Symmetry in Multivariate Analysis

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 March 2023) | Viewed by 13747

Special Issue Editors


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Guest Editor
Department of Applied Mathematics, Institute of Statistics, National Chung Hsing University, Taichung 402, Taiwan
Interests: multivariate analysis; computational statistics; Bayesian analysis

E-Mail Website
Guest Editor
Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran
Interests: shrinkage estimation; statistical machine learning; high-dimensional data

Special Issue Information

Dear Colleagues,

Multivariate statistical theory and methodology constitute essential requirements and necessary ingredients for successful research in the new era of data science. Because of computer technology advancements, gathering and analyzing complex and non-regular data with more than one feature have become feasible.

Complex data are collected from various scientific fields, such as atmospheric environmental science, social science, psychological and biomedical studies, genetics in bioinformatics, epidemiology, digital imaging information, animal behavior studies, and machine learning, to mention a few. Complex data usually include data sets that are extremely large and/or have complex structures. It becomes challenging to conduct data analysis using conventional statistical methods, software packages, and/or commonly used statistical tools. In developing workable processes to address these issues, statisticians play a vital role.

This Special Issue’s primary purpose is to propose novel computational and methodological multivariate statistical strategies for analyzing so-called “big/high-dimensional/complex/correlated data” and develop multivariate distributions and models to broaden the frontiers of analyzing complex data. From a distributional viewpoint, the error of the underlying analytical model can be symmetric or skew. 

Prof. Dr. Tsung-l Lin
Prof. Dr. Mohammad Arashi
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • high-dimensional
  • big data
  • data science
  • penalized estimation
  • mixed-effects models
  • correlated data
  • finite mixture models
  • shrinkage estimation
  • nonparametrics
  • machine learning
  • directional statistics
  • flexible modeling
  • multivariate distribution
  • non-normal errors

Published Papers (8 papers)

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Research

21 pages, 1597 KiB  
Article
Generalized Support Vector Regression and Symmetry Functional Regression Approaches to Model the High-Dimensional Data
by Mahdi Roozbeh, Arta Rouhi, Nur Anisah Mohamed and Fatemeh Jahadi
Symmetry 2023, 15(6), 1262; https://doi.org/10.3390/sym15061262 - 15 Jun 2023
Viewed by 929
Abstract
The analysis of the high-dimensional dataset when the number of explanatory variables is greater than the observations using classical regression approaches is not applicable and the results may be misleading. In this research, we proposed to analyze such data by introducing modern and [...] Read more.
The analysis of the high-dimensional dataset when the number of explanatory variables is greater than the observations using classical regression approaches is not applicable and the results may be misleading. In this research, we proposed to analyze such data by introducing modern and up-to-date techniques such as support vector regression, symmetry functional regression, ridge, and lasso regression methods. In this study, we developed the support vector regression approach called generalized support vector regression to provide more efficient shrinkage estimation and variable selection in high-dimensional datasets. The generalized support vector regression can improve the performance of the support vector regression by employing an accurate algorithm for obtaining the optimum value of the penalty parameter using a cross-validation score, which is an asymptotically unbiased feasible estimator of the risk function. In this regard, using the proposed methods to analyze two real high-dimensional datasets (yeast gene data and riboflavin data) and a simulated dataset, the most efficient model is determined based on three criteria (correlation squared, mean squared error, and mean absolute error percentage deviation) according to the type of datasets. On the basis of the above criteria, the efficiency of the proposed estimators is evaluated. Full article
(This article belongs to the Special Issue Symmetry in Multivariate Analysis)
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13 pages, 709 KiB  
Article
Small Area Estimation Using a Semiparametric Spatial Model with Application in Insurance
by Seyede Elahe Hosseini, Davood Shahsavani, Mohammad Reza Rabiei, Mohammad Arashi and Hossein Baghishani
Symmetry 2022, 14(10), 2194; https://doi.org/10.3390/sym14102194 - 18 Oct 2022
Cited by 1 | Viewed by 4177
Abstract
Additional information and borrowing strength from the related sites and other sources will improve estimation in small areas. Generalized linear mixed-effects models (GLMMs) have been frequently used in small area estimation; however, the relationship between the response variable and some covariates may not [...] Read more.
Additional information and borrowing strength from the related sites and other sources will improve estimation in small areas. Generalized linear mixed-effects models (GLMMs) have been frequently used in small area estimation; however, the relationship between the response variable and some covariates may not be linear in many cases. In such cases, using semiparametric modeling, incorporating some nonlinear symmetric/asymmetric functions to the predictor seems more appropriate due to their flexibility. In addition, spatial dependence is observed between areas in many cases. Thus, using the semiparametric spatial models for small areas is of interest. This paper presents semiparametric spatial GLMMs and approximates the nonlinear component using splines to estimate the linear part. We apply our proposal for analyzing insurance data obtained from an Iranian insurance company. Our numerical illustrations will support the use of our proposal in situations where the spatial GLMMs may not be appropriate. Full article
(This article belongs to the Special Issue Symmetry in Multivariate Analysis)
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19 pages, 3652 KiB  
Article
Tsallis and Other Generalised Entropy Forms Subject to Dirichlet Mixture Priors
by Johannes T. Ferreira, Tanita Botha and Andriette Bekker
Symmetry 2022, 14(6), 1110; https://doi.org/10.3390/sym14061110 - 28 May 2022
Viewed by 1273
Abstract
Entropy indicates a measure of information contained in a complex system, and its estimation continues to receive ongoing focus in the case of multivariate data, particularly that on the unit simplex. Oftentimes the Dirichlet distribution is employed as choice of prior in a [...] Read more.
Entropy indicates a measure of information contained in a complex system, and its estimation continues to receive ongoing focus in the case of multivariate data, particularly that on the unit simplex. Oftentimes the Dirichlet distribution is employed as choice of prior in a Bayesian framework conjugate to the popular multinomial likelihood with K distinct classes, where consideration of Shannon- and Tsallis entropy is of interest for insight detection within the data on the simplex. However, this prior choice only accounts for negatively correlated data, therefore this paper incorporates previously unconsidered mixtures of Dirichlet distributions as potential priors for the multinomial likelihood which addresses the drawback of negative correlation. The power sum functional, as the product moment of the mixture of Dirichlet distributions, is of direct interest in the multivariate case to conveniently access the Tsallis- and other generalized entropies that is incorporated within an estimation perspective of the posterior distribution using real economic data. A prior selection method is implemented to suggest a suitable prior for the consideration of the practitioner; empowering the user in future for consideration of suitable priors incorporating entropy within the estimation environment as well as having the option of certain mixture of Dirichlet distributions that may require positive correlation. Full article
(This article belongs to the Special Issue Symmetry in Multivariate Analysis)
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16 pages, 1366 KiB  
Article
Fitting Non-Parametric Mixture of Regressions: Introducing an EM-Type Algorithm to Address the Label-Switching Problem
by Sphiwe B. Skhosana, Frans H. J. Kanfer and Salomon M. Millard
Symmetry 2022, 14(5), 1058; https://doi.org/10.3390/sym14051058 - 21 May 2022
Cited by 1 | Viewed by 1353
Abstract
The non-parametric Gaussian mixture of regressions (NPGMRs) model serves as a flexible approach for the determination of latent heterogeneous regression relationships. This model assumes that the component means, variances and mixing proportions are smooth unknown functions of the covariates where the error distribution [...] Read more.
The non-parametric Gaussian mixture of regressions (NPGMRs) model serves as a flexible approach for the determination of latent heterogeneous regression relationships. This model assumes that the component means, variances and mixing proportions are smooth unknown functions of the covariates where the error distribution of each component is assumed to be Gaussian and hence symmetric. These functions are estimated over a set of grid points using the Expectation-Maximization (EM) algorithm to maximise the local-likelihood functions. However, maximizing each local-likelihood function separately does not guarantee that the local responsibilities and corresponding labels, obtained at the E-step of the EM algorithm, align at each grid point leading to a label-switching problem. This results in non-smooth estimated component regression functions. In this paper, we propose an estimation procedure to account for label switching by tracking the roughness of the estimated component regression functions. We use the local responsibilities to obtain a global estimate of the responsibilities which are then used to maximize each local-likelihood function. The performance of the proposed procedure is demonstrated using a simulation study and through an application using real world data. In the case of well-separated mixture regression components, the procedure gives similar results to competitive methods. However, in the case of poorly separated mixture regression components, the procedure outperforms competitive methods. Full article
(This article belongs to the Special Issue Symmetry in Multivariate Analysis)
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15 pages, 3530 KiB  
Article
Mixtures of Semi-Parametric Generalised Linear Models
by Salomon M. Millard and Frans H. J. Kanfer
Symmetry 2022, 14(2), 409; https://doi.org/10.3390/sym14020409 - 18 Feb 2022
Viewed by 1128
Abstract
The mixture of generalised linear models (MGLM) requires knowledge about each mixture component’s specific exponential family (EF) distribution. This assumption is relaxed and a mixture of semi-parametric generalised linear models (MSPGLM) approach is proposed, which allows for unknown distributions of the EF for [...] Read more.
The mixture of generalised linear models (MGLM) requires knowledge about each mixture component’s specific exponential family (EF) distribution. This assumption is relaxed and a mixture of semi-parametric generalised linear models (MSPGLM) approach is proposed, which allows for unknown distributions of the EF for each mixture component while much of the parametric structure of the traditional MGLM is retained. Such an approach inherently allows for both symmetric and non-symmetric component distributions, frequently leading to non-symmetrical response variable distributions. It is assumed that the random component of each mixture component follows an unknown distribution of the EF. The specific member can either be from the standard class of distributions or from the broader set of admissible distributions of the EF which is accessible through the semi-parametric procedure. Since the inverse link functions of the mixture components are unknown, the MSPGLM estimates each mixture component’s inverse link function using a kernel smoother. The MSPGLM algorithm alternates the estimation of the regression parameters with the estimation of the inverse link functions. The properties of the proposed MSPGLM are illustrated through a simulation study on the separable individual components. The MSPGLM procedure is also applied on two data sets. Full article
(This article belongs to the Special Issue Symmetry in Multivariate Analysis)
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27 pages, 416 KiB  
Article
Mean Equality Tests for High-Dimensional and Higher-Order Data with k-Self Similar Compound Symmetry Covariance Structure
by Ricardo Leiva and Anuradha Roy
Symmetry 2022, 14(2), 291; https://doi.org/10.3390/sym14020291 - 01 Feb 2022
Viewed by 915
Abstract
An extension of the D2 test statistic to test the equality of mean for high-dimensional and k-th order array-variate data using k-self similar compound symmetry (k-SSCS) covariance structure is derived. The k-th order data appear in many [...] Read more.
An extension of the D2 test statistic to test the equality of mean for high-dimensional and k-th order array-variate data using k-self similar compound symmetry (k-SSCS) covariance structure is derived. The k-th order data appear in many scientific fields including agriculture, medical, environmental and engineering applications. We discuss the property of this k-SSCS covariance structure, namely, the property of Jordan algebra. We formally show that our D2 test statistic for k-th order data is an extension or the generalization of the D2 test statistic for second-order data and for third-order data, respectively. We also derive the D2 test statistic for third-order data and illustrate its application using a medical dataset from a clinical trial study of the eye disease glaucoma. The new test statistic is very efficient for high-dimensional data where the estimation of unstructured variance-covariance matrix is not feasible due to small sample size. Full article
(This article belongs to the Special Issue Symmetry in Multivariate Analysis)
16 pages, 430 KiB  
Article
Capturing a Change in the Covariance Structure of a Multivariate Process
by Andriette Bekker, Johannes T. Ferreira, Schalk W. Human and Karien Adamski
Symmetry 2022, 14(1), 156; https://doi.org/10.3390/sym14010156 - 13 Jan 2022
Cited by 1 | Viewed by 1260
Abstract
This research is inspired from monitoring the process covariance structure of q attributes where samples are independent, having been collected from a multivariate normal distribution with known mean vector and unknown covariance matrix. The focus is on two matrix random variables, constructed from [...] Read more.
This research is inspired from monitoring the process covariance structure of q attributes where samples are independent, having been collected from a multivariate normal distribution with known mean vector and unknown covariance matrix. The focus is on two matrix random variables, constructed from different Wishart ratios, that describe the process for the two consecutive time periods before and immediately after the change in the covariance structure took place. The product moments of these constructed random variables are highlighted and set the scene for a proposed measure to enable the practitioner to calculate the run-length probability to detect a shift immediately after a change in the covariance matrix occurs. Our results open a new approach and provides insight for detecting the change in the parameter structure as soon as possible once the underlying process, described by a multivariate normal process, encounters a permanent/sustained upward or downward shift. Full article
(This article belongs to the Special Issue Symmetry in Multivariate Analysis)
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19 pages, 377 KiB  
Article
High-Dimensional Radial Symmetry of Copula Functions: Multiplier Bootstrap vs. Randomization
by Monica Billio, Lorenzo Frattarolo and Dominique Guégan
Symmetry 2022, 14(1), 97; https://doi.org/10.3390/sym14010097 - 07 Jan 2022
Cited by 3 | Viewed by 1334
Abstract
We use a recently proposed fast test of copula radial symmetry based on multiplier bootstrap and obtain an equivalent randomization test. The literature shows the statistical superiority of the randomization approach in the bivariate case. We extend the comparison of statistical performance focusing [...] Read more.
We use a recently proposed fast test of copula radial symmetry based on multiplier bootstrap and obtain an equivalent randomization test. The literature shows the statistical superiority of the randomization approach in the bivariate case. We extend the comparison of statistical performance focusing on the high-dimensional regime in a simulation study. We document radial asymmetry in the joint distribution of the percentage changes of sectorial industrial production indices of the European Union. Full article
(This article belongs to the Special Issue Symmetry in Multivariate Analysis)
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