Symmetry

Research

21 pages, 1597 KiB

Open AccessArticle

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

Cited by 1 | Viewed by 1128

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

Open AccessArticle

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 4419

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

Open AccessArticle

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 1465

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

Open AccessArticle

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 2 | Viewed by 1603

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

Open AccessArticle

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

Cited by 1 | Viewed by 1333

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

Open AccessArticle

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 - 1 Feb 2022

Viewed by 1068

Abstract

An extension of the

D^{2}

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

D^{2}

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

D^{2}

test statistic for k-th order data is an extension or the generalization of the

D^{2}

test statistic for second-order data and for third-order data, respectively. We also derive the

D^{2}

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

Open AccessArticle

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 1459

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

Open AccessArticle

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 - 7 Jan 2022

Cited by 3 | Viewed by 1520

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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