Mathematical Models and Methods in Various Sciences

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

Deadline for manuscript submissions: closed (31 July 2023) | Viewed by 21559

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Department of Mathematics and Center of Mathematics, University of Beira Interior, 6200 Covilhã, Portugal
Interests: applied statistics; data science; probability theory; statistical inference; statistical modeling
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Symmetry describes the shape of a data distribution and is a central notion in statistics and data science. For example, when working with symmetric models, it is only necessary to model half of the model.

The aim of this Special Issue is to assemble innovative papers on theory, methodology, and applications of symmetric/asymmetric mathematical models and methods, applied to various areas of science.

We encourage submissions presenting original works with high scientific merit on statistical, computational, and mathematical approaches with an emphasis on behavioral science, biology, biomedical sciences, computer science, data analytics, economics and management, engineering, epidemiology, genomics and genetics, and medicine and social science.

Innovative applications and case studies in any scientific fields are welcome. Contributed papers covering, but not limited to, the following topics are expected:

  • Big data analysis
  • Biostatistics
  • Computational methods
  • Data examples
  • Economic statistics
  • Engineering statistics
  • Experimental design
  • Mathematical models
  • Mathematics and Statistics in Biology
  • Matrix Analysis
  • Multivariate Analysis
  • Simulation experiments
  • Survival analysis
  • Statistical Classification
  • Statistical Inference
  • Statistics in Medicine
  • Symmetrical/asymmetric Distributions
  • Symmetric/asymmetric models.

Dr. Dário Ferreira
Guest Editor

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

  • applied mathematics
  • applied statistics
  • data science
  • mathematical statistics
  • model theory
  • probability theory
  • statistical inference
  • statistical modeling

Published Papers (15 papers)

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Editorial

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3 pages, 190 KiB  
Editorial
Editorial Summary: Mathematical Models and Methods in Various Sciences
by Dário Ferreira
Symmetry 2023, 15(12), 2132; https://doi.org/10.3390/sym15122132 - 30 Nov 2023
Viewed by 565
Abstract
The Special Issue of Symmetry, “Mathematical Models and Methods in Various Sciences”, aimed to bring together innovative papers on the theory, methodology, and applications of symmetric/asymmetric mathematical models and methods in various areas of science [...] Full article
(This article belongs to the Special Issue Mathematical Models and Methods in Various Sciences)

Research

Jump to: Editorial

14 pages, 580 KiB  
Article
SIMEX Estimation of Partially Linear Multiplicative Regression Model with Mismeasured Covariates
by Wei Chen and Mingzhen Wan
Symmetry 2023, 15(10), 1833; https://doi.org/10.3390/sym15101833 - 27 Sep 2023
Cited by 1 | Viewed by 702
Abstract
In many practical applications, such as the studies of financial and biomedical data, the response variable usually is positive, and the commonly used criteria are based on absolute errors, which is not desirable. Rather, the relative errors are more of concern. We consider [...] Read more.
In many practical applications, such as the studies of financial and biomedical data, the response variable usually is positive, and the commonly used criteria are based on absolute errors, which is not desirable. Rather, the relative errors are more of concern. We consider statistical inference for a partially linear multiplicative regression model when covariates in the linear part are measured with error. The simulation–extrapolation (SIMEX) estimators of parameters of interest are proposed based on the least product relative error criterion and B-spline approximation, where two kinds of relative errors are both introduced and the symmetry emerges in the loss function. Extensive simulation studies are conducted and the results show that the proposed method can effectively eliminate the bias caused by the measurement errors. Under some mild conditions, the asymptotic normality of the proposed estimator is established. Finally, a real example is analyzed to illustrate the practical use of our proposed method. Full article
(This article belongs to the Special Issue Mathematical Models and Methods in Various Sciences)
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12 pages, 1444 KiB  
Article
Scaled-Invariant Extended Quasi-Lindley Model: Properties, Estimation, and Application
by Mohamed Kayid, Abdulrahman Abouammoh and Ghadah Alomani
Symmetry 2023, 15(9), 1780; https://doi.org/10.3390/sym15091780 - 18 Sep 2023
Cited by 1 | Viewed by 614
Abstract
In many research fields, statistical probability models are often used to analyze real-world data. However, data from many fields, such as the environment, economics, and health care, do not necessarily fit traditional models. New empirical models need to be developed to improve the [...] Read more.
In many research fields, statistical probability models are often used to analyze real-world data. However, data from many fields, such as the environment, economics, and health care, do not necessarily fit traditional models. New empirical models need to be developed to improve the fit. In this study, we investigated a further extension of the quasi-Lindley model. This extension was asymmetrically distributed on the positive real number line. Maximum likelihood, least square error, Anderson–Darling, and expectation maximization algorithms were used to estimate the parameters studied. All techniques provided accurate and reliable estimates of the parameters. However, the mean square error of the expectation-maximization approach was lower. The usefulness of the proposed model was demonstrated by analyzing a reliability data set, and the analysis showed that it outperformed all other alternative models. Full article
(This article belongs to the Special Issue Mathematical Models and Methods in Various Sciences)
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18 pages, 3616 KiB  
Article
Illustration of Convective Boundary Conditions on the Darcy–Forchheimer Flow of Nanofluid with the Impact of Chemical Reaction
by Priyashree Chandini Pattanaik, Swarnalata Jena, Satya Ranjan Mishra, Mansoor Alshehri and Nehad Ali Shah
Symmetry 2023, 15(9), 1728; https://doi.org/10.3390/sym15091728 - 8 Sep 2023
Cited by 8 | Viewed by 817
Abstract
The application of convective heat transport holds great significance in physiological studies, particularly in preventing the overheating of birds and mammals living in warm climates. This process involves the transfer of heated blood from the body’s core to the nearest blood vessels, effectively [...] Read more.
The application of convective heat transport holds great significance in physiological studies, particularly in preventing the overheating of birds and mammals living in warm climates. This process involves the transfer of heated blood from the body’s core to the nearest blood vessels, effectively dissipating the excess heat into the environment. As a result, analyzing convective boundary conditions becomes crucial for understanding heat and solutal profiles in the flow of a two-phase nanofluid model (Darcy–Forchheimer), which also takes into account heat sources and chemical reactions. This model encompasses the combined effects of Brownian and thermophoresis phenomena on flow behavior. The development of a three-dimensional model leads to a set of nonlinear ODEs, which can be tackled using appropriate similarity variables and traditional numerical techniques, i.e., the Runge–Kutta fourth-order combined with shooting technique is adopted to obtain the solutions. To ensure the model’s accuracy, physical parameters are carefully chosen within their appropriate ranges to reflect real-world behavior. This approach helps to capture the physical essence of the system under study. It is observed that the streamlines for the proposed stream function shows the flow pattern of the fluid particles within the domain for the variation of the kinematic viscosity and stream values, and enhanced Brownian motion controls the fluid concentration. Full article
(This article belongs to the Special Issue Mathematical Models and Methods in Various Sciences)
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31 pages, 8049 KiB  
Article
Two-Sided Beneficial Value-Added Service Investment and Pricing Strategies in Asymmetric/Symmetric Investment Scenarios
by Fengyue An and Gang Liu
Symmetry 2023, 15(6), 1246; https://doi.org/10.3390/sym15061246 - 12 Jun 2023
Cited by 2 | Viewed by 1495
Abstract
We explore media platforms’ investment strategies for two-sided beneficial value-added services, which can directly benefit both consumers and advertisers, and ad pricing strategies by using game theory. We consider an asymmetric investment scenario, scenario A, and a symmetric investment scenario, scenario S, and [...] Read more.
We explore media platforms’ investment strategies for two-sided beneficial value-added services, which can directly benefit both consumers and advertisers, and ad pricing strategies by using game theory. We consider an asymmetric investment scenario, scenario A, and a symmetric investment scenario, scenario S, and develop game models under each scenario. First, we obtain the equilibrium investments, prices and profits and analyze the influence of the three important parameters—marginal investment costs, positive consumer effects and negative advertising effects, on the equilibrium outcomes in each scenario. Then we compare these equilibrium outcomes between both scenarios. Finally, we conduct numerical simulations to verify the conclusions obtained in both scenarios. We show that in scenario A, the value-added service levels and ad prices of the investment platforms remain constant and then decrease with marginal investment costs. The ad prices and profits of the investment platforms increase (decrease) with positive consumer effects (negative advertising effects). The same change is true for the value-added service levels only under certain conditions. In scenario S, the value-added service levels of the investment platforms change with positive consumer effects or negative advertising effects only when marginal investment costs are high. The ad prices of the platforms always increase with positive consumer effects but increase with negative advertising effects only when marginal costs are low. The profits of the platforms vary monotonically with negative advertising effects, but not necessarily with positive consumer effects. Compared to scenario S, the ad prices of the investment platforms in scenario A are higher, but that is not always true for the value-added service levels. Full article
(This article belongs to the Special Issue Mathematical Models and Methods in Various Sciences)
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21 pages, 3071 KiB  
Article
Enhanced Dispersion Monitoring Structures Based on Modified Successive Sampling: Application to Fertilizer Production Process
by Mehvish Hyder, Syed Muhammad Muslim Raza, Tahir Mahmood and Nasir Abbas
Symmetry 2023, 15(5), 1108; https://doi.org/10.3390/sym15051108 - 18 May 2023
Cited by 3 | Viewed by 1203
Abstract
In this era of Industry 4.0, efficient and affordable monitoring solutions are needed for the surveillance of manufacturing/service operations. In general, memory-type control charts outperform memoryless control charts when it comes to determining the changes in location and dispersion parameters of symmetrically distributed [...] Read more.
In this era of Industry 4.0, efficient and affordable monitoring solutions are needed for the surveillance of manufacturing/service operations. In general, memory-type control charts outperform memoryless control charts when it comes to determining the changes in location and dispersion parameters of symmetrically distributed processes. Before monitoring the process location, it is essential to monitor the process dispersion, since the latter presumes that the process variance remains stable. In practice, the modified successive sampling (MSS) mechanism is preferred over simple random sampling for its cost-effectiveness and efficiency. This study was designed in order to propose moving average and double moving average control charts based on the MSS mechanism for monitoring the dispersion parameter. The performance of the proposed charts is evaluated using run-length measures, and a comparison is made with an existing control chart based on MSS and repetitive sampling. Furthermore, the application of the designed moving and double moving average charts is demonstrated using a case study related to fertilizer production. It is observed that the proposed double moving average control chart performs better than the other control charts designed under the MSS and repetitive sampling schemes. Full article
(This article belongs to the Special Issue Mathematical Models and Methods in Various Sciences)
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15 pages, 1458 KiB  
Article
Daily Semiparametric GARCH Model Estimation Using Intraday High-Frequency Data
by Fangrou Chai, Yuan Li, Xingfa Zhang and Zhongxiu Chen
Symmetry 2023, 15(4), 908; https://doi.org/10.3390/sym15040908 - 13 Apr 2023
Cited by 2 | Viewed by 1422
Abstract
The GARCH model is one of the most influential models for characterizing and predicting fluctuations in economic and financial studies. However, most traditional GARCH models commonly use daily frequency data to predict the return, correlation, and risk indicator of financial assets, without taking [...] Read more.
The GARCH model is one of the most influential models for characterizing and predicting fluctuations in economic and financial studies. However, most traditional GARCH models commonly use daily frequency data to predict the return, correlation, and risk indicator of financial assets, without taking data with other frequencies into account. Hence, financial market information may not be sufficiently applied to the estimation of GARCH-type models. To partially solve this problem, this paper introduces intraday high-frequency data to improve estimation of the volatility function of a semiparametric GARCH model. To achieve this objective, a semiparametric volatility proxy model was proposed, which includes both symmetric and asymmetric cases. Under mild conditions, the asymptotic normality of estimators was established. Furthermore, we also discuss the impact of different volatility proxies on estimation precision. Both the simulation and empirical results showed that estimation of the volatility function could be improved by the introduction of high-frequency data. Full article
(This article belongs to the Special Issue Mathematical Models and Methods in Various Sciences)
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28 pages, 2172 KiB  
Article
Survival Analysis and Applications of Weighted NH Parameters Using Progressively Censored Data
by Ahmed Elshahhat and Heba S. Mohammed
Symmetry 2023, 15(3), 735; https://doi.org/10.3390/sym15030735 - 15 Mar 2023
Cited by 1 | Viewed by 1028
Abstract
A new weighted Nadarajah–Haghighi (WNH) distribution, as an alternative competitor model to gamma, standard half-logistic, generalized-exponential, Weibull, and other distributions, is considered. This paper explores both maximum likelihood and Bayesian estimation approaches for estimating the parameters, reliability, and hazard rate functions of the [...] Read more.
A new weighted Nadarajah–Haghighi (WNH) distribution, as an alternative competitor model to gamma, standard half-logistic, generalized-exponential, Weibull, and other distributions, is considered. This paper explores both maximum likelihood and Bayesian estimation approaches for estimating the parameters, reliability, and hazard rate functions of the WNH distribution when the sample type is Type-II progressive censored order statistics. In the classical interval setup, both asymptotic and bootstrap intervals of each unknown parameter are constructed. Using independent gamma priors and symmetric squared-error loss, the Bayes estimators cannot be obtained theoretically. Thus, two approximation techniques, namely: Lindley and Markov-Chain Monte Carlo (MCMC) methods, are used. From MCMC variates, the Bayes credible and highest posterior density intervals of all unknown parameters are also created. Extensive Monte Carlo simulations are implemented to compare the performance of the proposed methodologies. Numerical evaluations showed that the estimates developed by the MCMC sampler performed better than the Lindley estimates, and both behaved significantly better than the frequentist estimates. To choose the optimal censoring scheme, several optimality criteria are considered. Three engineering applications, including vehicle fatalities, electronic devices, and electronic components data sets, are provided. These applications demonstrated how the proposed methodologies could be applied in real practice and showed that the proposed model provides a satisfactory fit compared to three new weighted models, namely: weighted exponential, weighted Gompertz, and new weighted Lindley distributions. Full article
(This article belongs to the Special Issue Mathematical Models and Methods in Various Sciences)
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16 pages, 845 KiB  
Article
A Gamma Process with Three Sources of Variability
by Luis Alberto Rodríguez-Picón, Luis Carlos Méndez-González, Iván Juan Carlos Pérez-Olguín and Jesús Israel Hernández-Hernández
Symmetry 2023, 15(1), 162; https://doi.org/10.3390/sym15010162 - 5 Jan 2023
Cited by 3 | Viewed by 1301
Abstract
Degradation modeling requires to consider the complexity of both the internal structure of highly reliable products and the environmental conditions, to define appropriate models to obtain estimations about the reliability and quality. These conditions reflect different sources of variability that need to be [...] Read more.
Degradation modeling requires to consider the complexity of both the internal structure of highly reliable products and the environmental conditions, to define appropriate models to obtain estimations about the reliability and quality. These conditions reflect different sources of variability that need to be considered in the aims of obtaining accurate estimations. Although several models have been proposed in the literature, few of them consider several simultaneous sources of variability. In this paper, we propose a model based on the gamma process that considers three sources of variability, specifically in the threshold, the initial level of degradation, and in the scale parameter of the gamma process. The model considers a convolution operation of the threshold and the initial level to then be characterized via numerical integration with the gamma process with random scale. The obtained results showed that the model can be used to model the degradation of products with these sources of variability, which means that it can used for case studies where both the initial level and threshold are inherently random and the randomness in the scale parameter can be proved. The performance is illustrated with a comprehensive simulation study and with the application in a case study. Full article
(This article belongs to the Special Issue Mathematical Models and Methods in Various Sciences)
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14 pages, 366 KiB  
Article
Queue-Size Distribution in a Discrete-Time Finite-Capacity Model with a Single Vacation Mechanism
by Wojciech M. Kempa
Symmetry 2022, 14(11), 2350; https://doi.org/10.3390/sym14112350 - 8 Nov 2022
Cited by 3 | Viewed by 1207
Abstract
In the paper a finite-capacity discrete-time queueing system with geometric interarrival times and generally distributed processing times is studied. Every time when the service station becomes idle it goes for a vacation of random duration that can be treated as a power-saving mechanism. [...] Read more.
In the paper a finite-capacity discrete-time queueing system with geometric interarrival times and generally distributed processing times is studied. Every time when the service station becomes idle it goes for a vacation of random duration that can be treated as a power-saving mechanism. Application of a single vacation policy is one way for the system to achieve symmetry in terms of system operating costs. A system of differential equations for the transient conditional queue-size distribution is established. The solution of the corresponding system written for double probability generating functions is found using the analytical method based on a linear algebraic approach. Moreover, the representation for the probability-generating function of the stationary queue-size distribution is obtained. Numerical study illustrating theoretical results is attached as well. Full article
(This article belongs to the Special Issue Mathematical Models and Methods in Various Sciences)
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50 pages, 759 KiB  
Article
Synergies vs. Clustering Only of Depressive Symptoms in Diabetes and Co-Occurring Conditions: Symmetric Indicators with Asymmetric, Bidirectional Influences in MIMIC Models
by Richard B. Francoeur
Symmetry 2022, 14(11), 2275; https://doi.org/10.3390/sym14112275 - 30 Oct 2022
Cited by 1 | Viewed by 2362
Abstract
I extend multiple indicators multiple causes (MIMIC) models to unveil unbiased, asymmetric, bidirectional influences using indicators of the same items within variable-defined subgroups. The strategy discerns (1) item-variation in interaction (and derivative) terms that capture synergies and cluster together (formative or causal indicators) [...] Read more.
I extend multiple indicators multiple causes (MIMIC) models to unveil unbiased, asymmetric, bidirectional influences using indicators of the same items within variable-defined subgroups. The strategy discerns (1) item-variation in interaction (and derivative) terms that capture synergies and cluster together (formative or causal indicators) from (2) item-variation in duplicate terms when items lack synergy and cluster together only (reflective or effect indicators). An item may reveal either or both influences. These symmetric indicators yield estimates of (1) the unique variation and synergy of each formative indicator within the structural model portion of the MIMIC model (based on moderated regression) and (2) the remaining shared variation in the reflective indicator within the measurement model portion (based on confirmatory factor analysis). I reveal two patterns of comorbidity in disease subgroups of a specific co-occurring condition across a community sample of older adults and in age and gender subsamples. First, as structural model indicators, depressive symptoms may display different synergies as they cluster within a disease subgroup of diabetes and a specific co-occurring condition. As measurement model indicators, depressive symptoms capture non-synergistic clustering within the disease subgroup. Second, diabetes may mediate the co-occurring condition when depressive symptoms lack synergies but cluster within the disease subgroup. Researchers should distinguish both comorbidity patterns, which have different implications. I offer insights for adaptive modeling, conceptualizing and screening symptom clusters, metabolomics, and economic or social monitoring. Full article
(This article belongs to the Special Issue Mathematical Models and Methods in Various Sciences)
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12 pages, 397 KiB  
Article
Bivariate Proportional Hazard Models: Structure and Inference
by Barry C. Arnold, Guillermo Martínez-Flórez and Héctor W. Gómez
Symmetry 2022, 14(10), 2073; https://doi.org/10.3390/sym14102073 - 6 Oct 2022
Cited by 2 | Viewed by 1047
Abstract
We focus on a variety of bivariate models with proportional hazard components. Models with proportional hazard marginals are described together with a selection of models with propor- tional hazard conditional distributions. The bivariate distributions with marginal proportional hazards distributions are shown to be [...] Read more.
We focus on a variety of bivariate models with proportional hazard components. Models with proportional hazard marginals are described together with a selection of models with propor- tional hazard conditional distributions. The bivariate distributions with marginal proportional hazards distributions are shown to be closely related to certain known bivariate exponential models. Two distinct kinds of conditional specification are investigated. Discussion is provided of cases with hazard function components that are (1) completely unknown, (2) known to belong to given parametric families and (3) completely known. Since the models are designed for use with survival data, it is inevitable that the marginal and conditional distributions will be asymmetric. However, logarithmic transformations in some cases will result in symmetric component distributions. Full article
(This article belongs to the Special Issue Mathematical Models and Methods in Various Sciences)
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22 pages, 313 KiB  
Article
Suitability of the Single Transferable Vote as a Replacement for Largest Remainder Proportional Representation
by Martynas Patašius
Symmetry 2022, 14(8), 1648; https://doi.org/10.3390/sym14081648 - 10 Aug 2022
Cited by 1 | Viewed by 1324
Abstract
There are two main approaches to achieving proportional representation in elections: the single transferable vote and methods based on party lists. This paper discusses ways to use the single transferable vote while using some of the main features used with the largest remainder [...] Read more.
There are two main approaches to achieving proportional representation in elections: the single transferable vote and methods based on party lists. This paper discusses ways to use the single transferable vote while using some of the main features used with the largest remainder method, such as the electoral threshold. The investigation has shown that the Weighted Inclusive Gregory method is a suitable replacement for the largest remainder method when it is desirable to avoid wasted votes and to handle independent candidates in a straightforward way, but it is also desirable to keep the results as close to the ones achieved under the largest remainder method as possible. The investigation also led to the development of an algorithm for using the single transferable vote when preference lists are based on party lists, exploiting commonalities and symmetries between the patterns of preferences given in the votes. It has been shown that such an algorithm makes the calculations faster than the use of ordinary implementations of the single transferable vote when the numbers of seats and candidates are high, as commonly happens when methods based on party lists are used. Full article
(This article belongs to the Special Issue Mathematical Models and Methods in Various Sciences)
11 pages, 317 KiB  
Article
A Boundedly Rational Decision-Making Model Based on Weakly Consistent Preference Relations
by Xinlin Wu and Haiyan Xiao
Symmetry 2022, 14(5), 918; https://doi.org/10.3390/sym14050918 - 30 Apr 2022
Cited by 2 | Viewed by 1719
Abstract
Completeness is one of the basic assumptions about the rational preference relation in classical decision theory. Strongly and weakly consistent preferences are presented by abandoning the completeness of the rational preference relation. Some expansion and contraction conditions are proposed and the relationships between [...] Read more.
Completeness is one of the basic assumptions about the rational preference relation in classical decision theory. Strongly and weakly consistent preferences are presented by abandoning the completeness of the rational preference relation. Some expansion and contraction conditions are proposed and the relationships between these conditions of rationality are discussed. The relationships between the conditions of rationality and boundedly rational choice behavior based on strongly and weakly consistent preferences are analyzed and discussed. Furthermore, an example about the choices of chocolates with interval ordinal numbers is given to explain some of the main conclusions in this paper. The results can be used as references for the study of boundedly rational decisions. Full article
(This article belongs to the Special Issue Mathematical Models and Methods in Various Sciences)
26 pages, 801 KiB  
Article
Reliability Analysis with Wiener-Transmuted Truncated Normal Degradation Model for Linear and Non-Negative Degradation Data
by Isyaku Muhammad, Xingang Wang, Changyou Li, Mingming Yan, Mustapha Mukhtar and Mustapha Muhammad
Symmetry 2022, 14(2), 353; https://doi.org/10.3390/sym14020353 - 10 Feb 2022
Cited by 1 | Viewed by 1507
Abstract
The degradation rates of many engineering systems are always positive in practical applications. In some cases, the degradation process is approximately linear, such as train wheel wear degradation and an increase in the operating current of laser devices. In this paper we introduce [...] Read more.
The degradation rates of many engineering systems are always positive in practical applications. In some cases, the degradation process is approximately linear, such as train wheel wear degradation and an increase in the operating current of laser devices. In this paper we introduce a degradation modeling and reliability estimation method based on the Wiener process with a transmuted-truncated normal distribution (TTND). The TTND is employed to characterize unit-to-unit variability due to its flexibility in capturing symmetry and the asymmetrical behavior of the drift variable of the Wiener process. A Wiener process with TTND is applied to model the degradation processes of the deteriorating systems in two cases. In the first case, we assume that the degradation process is unaffected by measurement uncertainties, whereas in the second case, the measurement uncertainties are considered. The exact and explicit expressions of PDFs and the reliability functions are derived based on the concept of first hitting time. The Gibbs sampling technique and the Metropolis–Hastings algorithm are employed to obtain the Bayesian estimates of the model’s parameters. The efficiency and the feasibility of the presented approach are demonstrated through numerical examples, with Monte Carlo simulation and a practical application with laser degradation data. Furthermore, deviance information criteria (DIC) are used to compare the proposed model and some existing models. The result indicated that the proposed approach provided better reliability estimation results. Full article
(This article belongs to the Special Issue Mathematical Models and Methods in Various Sciences)
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