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One of the useful distributions in modeling mortality (or failure) data is the univariate Gompertz–Makeham distribution. To examine the relationship between the two variables, the extended bivariate Gompertz–Makeham distribution is introduced, and its properties are provided. Also, some reliability indices, including aging intensity and stress-strength reliability, are calculated for the proposed model. Here, a new copula function is constructed based on the extended bivariate Gompertz–Makeham distribution. Some of its features including dependency properties, such as dependence structure, some measures of dependence, and tail dependence, are studied. The estimation of the parameters of new copula is presented, and at the end, a simulation study and a performance analysis based on the real data are presented. So, by analyzing the mortality data due to COVID-19, the appropriateness of the proposed model is examined. [ABSTRACT FROM AUTHOR]

Many highly reliable products (systems) usually have multiple dependent degradation processes because of their complex structure. Therefore, it is important to investigate the multivariate degradation models along with the reliability assessment in the stochastic modeling. This article proposes a framework for bivariate degradation modeling based on hybrid stochastic processes for products with two performance characteristics (PCs), the dependence of which is captured by copula functions. Considering heterogeneities among product units, different random effects are introduced in marginal stochastic processes. Then two classes of reliability functions based on joint posterior distributions are derived. In addition, a simulation study indicates that the holistic Bayesian parameter estimation method is better than the two-step Bayesian parameter estimation method. Finally, this article concludes with a case application to demonstrate the effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]

In order to accurately calculate the reliability of mechanical components and systems with multiple correlated failure modes and to reduce the computational complexity of these calculations, the Copula function is used to represent related structures among failure modes. Based on a correlation analysis of the failure modes of parts of a system, a life distribution model of components is constructed using the Copula function. The type of Copula model was initially selected using a binary frequency histogram of the life empirical distribution between the two components. The unknown parameters in the Copula model were estimated using the maximum likelihood estimation method and the most suitable Copula model was determined by calculating the square Euclidean distance. The reliability of series, parallel, and series–parallel systems was analyzed based on the Copula function, where life was used as a variable to measure the correlation between components. Thus, a reliability model of a system with life correlations was established. Reliability calculation of a particular diesel crank and connecting rod mechanism was taken as a practical example to illustrate the feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

Copula functions, which are flexible tools for the derivation of joint probability distributions, have been widely and effectively used to deal with multi variable analysis problems in the field of hydrology and water resources. Particularly in recent decades, this method has received great attention from researchers and engineers because of its advantages in multivariate analysis where the marginals of the data are not normally distributed. This paper provides a state-of-the-art review of the last decade of copula functions in multivariate hydrological frequency analysis, coinci dence and combination analysis of hydrological events, hydrological stochastic simulation, and hydrological modeling and forecasting, etc. More specifically, application issues such as parameter estimation, variable asymmetry, fitting optimization, tail characteristics, multivariable return period selection, and sampling error are described. Finally, the key points and development direction of Copula functions in the field of hydrology and water resources are consid ered, which provides guidance for the application of this method in the future. [ABSTRACT FROM AUTHOR]

In recent years, the use of copulas has grown rapidly, especially in survival analysis. In this paper, we introduce a bivariate modified Weibull distribu-tion derived from the Farlie{Gumbel{Morgenstern (FGM), a copula function commonly used to model very weak linear dependences. Considering the presence of non censored data and censored data, an extensive simulation study was developed to check the performance of the maximum likelihood method in estimating the parameters of the proposed model. Maximum like-lihood and Bayesian approaches for the estimation of the model parameters are presented. In the Bayesian analysis, the posterior distributions of the pa-rameters are estimated using Markov chain Monte Carlo (MCMC) method-ology. An example, considering a real data set, is introduced to illustrate the proposed methodology. [ABSTRACT FROM AUTHOR]

In the literature, up to now, it is common that for Gumbel, Clayton and Frank calculated Kendall Distribution function K(u) and to the extent those applications have been made. Kendall distribution functions show stochastic orderings of random vectors. The aim of Kendall distribution function is selected suitable copula function for using data set. For dependence structures of the data set, we calculated Kendall Tau and Spearman Rho values which are nonparametric. Based on this method, parameters of copula are obtained. In this paper, we are made Kendall Distribution function which obtained with the help of generator function of Archimedean copula calculation for Ali Mikhail Haq and Joe and in relation with that simulation study. We used data set which generated dependent generalized pareto distribution (Gp(3,3,3)) for this study. For dependency among these variables, we used Archimedean copula. In connection with this, we define basic properties of copulas and nonparametric methods Kendall Tau, Spearman Rho are given. In this study, to explain the relationship among the variables, five Archimedean copula are selected; Gumbel, Clayton, Frank Joe and Ali Mikhail Haq. Afterwards, we are obtained nonparametric estimation of parameters of these copulas with the help of Kendall Tau. With Kendall distribution function values, we found the suitable Archimedean copula family for this data set. [ABSTRACT FROM AUTHOR]

• A statistical framework for identifying the J-C model parameters is proposed. • The proposed method acquires the efficiency of the J-C model parameter estimation. • Quantifying as a copula form to reflect correlations of the J-C model parameters. • The proposed method accurately estimates the inherent variability of the material. The Johnson-Cook (J-C) model, including a constitutive model and a failure criterion, is an empirical model that is widely used in various fields to implement large deformation and damage of materials. Experimental methods are selected to determine the J-C model parameters considering the strain rate and condition where the material is exposed. This study deals with a uniaxial tensile test performed to obtain the stress–strain response of a material under quasi-static strain rate conditions. However, a low coefficient of determination may appear in a linear regression that determines the strain rate sensitivity parameters. Furthermore, in terms of efficiency, it requires a large amount of test data to obtain a set of the J-C model parameters. The novelty of this study lies in the adoption of a probabilistic approach to identify the J-C model parameters, which resolves the limitations of the existing method while simultaneously improving efficiency and accuracy. Namely, the efficiency is secured since all the J-C model parameters derived from each uniaxial tensile test response are used to estimate the probability model without the dissipation of the test data. Moreover, the proposed method is also efficient since it is capable of sampling as many of the J-C model parameters as necessary from the estimated probability model utilizing D-vine. Lastly, the inherent variability of material properties is accurately modeled by estimating the joint probability distribution as copulas, which can reflect the correlation between the J-C model parameters. The implemented statistical strategies are divided into two stages: (1) estimating the marginal probability distribution through the maximum likelihood estimate (MLE) and (2) the joint probability distribution through the Bayesian method. The two numerical examples adopted to validate the proposed method demonstrate that the proposed method better matches the experimental data rather than the existing deterministic method or the probabilistic method whose correlation has not been considered. [Display omitted] [ABSTRACT FROM AUTHOR]

In this paper, we consider that the degradation of two performance characteristics of a product can be modelled by stochastic processes and jointly by copula functions, but different stochastic processes govern the behaviour of each performance characteristic (PC) degradation. Different heterogeneous and homogeneous models are presented considering copula functions and different combinations of the most used stochastic processes in degradation analysis as marginal distributions. This is an important aspect to consider because the behaviour of the degradation of each PC may be different in its nature. As the joint distributions of the proposed models result in complex distributions, the estimation of the parameters of interest is performed via MCMC. A simulation study is performed to compare heterogeneous and homogeneous models. In addition, the proposed models are implemented to crack propagation data of two terminals of an electronic device, and some insights are provided about the product reliability under heterogeneous models. [ABSTRACT FROM AUTHOR]

The regional bivariate modeling of drought characteristics using the copulas provides valuable information for water resources management and drought risk assessment. The regional frequency analysis (RFA) can specify the similar sites within a region using L-comoments approach. One of the important steps in the RFA is estimating regional parameters of the copula function. In the present study, an optimization-based method along with the adjusted charged system search are introduced and applied to estimate the regional parameters of the copula models. The capability of the proposed methodology is illustrated by copula functions on drought events. Three commonly used copulas containing Clayton, Frank and Gumbel are employed to derive the joint distribution of drought severity and duration. The result of the new method are compared to the method of moments and after applying several goodness-of-fit tests, the results indicate that the new method provides higher accuracy than the classic one. Furthermore, the results of the upper tail dependence coefficient indicate that the Gumbel copula is the best-fitted copula among the other ones for modeling drought characteristics. [ABSTRACT FROM AUTHOR]

In the traditional Stress-strength interference (SSI) model, stress and strength must satisfy the basic assumption of mutual independence. However, a complex dependence between stress and strength exists in practical engineering. To evaluate structural reliability under the case that stress and strength are dependent, a mixed copula function is introduced to a new dependent SSI model. This model can fully characterize the dependence between stress and strength. The residual square sum method and genetic algorithm are also used to estimate the unknown parameters of the model. Finally, the validity of the proposed model is demonstrated via a practical case. Results show that traditional SSI model ignoring the dependence between stress and strength more easily overestimates product reliability than the new dependent SSI model. [ABSTRACT FROM AUTHOR]

Multivariate beta regression models for jointly modelling two or more variables whose values belong in the (0,1) interval, such as indexes, rates or proportions, are proposed for making small area predictions. The multivariate model can help the estimation process by borrowing strength between units and obtaining more precise estimates, especially for small samples. Each response variable is assumed to have a beta distribution so the models could accommodate multivariate asymmetric data. Copula functions are used to construct the joint distribution of the dependent variables; all the marginal distributions are fixed as beta. A hierarchical beta regression model is additionally proposed with correlated random effects. We present an illustration of the proposed approach by estimating two indexes of educational attainment at school level in a Brazilian state. Our predictions are compared with separate univariate beta regressions. The inference process was conducted using a full Bayesian approach. [ABSTRACT FROM AUTHOR]

Modern engineering systems are generally composed of multicomponents and are characterized as multifunctional. Condition monitoring and health management of these systems often confronts the difficulty of degradation analysis with multiple performance characteristics. Degradation observations generally exhibit an s-dependent nature and sometimes experience incomplete measurements. These issues necessitate investigating multiple s-dependent degradations analysis with incomplete observations. In this paper, a new type of bivariate degradation model based on inverse Gaussian processes and copulas is proposed. A two-stage Bayesian method is introduced to implement parameter estimation for the bivariate degradation model by treating the degradation processes and copula function separately. Degradation inferences for missing observation points, and for future observation points are investigated. A simulation study is presented to study the effectiveness of the dependence modeling and degradation inference of the proposed method. For demonstration, a bivariate degradation analysis of positioning accuracy and output power of heavy machine tools subject to incomplete measurements is provided. [ABSTRACT FROM AUTHOR]

Summary Multisite stochastic simulation of streamflow sequences is needed for water resources planning and management. In this study, a new copula-based method is proposed for generating long-term multisite monthly and daily streamflow data. A multivariate copula, which is established based on bivariate copulas and conditional probability distributions, is employed to describe temporal dependences (single site) and spatial dependences (between sites). Monthly or daily streamflows at multiple sites are then generated by sampling from the conditional copula. Three tributaries of Colorado River and the upper Yangtze River are selected to evaluate the proposed methodology. Results show that the generated data at both higher and lower time scales can capture the distribution properties of the single site and preserve the spatial correlation of streamflows at different locations. The main advantage of the method is that the trivairate copula can be established using three bivariate copulas and the model parameters can be easily estimated using the Kendall tau rank correlation coefficient, which makes it possible to generate daily streamflow data. The method provides a new tool for multisite stochastic simulation. [ABSTRACT FROM AUTHOR]

Due to the growing importance in maintenance scheduling, the issue of residual life (RL) estimation for some high reliable products based on degradation data has been studied quite extensively. However, most of the existing work only deals with one-dimensional degradation data, which may not be realistic in some cases. Here, an adaptive method of RL estimation is developed based on two-dimensional degradation data. It is assumed that a product has two performance characteristics (PCs) and that the degradation of each PC over time is governed by a non-stationary gamma degradation process. From a practical consideration, it is further assumed that these two PCs are dependent and that their dependency can be characterized by a copula function. As the likelihood function in such a situation is complicated and computationally quite intensive, a two-stage method is used to estimate the unknown parameters of the model. Once new degradation information of the product being monitored becomes available, random effects are first updated by using the Bayesian method. Following that, the RL at current time is estimated accordingly. As the degradation data information accumulates, the RL can be re-estimated in an adaptive manner. Finally, a numerical example about fatigue cracks is presented in order to illustrate the proposed model and the developed inferential method. [ABSTRACT FROM AUTHOR]

Modern highly reliable products usually have complex structure and many functions. This means that they may have two or more performance characteristics. All the performance characteristics can reflect the product's performance degradation over time, and they may be independent or dependent. If the performance characteristics are independent, they can be modelled separately. But if they are not independent, it is very important to find the joint distribution function of the performance characteristics for estimating the reliability of the product as accurately as possible. Here, we suppose that a product has two performance characteristics and the degradation paths of these two performance characteristics can be governed by a Wiener process with a time-scale transformation, and that the dependency of the performance characteristics can be described by a copula function. The parameters of the two performance characteristics and the copula function can be estimated jointly. The model in such a situation is very complicated and analytically intractable and becomes cumbersome from a computational viewpoint. For this reason, the Bayesian Markov chain Monte Carlo method is developed for this problem that allows the maximum-likelihood estimates of the parameters to be determined in an efficient manner. For an illustration of the proposed model, a numerical example about fatigue cracks is presented. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, a novel method based on extreme learning machine (ELM) and Copula function is proposed to predict the damages to electricity transmission facilities during ice storms. The ELM is firstly trained based on the historical data of wind speed, freezing precipitation, temperature, as well as the distribution parameters of wind and ice loads. The ELM can then be employed to predict the distributions of the real-time wind and ice loads on electricity transmission facilities. Furthermore, the correlation between wind load and ice load is modeled with Copula functions. On the basis of ELM and Copula function, the joint probability distribution of wind and ice loads can be finally formulated and applied to predict the potential damages to electricity transmission facilities such as transmission lines and towers. The proposed method is tested with a real dataset to demonstrate its effectiveness. [Copyright &y& Elsevier]