Mathematics

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21 pages, 3576 KiB

Open AccessArticle

Alias Structures and Sequential Experimentation for Mixed-Level Designs

by Armando Javier Ríos-Lira, Yaquelin Verenice Pantoja-Pacheco, José Antonio Vázquez-López, José Alfredo Jiménez-García, Martha Laura Asato-España and Moisés Tapia-Esquivias

Mathematics 2021, 9(23), 3053; https://doi.org/10.3390/math9233053 - 28 Nov 2021

Cited by 2 | Viewed by 2791

Abstract

Alias structures for two-level fractional designs are commonly used to describe the correlations between different terms. The concept of alias structures can be extended to other types of designs such as fractional mixed-level designs. This paper proposes an algorithm that uses the Pearson’s [...] Read more.

Alias structures for two-level fractional designs are commonly used to describe the correlations between different terms. The concept of alias structures can be extended to other types of designs such as fractional mixed-level designs. This paper proposes an algorithm that uses the Pearson’s correlation coefficient and the correlation matrix to construct alias structures for these designs, which can help experimenters to more easily visualize which terms are correlated (or confounded) in the mixed-level fraction and constitute the basis for efficient sequential experimentation. Full article

(This article belongs to the Special Issue Advances in Mathematics and Statistics with Applications in Engineering and Industry)

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24 pages, 18175 KiB

Open AccessArticle

Möbius Transformation-Induced Distributions Provide Better Modelling for Protein Architecture

by Mohammad Arashi, Najmeh Nakhaei Rad, Andriette Bekker and Wolf-Dieter Schubert

Mathematics 2021, 9(21), 2749; https://doi.org/10.3390/math9212749 - 29 Oct 2021

Viewed by 1909

Abstract

Proteins are found in all living organisms and constitute a large group of macromolecules with many functions. Proteins achieve their operations by adopting distinct three-dimensional structures encoded within the sequence of the constituent amino acids in one or more polypeptides. New, more flexible [...] Read more.

Proteins are found in all living organisms and constitute a large group of macromolecules with many functions. Proteins achieve their operations by adopting distinct three-dimensional structures encoded within the sequence of the constituent amino acids in one or more polypeptides. New, more flexible distributions are proposed for the MCMC sampling method for predicting protein 3D structures by applying a Möbius transformation to the bivariate von Mises distribution. In addition to this, sine-skewed versions of the proposed models are introduced to meet the increasing demand for modelling asymmetric toroidal data. Interestingly, the marginals of the new models lead to new multimodal circular distributions. We analysed three big datasets consisting of bivariate information about protein domains to illustrate the efficiency and behaviour of the proposed models. These newly proposed models outperformed mixtures of well-known models for modelling toroidal data. A simulation study was carried out to find the best method for generating samples from the proposed models. Our results shed new light on proposal distributions in the MCMC sampling method for predicting the protein structure environment. Full article

(This article belongs to the Special Issue Advances in Mathematics and Statistics with Applications in Engineering and Industry)

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12 pages, 1756 KiB

Open AccessArticle

An Improved Variable Kernel Density Estimator Based on L₂ Regularization

by Yi Jin, Yulin He and Defa Huang

Mathematics 2021, 9(16), 2004; https://doi.org/10.3390/math9162004 - 21 Aug 2021

Cited by 5 | Viewed by 2654

Abstract

The nature of the kernel density estimator (KDE) is to find the underlying probability density function (p.d.f) for a given dataset. The key to training the KDE is to determine the optimal bandwidth or Parzen window. All the data points share [...] Read more.

The nature of the kernel density estimator (KDE) is to find the underlying probability density function (p.d.f) for a given dataset. The key to training the KDE is to determine the optimal bandwidth or Parzen window. All the data points share a fixed bandwidth (scalar for univariate KDE and vector for multivariate KDE) in the fixed KDE (FKDE). In this paper, we propose an improved variable KDE (IVKDE) which determines the optimal bandwidth for each data point in the given dataset based on the integrated squared error (ISE) criterion with the

L_{2}

regularization term. An effective optimization algorithm is developed to solve the improved objective function. We compare the estimation performance of IVKDE with FKDE and VKDE based on ISE criterion without

L_{2}

regularization on four univariate and four multivariate probability distributions. The experimental results show that IVKDE obtains lower estimation errors and thus demonstrate the effectiveness of IVKDE. Full article

(This article belongs to the Special Issue Advances in Mathematics and Statistics with Applications in Engineering and Industry)

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17 pages, 10468 KiB

Open AccessArticle

Alternative Dirichlet Priors for Estimating Entropy via a Power Sum Functional

by Tanita Botha, Johannes Ferreira and Andriette Bekker

Mathematics 2021, 9(13), 1493; https://doi.org/10.3390/math9131493 - 25 Jun 2021

Cited by 3 | Viewed by 2287

Abstract

Entropy is a functional of probability and is a measurement of information contained in a system; however, the practical problem of estimating entropy in applied settings remains a challenging and relevant problem. The Dirichlet prior is a popular choice in the Bayesian framework [...] Read more.

Entropy is a functional of probability and is a measurement of information contained in a system; however, the practical problem of estimating entropy in applied settings remains a challenging and relevant problem. The Dirichlet prior is a popular choice in the Bayesian framework for estimation of entropy when considering a multinomial likelihood. In this work, previously unconsidered Dirichlet type priors are introduced and studied. These priors include a class of Dirichlet generators as well as a noncentral Dirichlet construction, and in both cases includes the usual Dirichlet as a special case. These considerations allow for flexible behaviour and can account for negative and positive correlation. Resultant estimators for a particular functional, the power sum, under these priors and assuming squared error loss, are derived and represented in terms of the product moments of the posterior. This representation facilitates closed-form estimators for the Tsallis entropy, and thus expedite computations of this generalised Shannon form. Select cases of these proposed priors are considered to investigate the impact and effect on the estimation of Tsallis entropy subject to different parameter scenarios. Full article

(This article belongs to the Special Issue Advances in Mathematics and Statistics with Applications in Engineering and Industry)

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17 pages, 341 KiB

Open AccessArticle

Tail Conditional Expectations Based on Kumaraswamy Dispersion Models

by Indranil Ghosh and Filipe J. Marques

Mathematics 2021, 9(13), 1478; https://doi.org/10.3390/math9131478 - 24 Jun 2021

Cited by 1 | Viewed by 1674

Abstract

Recently, there seems to be an increasing amount of interest in the use of the tail conditional expectation (TCE) as a useful measure of risk associated with a production process, for example, in the measurement of risk associated with stock returns corresponding to [...] Read more.

Recently, there seems to be an increasing amount of interest in the use of the tail conditional expectation (TCE) as a useful measure of risk associated with a production process, for example, in the measurement of risk associated with stock returns corresponding to the manufacturing industry, such as the production of electric bulbs, investment in housing development, and financial institutions offering loans to small-scale industries. Companies typically face three types of risk (and associated losses from each of these sources): strategic (S); operational (O); and financial (F) (insurance companies additionally face insurance risks) and they come from multiple sources. For asymmetric and bounded losses (properly adjusted as necessary) that are continuous in nature, we conjecture that risk assessment measures via univariate/bivariate Kumaraswamy distribution will be efficient in the sense that the resulting TCE based on bivariate Kumaraswamy type copulas do not depend on the marginals. In fact, almost all classical measures of tail dependence are such, but they investigate the amount of tail dependence along the main diagonal of copulas, which has often little in common with the concentration of extremes in the copula’s domain of definition. In this article, we examined the above risk measure in the case of a univariate and bivariate Kumaraswamy (KW) portfolio risk, and computed TCE based on bivariate KW type copulas. For illustrative purposes, a well-known Stock indices data set was re-analyzed by computing TCE for the bivariate KW type copulas to determine which pairs produce minimum risk in a two-component risk scenario. Full article

(This article belongs to the Special Issue Advances in Mathematics and Statistics with Applications in Engineering and Industry)

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20 pages, 1408 KiB

Open AccessArticle

One Note for Fractionation and Increase for Mixed-Level Designs When the Levels Are Not Multiple

by Yaquelin Verenice Pantoja-Pacheco, Armando Javier Ríos-Lira, José Antonio Vázquez-López, José Alfredo Jiménez-García, Martha Laura Asato-España and Moisés Tapia-Esquivias

Mathematics 2021, 9(13), 1455; https://doi.org/10.3390/math9131455 - 22 Jun 2021

Cited by 2 | Viewed by 2153

Abstract

Mixed-level designs have a wide application in the fields of medicine, science, and agriculture, being very useful for experiments where there are both, quantitative, and qualitative factors. Traditional construction methods often make use of complex programing specialized software and powerful computer equipment. This [...] Read more.

Mixed-level designs have a wide application in the fields of medicine, science, and agriculture, being very useful for experiments where there are both, quantitative, and qualitative factors. Traditional construction methods often make use of complex programing specialized software and powerful computer equipment. This article is focused on a subgroup of these designs in which none of the factor levels are multiples of each other, which we have called pure asymmetrical arrays. For this subgroup we present two algorithms of zero computational cost: the first with capacity to build fractions of a desired size; and the second, a strategy to increase these fractions with M additional new runs determined by the experimenter; this is an advantage over the folding methods presented in the literature in which at least half of the initial runs are required. In both algorithms, the constructed fractions are comparable to those showed in the literature as the best in terms of balance and orthogonality. Full article

(This article belongs to the Special Issue Advances in Mathematics and Statistics with Applications in Engineering and Industry)

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19 pages, 2796 KiB

Open AccessArticle

A New Class of Estimators Based on a General Relative Loss Function

by Tao Hu and Baosheng Liang

Mathematics 2021, 9(10), 1138; https://doi.org/10.3390/math9101138 - 18 May 2021

Cited by 1 | Viewed by 1868

Abstract

Motivated by the relative loss estimator of the median, we propose a new class of estimators for linear quantile models using a general relative loss function defined by the Box–Cox transformation function. The proposed method is very flexible. It includes a traditional quantile [...] Read more.

Motivated by the relative loss estimator of the median, we propose a new class of estimators for linear quantile models using a general relative loss function defined by the Box–Cox transformation function. The proposed method is very flexible. It includes a traditional quantile regression and median regression under the relative loss as special cases. Compared to the traditional linear quantile estimator, the proposed estimator has smaller variance and hence is more efficient in making statistical inferences. We show that, in theory, the proposed estimator is consistent and asymptotically normal under appropriate conditions. Extensive simulation studies were conducted, demonstrating good performance of the proposed method. An application of the proposed method in a prostate cancer study is provided. Full article

(This article belongs to the Special Issue Advances in Mathematics and Statistics with Applications in Engineering and Industry)

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20 pages, 3733 KiB

Open AccessArticle

An Analytical EM Algorithm for Sub-Gaussian Vectors

by Audrius Kabašinskas, Leonidas Sakalauskas and Ingrida Vaičiulytė

Mathematics 2021, 9(9), 945; https://doi.org/10.3390/math9090945 - 23 Apr 2021

Viewed by 1804

Abstract

The area in which a multivariate

α

-stable distribution could be applied is vast; however, a lack of parameter estimation methods and theoretical limitations diminish its potential. Traditionally, the maximum likelihood estimation of parameters has been considered using a representation of the multivariate [...] Read more.

The area in which a multivariate

α

-stable distribution could be applied is vast; however, a lack of parameter estimation methods and theoretical limitations diminish its potential. Traditionally, the maximum likelihood estimation of parameters has been considered using a representation of the multivariate stable vector through a multivariate normal vector and an

α

-stable subordinator. This paper introduces an analytical expectation maximization (EM) algorithm for the estimation of parameters of symmetric multivariate

α

-stable random variables. Our numerical results show that the convergence of the proposed algorithm is much faster than that of existing algorithms. Moreover, the likelihood ratio (goodness-of-fit) test for a multivariate

α

-stable distribution was implemented. Empirical examples with simulated and real world (stocks, AIS and cryptocurrencies) data showed that the likelihood ratio test can be useful for assessing goodness-of-fit. Full article

(This article belongs to the Special Issue Advances in Mathematics and Statistics with Applications in Engineering and Industry)

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16 pages, 680 KiB

Open AccessFeature PaperArticle

A New Compromise Design Plan for Accelerated Failure Time Models with Temperature as an Acceleration Factor

by Irene Mariñas-Collado, M. Jesús Rivas-López, Juan M. Rodríguez-Díaz and M. Teresa Santos-Martín

Mathematics 2021, 9(8), 836; https://doi.org/10.3390/math9080836 - 12 Apr 2021

Cited by 5 | Viewed by 2274

Abstract

An accelerated life test of a product or material consists of the observation of its failure time when it is subjected to conditions that stress the usual ones. The purpose is to obtain the parameters of the distribution of the time-to-failure for usual [...] Read more.

An accelerated life test of a product or material consists of the observation of its failure time when it is subjected to conditions that stress the usual ones. The purpose is to obtain the parameters of the distribution of the time-to-failure for usual conditions through the observed failure times. A widely used method to provoke an early failure in a mechanism is to modify the temperature at which it is used. In this paper, the statistically optimal plan for Accelerated Failure Time (AFT) models, when the accelerated failure process is described making use of Arrhenius or Eyring equations, was calculated. The result was a design that had only two stress levels, as is common in other AFT models and that is not always practical. A new compromise plan was presented as an alternative to the widely used “4:2:1 plan”. The three-point mixture design proposed specified a support point in the interval that was optimal for the estimation of the parameters in AFT models, rather than simply the middle point. It was studied in comparison to different commonly used designs, and it proved to have a higher D-efficiency than the others. Full article

(This article belongs to the Special Issue Advances in Mathematics and Statistics with Applications in Engineering and Industry)

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18 pages, 1499 KiB

Open AccessArticle

Multivariate Pattern Recognition in MSPC Using Bayesian Inference

by Jose Ruiz-Tamayo, Jose Antonio Vazquez-Lopez, Edgar Augusto Ruelas-Santoyo, Aidee Hernandez-Lopez, Ismael Lopez-Juarez and Armando Javier Rios-Lira

Mathematics 2021, 9(4), 306; https://doi.org/10.3390/math9040306 - 4 Feb 2021

Cited by 4 | Viewed by 2741

Abstract

Multivariate Statistical Process Control (MSPC) seeks to monitor several quality characteristics simultaneously. However, it has limitations derived from its inability to identify the source of special variation in the process. In this research, a proposed model that does not have this limitation is [...] Read more.

Multivariate Statistical Process Control (MSPC) seeks to monitor several quality characteristics simultaneously. However, it has limitations derived from its inability to identify the source of special variation in the process. In this research, a proposed model that does not have this limitation is presented. In this paper, data from two scenarios were used: (A) data created by simulation and (B) random variable data obtained from the analysed product, which in this case corresponds to cheese production slicing process in the dairy industry. The model includes a dimensional reduction procedure based on the centrality and data dispersion. The goal is to recognise a multivariate pattern from the conjunction of univariate variables with variation patterns so that the model indicates the univariate patterns from the multivariate pattern. The model consists of two stages. The first stage is concerned with the identification process and uses Moving Windows (MWs) for data segmentation and pattern analysis. The second stage uses Bayesian Inference techniques such as conditional probabilities and Bayesian Networks. By using these techniques, the univariate variable that contributed to the pattern found in the multivariate variable is obtained. Furthermore, the model evaluates the probability of the patterns of the individual variables generating a specific pattern in the multivariate variable. This probability is interpreted as a signal of the performance of the process that allows to identify in the process a multivariate out-of-control state and the univariate variable that causes the failure. The efficiency results of the proposed model compared favourably with respect to the results obtained using the Hotelling’s

T^{2}

chart, which validates our model. Full article

(This article belongs to the Special Issue Advances in Mathematics and Statistics with Applications in Engineering and Industry)

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