Lo-BFN and Ho-BFN features and revealed that those extracted from different networks can provide complementary information, which is beneficial for MDD diagnosis.

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

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 this article, we introduce 24 matrix-variate CWMs which are obtained by allowing both the responses and covariates in each cluster to be modelled by one of four existing skewed matrix-variate distributions or the matrix-variate normal distribution. Endowed with 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-variate normal CWM and matrix-variate FMRs, are applied to two real datasets for illustrative purposes.

Estimating High-Order Brain Functional Networks in Bayesian View for Autism Spectrum Disorder Identification.

Brain functional network (BFN) has become an increasingly important tool to understand the inherent organization of the brain and explore informative biomarkers of neurological disorders. Pearson’s correlation (PC) is the most widely accepted method for constructing BFNs and provides a basis for designing new BFN estimation schemes. Particularly, a recent study proposes to use two sequential PC operations, namely, correlation’s correlation (CC), for constructing the high-order BFN. Despite its empirical effectiveness in identifying neurological disorders and detecting subtle changes of connections in different subject groups, CC is defined intuitively without a solid and sustainable theoretical foundation. For understanding CC more rigorously and providing a systematic BFN learning framework, in this paper, we reformulate it in the Bayesian view with a prior of matrix-variate normal distribution. As a result, we obtain a probabilistic explanation of CC. In addition, we develop a Bayesian high-order method (BHM) to automatically and simultaneously estimate the high- and low-order BFN based on the probabilistic framework. An efficient optimization algorithm is also proposed. Finally, we evaluate BHM in identifying subjects with autism spectrum disorder (ASD) from typical controls based on the estimated BFNs. Experimental results suggest that the automatically learned high- and low-order BFNs yield a superior performance over the artificially defined BFNs via conventional CC and PC.

Soft thresholding wavelet shrinkage estimation for mean matrix of matrix-variate normal distribution: low and high dimensional

Soft thresholding wavelet shrinkage estimation for mean matrix of matrix-variate normal distribution: low and high dimensional

Hierarchical Synchronization Estimation of Low- and High-Order Functional Connectivity Based on Sub-Network Division for the Diagnosis of Autism Spectrum Disorder.

Functional connectivity network (FCN) calculated by resting-state functional magnetic resonance imaging (rs-fMRI) plays an increasingly important role in the exploration of neurologic and mental diseases. Among the presented researches, the method of constructing FCN based on Matrix Variate Normal Distribution (MVND) theory provides a novel perspective, which can capture both low- and high-order correlations simultaneously with a clear mathematical interpretability. However, when fitting MVND model, the dimension of the parameters (i.e., population mean and population covariance) to be estimated is too high, but the number of samples is relatively quite small, which is insufficient to achieve accurate fitting. To address the issue, we divide the brain network into several sub-networks, and then the MVND based FCN construction algorithm is implemented in each sub-network, thus the spatial dimension of MVND is reduced and more accurate estimates of low- and high-order FCNs is obtained. Furthermore, for making up the functional connectivity which is lost because of the sub-network division, the rs-fMRI mean series of all sub-networks are calculated, and the low- and high-order FCN across sub-networks are estimated with the MVND based FCN construction method. In order to prove the superiority and effectiveness of this method, we design and conduct classification experiments on ASD patients and normal controls. The experimental results show that the classification accuracy of “hierarchical sub-network method” is greatly improved, and the sub-network found most related to ASD in our experiment is consistent with other related medical researches.

Relationship between incidence and mortality of asthma with PM2.5, ozone, and household air pollution from 1990 to 2106 in the world: An ecological study

Background Asthma is a chronic airway disease characterized by an overreaction of the airways to a wide range of stimuli. Objective To determine the relationship between the incidence and mortality rate of asthma and the levels of PM2.5, ozone, and household air pollution (HAP) from 1990 to 2016 in the world. Materials and methods This ecological study was conducted across 194 countries. The asthma incidence and mortality rates were extracted from IHME Global Burden of Disease databases. Data on average annual population-weighted HAP, PM2.5, and ozone were obtained from the State of Global Air website. A model-based clustering with a finite mixture of matrix-variate normal distributions was used to identify countries with a similar pattern of three air pollution index values. The number of clusters was determined by the Bayesian information criterion, and R software was used for integrated completed likelihood criterion. The random-effects model was applied to evaluate the relationship between asthma incidence in groups and different years by using the SAS software. Results The countries were classified into five clusters in terms of the mean of PM2.5, ozone, and HAP. The incidence rates had a decreasing trend during 1990–2005 and an increasing trend during 2011–2016 in the clusters. The lowest incidence and mortality rates were observed for countries of cluster 3. The highest incidence and mortality rates were observed for countries in clusters 1 and 2, respectively. The incidence and mortality ratio trends experienced an increase in cluster 1 and a decrease in other clusters. Conclusions Depending on the mortality and incidence trends of asthma in countries in each cluster, effective national and international intervention measures are recommended to deal with such pollutants.

Upper Bounds for the Capacity for Severely Fading MIMO Channels under a Scale Mixture Assumption.

A cornerstone in the modeling of wireless communication is MIMO systems, where a complex matrix variate normal assumption is often made for the underlying distribution of the propagation matrix. A popular measure of information, namely capacity, is often investigated for the performance of MIMO designs. This paper derives upper bounds for this measure of information for the case of two transmitting antennae and an arbitrary number of receiving antennae when the propagation matrix is assumed to follow a scale mixture of complex matrix variate normal distribution. Furthermore, noncentrality is assumed to account for LOS scenarios within the MIMO environment. The insight of this paper illustrates the theoretical form of capacity under these key assumptions and paves the way for considerations of alternative distributional choices for the channel propagation matrix in potential cases of severe fading, when the assumption of normality may not be realistic.

Two new matrix-variate distributions with application in model-based clustering

Two matrix-variate distributions, both elliptical heavy-tailed generalization of the matrix-variate normal distribution, are introduced. They belong to the normal scale mixture family, and are respectively obtained by choosing a convenient shifted exponential or uniform as mixing distribution. Moreover, they have a closed-form for the probability density function that is characterized by only one additional parameter, with respect to the nested matrix-variate normal, governing the tail-weight. Both distributions are then used for model-based clustering via finite mixture models. The resulting mixtures, being able to handle data with atypical observations in a better way than the matrix-variate normal mixture, can avoid the disruption of the true underlying group structure. Different EM-based algorithms are implemented for parameter estimation and tested in terms of computational times and parameter recovery. Furthermore, these mixture models are fitted to simulated and real datasets, and their fitting and clustering performances are analyzed and compared to those obtained by other well-established competitors.

Hypothesis Testing for the Covariance Matrix in High-Dimensional Transposable Data with Kronecker Product Dependence Structure

The matrix-variate normal distribution is a popular model for highdimensional transposable data because it decomposes the dependence structure of the random matrix into the Kronecker product of two covariance matrices: one for each of the row and column variables. We develop tests for assessing the form of the row (column) covariance matrix in high-dimensional settings while treating the column (row) dependence structure as a nuisance. Our tests are robust to normality departures provided that the Kronecker product dependence structure holds. In simulations, we observe that the proposed tests maintain the nominal level and are powerful against the alternative hypotheses tested. We illustrate the utility of our approach by examining whether genes associated with a given signalling network show correlated patterns of expression in different tissues and by studying correlation patterns within measurements of brain activity collected using electroencephalography.

Novel Study of Model-Based Clustering Time Series Gene Expression in Different Tissues: Applications to Aging Process.

Aging is an organized biological process that is regulated by highly interconnected pathways between different cells and tissues in the living organism. Identification of similar genes between tissues in different ages may also help to discover the general mechanism of aging or to discover more effective therapeutic decisions. According to the wide application of model-based clustering techniques, the aim is to evaluate the performance of the Mixture of Multivariate Normal Distributions (MMNDs) as a valid method for clustering time series gene expression data with the Mixture of Matrix-Variate Normal Distributions (MMVNDs). In this study, the expression of aging data from NCBI's Gene Expression Omnibus was elaborated to utilize proper data. A set of common genes which were differentially expressed between different tissues were selected and then clustered together through two methods. Finally, the biological significance of clusters was evaluated, using their ability to find genes in the cell using Enricher. The MMVNDs is more efficient to find co-express genes. Six clusters of genes were observed using the MMVNDs. According to the functional analysis, most genes in clusters 1-6 are related to the B-cell receptors and IgG immunoglobulin complex, proliferating cell nuclear antigen complex, the metabolic pathways of iron, fat, and body mass control, the defense against bacteria, the cancer development incidence, and the chronic kidney failure, respectively. Results showed that most biological changes of aging between tissues are related to the specific components of immune cells. Also, the application of MMVNDs can increase the ability to find similar genes.

Analyzing multiple vector autoregressions through matrix-variate normal distribution with two covariance matrices

This article proposes a new approach to analyze multiple vector autoregressive (VAR) models that render us a newly constructed matrix autoregressive (MtAR) model based on a matrix-variate normal distribution with two covariance matrices. The MtAR is a generalization of VAR models where the two covariance matrices allow the extension of MtAR to a structural MtAR analysis. The proposed MtAR can also incorporate different lag orders across VAR systems that provide more flexibility to the model. The estimation results from a simulation study and an empirical study on macroeconomic application show favorable performance of our proposed models and method.

Three skewed matrix variate distributions

Three-way data can be conveniently modelled by using matrix variate distributions. Although there has been a lot of work for the matrix variate normal distribution, there is little work in the area of matrix skew distributions. Three matrix variate distributions that incorporate skewness, as well as other flexible properties such as concentration, are discussed. Equivalences to multivariate analogues are presented, and moment generating functions are derived. Maximum likelihood parameter estimation is discussed, and simulated data is used for illustration.

Minimax Estimation of the Mean Matrix of the Matrix Variate Normal Distribution under the Divergence Loss Function

The problem of estimating the mean matrix of a matrix-variate normal distribution with a covariance matrix is considered under two loss functions. We construct a class of empirical Bayes estimators which are better than the maximum likelihood estimator under the first loss function and hence show that the maximum likelihood estimator is inadmissible. We find a general class of minimax estimators. Also we give a class of estimators that improve on the maximum likelihood estimator under the second loss function and hence show that the maximum likelihood estimator is inadmissible.

Simultaneous Estimation of Low- and High-Order Functional Connectivity for Identifying Mild Cognitive Impairment.

Functional connectivity (FC) network has been becoming an increasingly useful tool for understanding the cerebral working mechanism and mining sensitive biomarkers for neural/mental disease diagnosis. Currently, Pearson's Correlation (PC) is the simplest and most commonly used scheme in FC estimation. Despite its empirical effectiveness, PC only encodes the low-order (i.e., second-order) statistics by calculating the pairwise correlations between network nodes (brain regions), which fails to capture the high-order information involved in FC (e.g., the correlations among different edges in a network). To address this issue, we propose a novel FC estimation method based on Matrix Variate Normal Distribution (MVND), which can capture both low- and high-order correlations simultaneously with a clear mathematical interpretability. Specifically, we first generate a set of BOLD subseries by the sliding window scheme, and for each subseries we construct a temporal FC network by PC. Then, we employ the constructed FC networks as samples to estimate the final low- and high-order FC networks by maximizing the likelihood of MVND. To illustrate the effectiveness of the proposed method, we conduct experiments to identify subjects with Mild Cognitive Impairment (MCI) from Normal Controls (NCs). Experimental results show that the fusion of low- and high-order FCs can generally help to improve the final classification performance, even though the high-order FC may contain less discriminative information than its low-order counterpart. Importantly, the proposed method for simultaneous estimation of low- and high-order FCs can achieve better classification performance than the two baseline methods, i.e., the original PC method and a recent high-order FC estimation method.

Proper Bayes and minimax predictive densities related to estimation of a normal mean matrix

This paper deals with the problem of estimating predictive densities of a matrix-variate normal distribution with known covariance matrix. Our main aim is to establish some Bayesian predictive densities related to matricial shrinkage estimators of the normal mean matrix. The Kullback–Leibler loss is used for evaluating decision-theoretic optimality of predictive densities. It is shown that a proper hierarchical prior yields an admissible and minimax predictive density. Also, some minimax predictive densities are derived from superharmonicity of prior densities.