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In this paper, sufficient conditions are presented for comparing the residual lifetimes and inactivity times of dependent random variables at different fixed times, with respect to hazard rate, mean residual life and likelihood ratio orders. The stochastic comparisons of residual lifetimes at different random times are discussed next. Several examples are presented to illustrate the established results. [ABSTRACT FROM AUTHOR]

The probabilistic dependence in a multi-variate random field consists of two parts: the spatial dependence of a random quantity at different positions and the probabilistic dependence between different random variables at the same position. The classical model cannot capture the possible non-Gaussian dependence or nonlinear dependence between different random variables. To this end, in this paper, an approach by synthesizing the spatial correlation function and the multi-variate copula function (SCFVCF) is proposed. In this model, the correlation function model is adopted to quantify the spatial dependence involved in the random field of the prime variable, and the copula function model is adopted to quantify the dependence configuration between subordinate variables. The properties of such multi-variate random fields are then studied. To generate samples of such multi-variate random fields, the spectral representation method is incorporated with the conditional sampling method. As an example, to illustrate the application of SCFVCF, the random field of the constitutive parameters of concrete is adopted. The results demonstrate that the proposed method can capture the spatial dependence of compressive strength, and at the same time the dependence configuration between different parameters is consistent with the test complete compressive stress-strain curves. [ABSTRACT FROM AUTHOR]

Both Reservoir Inflow (RI) and Irrigation Water Demand (IWD) depend mainly on the basin-wide climate. Hence, various unpredictable (stochastic) weather and climate have induced significant variation in the water supply and demand processes in recent years. Meanwhile, the spatial similarity of climate conditions was also required to consider the relevancy between water supply and demand. More importantly, the smaller the basin is, the more considerable the relevancy is. However, the traditional one-dimensional stochastic dynamic programming cannot concern the relevancy between the multiple variables of water resources. In this study, a novel multi-dimensional stochastic dynamic programming was developed to deal with the relevancy and stochasticity in the streamflow and demand for the reservoir operations. Firstly, the one-dimensional distribution was selected to characterize the stochasticity of individual RI and IWD processes. Secondly, a Copula function of several variables was constructed using the marginal distribution. Specifically, the relevancy involved two aspects, i.e., the RI or IWD between the adjacent time intervals, and between RI and IWD at the same time. The stochasticity was referred to the individual RI, IWD, as well as the two-dimensional RI and IWD processes. Finally, the Copula function was integrated into the stochastic dynamic programming for the optimal dispatching model of the reservoir. A case study was set as an annual regulating reservoir responsible for agricultural irrigation. The results indicate that: 1) The Generalized Extreme Value (GEV) distribution was performed better on the RI and IWD in months. 2) There was remarkable relevancy of RI or IWD between a few adjacent time intervals. Moreover, there was a strong significance in April, May, June, July, September, and October. 3) Various types of Copula function were selected to model the dependent structure of RI/IWD variables at the adjacent time intervals. A Frank Copula was also employed to describe the negative relationship of dependent structure between RI and IWD at the same time. 4) The reservoir operation model considering the relevancy between RI and IWD was performed better than not. Specifically, the case study showed that the total water shortage was 10.37 × 108 m³, when considering the relevancy and stochasticity of multiple variables, which was less than before (10.49×108 m³). Nevertheless, there was a little difference between the optimal total water shortages by the two multi-dimensional stochastic dynamic programmings. This trend was attributed to a little difference of state transition probability between with and without considering the relevancy of RI and IWD. Consequently, the newly-developed multi-dimensional stochastic dynamic programming can be widely expected to extend for the multi-dimensional stochastic optimal scheduling in fields, such as the multi-energy supplementary model considering the hydropower, wind, and light energy. Moreover, a better fitting can be suitable for the current relevancy and stochasticity in multi-dimensional processes, where the wind speed and solar radiation are stochastic variables in this case. [ABSTRACT FROM AUTHOR]

Managing failure dependence of complex systems with hybrid uncertainty is one of the hot problems in reliability assessment. Epistemic uncertainty is attributed to complex working environment, system structure, human factors, imperfect knowledge, etc. Probability‐box has powerful characteristics for uncertainty analysis and can be effectively adopted to represent epistemic uncertainty. However, arithmetic rules on probability‐box structures are mostly used among structures representing independent random variables. In most practical engineering applications, failure dependence is always introduced in system reliability analysis. Therefore, this paper proposes a developed Bayesian network combining copula method with probability‐box for system reliability assessment. There are four main steps involved in the reliability computation process: marginal distribution identification and estimation, copula function selection and parameter estimation, reliability analysis of components with correlations and Bayesian forward analysis. The benefits derived from the proposed approach are used to overcome the computational limitations of n‐dimensional integral operation, and the advantages of useful properties of copula function in reliability analysis of systems with correlations are adopted. To demonstrate the effectiveness of the developed Bayesian network, the proposed method is applied to a real large piston compressor. [ABSTRACT FROM AUTHOR]

We develop an empirical likelihood (EL) approach to test independence of two univariate random variables X and Y versus the alternative that X and Y are strictly positive quadrant dependent (PQD). Establishing this type of ordering between X and Y is of interest in many applications, including finance, insurance, engineering, and other areas. Adopting the framework in Einmahl and McKeague, we create a distribution-free test statistic that integrates a localized EL ratio test statistic with respect to the empirical joint distribution of X and Y. When compared to well-known existing tests and distance-based tests we develop by using copula functions, simulation results show the EL testing procedure performs well in a variety of scenarios when X and Y are strictly PQD. We use three datasets for illustration and provide an online R resource practitioners can use to implement the methods in this article. for this article are available online. [ABSTRACT FROM AUTHOR]

Currently, the increasing wind power penetration, with consequent randomness and variability, presents great challenges to power system planning and operation. Probabilistic power flow (PPF) has been developed to calculate the power flow under uncertain circumstances. However, the current wind power models are subject to specific probability distributions, limiting their accuracies in wider applications. Additionally, the cumulant method (CM)-based PPF, if nonlinear relationship is considered in, would face an impractically high computational complexity. To address these problems in modeling and cumulant calculation, this article proposes a novel generalized density/distribution fitting method (GDFM) combining with the Copula function to establish a joint probability model for wind power generation. A special impulse- mixed probability density (IMPD) integration method is also introduced to derive the input cumulants from the model. Finally, a fast cumulant method (FCM) is proposed to reduce the computational burden of output cumulant calculation while retaining a high accuracy in a nonlinear context. Case study on the IEEE-118 test system validates the effectiveness of the proposed methods, and a real application to a provincial power grid in China provides some useful power flow risk information for decision making. The whole FCM-based PPF scheme can be helpful for future power flow examination in power system planning and operation. [ABSTRACT FROM AUTHOR]

Mutual information (MI) can be viewed as a measure of multivariate association in a random vector. However, the estimation of MI is difficult since the estimation of the joint probability density function (PDF) of non-Gaussian distributed data is a hard problem. Copula function is an appropriate tool for estimating MI since the joint probability density function of random variables can be expressed as the product of the associated copula density function and marginal PDF's. With a little search, we find that the proposed copulas-based mutual information is much more accurate than conventional methods such as the joint histogram and Parzen window-based MI. In this paper, by using the copulas-based method, we compute MI for some family of bivariate distribution functions and study the relationship between Kendall's tau correlation and MI of bivariate distributions. Finally, using a real dataset, we illustrate the efficiency of this approach. [ABSTRACT FROM AUTHOR]

Operating high-speed trains imposes excessive vibrations to bridges raising concerns about their safety. In this context, it was shown that some conventional design methods such as those related to the running safety suffer from a vague scientific background questioning their reliability or optimality. Therefore, the current article is devoted to updating the conventional design methodology, using Reliability-Based Design Optimization (RBDO) to propose the minimum allowable mass and stiffness which assures satisfying the target reliability. These proposed minimum design values can conceptually replace the conventional partial safety factor-based design method for running safety without the need for dynamic analysis. If the mass and stiffness resulting from the control of other limit states meet the proposed minimum values, the desired target reliability for running safety will be assured. This is achieved by adaptively training Kriging meta-models as a surrogate for the computational models decoupling the RBDO problem. In this regard, a new stopping criteria is proposed using mis-classification ratio of the cross-validated model; which reduces the generalization error of the trained meta-model and consequently the estimated failure probability. Moreover, due to the dependence of the design variables, the Copula concept is used to refine the augmented space and reformulate the RBDO problem. • A novel and straightforward dynamic design method is proposed for railway bridges. • The Copula concept is adopted to reformulate RBDO problems with dependent variables. • A new stopping criterion is proposed for adaptive training of surrogate models. • An extensive data is collected for variables in dynamic behavior of railway bridges. • Proper PDFs are assigned for contributing random variables of railway bridges. [ABSTRACT FROM AUTHOR]

This paper introduces a weighted entropic copula from preliminary knowledge of dependence. Considering a copula with common distri- bution we formulate the weighted entropy dependence model (WMEC). We give an approximator for the copula function of this problem. Also, we discuss some asymptotical properties regarding the unknown param- eters of the model. [ABSTRACT FROM AUTHOR]

In the information geometry suggested by Amari (1985) and Amari et al. (1987), a parametric statistical model can be regarded as a differentiable manifold with the parameter space as a coordinate system. Note that the q -exponential distribution plays an important role in Tsallis statistics (see Tsallis, 2009), this paper investigates the geometry of the q -exponential distribution with dependent competing risks and accelerated life testing (ALT). A copula function based on the q -exponential function, which can be considered as the generalized Gumbel copula, is discussed to illustrate the structure of the dependent random variable. Employing two iterative algorithms, simulation results are given to compare the performance of estimations and levels of association under different hybrid progressively censoring schemes (HPCSs). [ABSTRACT FROM AUTHOR]

The scheme of type-II hybrid censoring is of great value in life-testing experiments. In the literature, type-II hybrid censored order statistics are assumed to arise from independent random variables. However, in real lifetime systems, it is quite common for the components to be dependent. In this paper, we study the properties of type-II hybrid censored order statistics in the case when the units are statistically dependent. Density, distribution and joint density functions of dependent type-II hybrid censored order statistics are derived under this set-up. For certain special cases, more explicit expressions are presented. Illustrative examples are also provided. [ABSTRACT FROM AUTHOR]

There is complicated correlations in mechanical system. By using the advantages of copula function to solve the related issues, this paper proposes the mechanical system reliability model based on copula function. And makes a detailed research for the serial and parallel mechanical system model and gets their reliability function respectively. Finally, the application research is carried out for serial mechanical system reliability model to prove its validity by example. Using Copula theory to make mechanical system reliability modeling and its expectation, studying the distribution of the random variables (marginal distribution) of the mechanical product' life and associated structure of variables separately, can reduce the difficulty of multivariate probabilistic modeling and analysis to make the modeling and analysis process more clearly. [ABSTRACT FROM AUTHOR]

Global sensitivity analysis (GSA) plays an important role in uncertainty analysis and quantification. Conventional GSA for structures requires tackling two main challenges: (1) the incomplete probability information of inputs and (2) the effects caused by the static/dynamic correlation of random variables or stochastic processes. In this paper, two kinds of novel copula-based methods for variance-based GSA are proposed to address these challenges. Based on the known samples, the proposed methods can choose the optimal copula function to construct the joint distribution of inputs, and compute the global sensitivity indices combined with Monte Carlo (MC) simulation. Time-variant copula function is used to generate the samples of time series which are both auto-correlated and cross-correlated, and the proposed methods are extended to develop time-variant GSA of dynamic structures with correlated random variables and stochastic processes. Four engineering examples are given to illustrate the good applicability and capability of the proposed methods for the dependent model functions under incomplete probability information. [ABSTRACT FROM AUTHOR]

The capability of a distributed renewable generator (DRG) in providing load low-voltage ride-through (LVRT) is examined. The harnessed renewable power, load demand, and the occurrences of low-voltage incidents are treated as random variables. The probability of successful load LVRT is assessed through the use of a copula function to quantify the stochastic dependency between the load and the renewable power. The analysis is then applied to the case of a series-connected photovoltaic DRG incorporated with capacitor energy storage, wherein the focus is to establish the analytical relationship between the probability of successful load LVRT and the rated power/energy capacities of the DRG capacitor. Determination of the optimal capacities of the DRG capacitor is achieved through maximization of the expected economic benefits obtained from the renewable energy harness and load LVRT minus the cost of the DRG capacitor. [ABSTRACT FROM AUTHOR]

In this paper the interest is in testing the null hypothesis of positive quadrant dependence (PQD) between two random variables. Such a testing problem is important since prior knowledge of PQD is a qualitative restriction that should be taken into account in further statistical analysis, for example, when choosing an appropriate copula function to model the dependence structure. The key methodology of the proposed testing procedures consists of evaluating a 'distance' between a nonparametric estimator of a copula and the independence copula, which serves as a reference case in the whole set of copulas having the PQD property. Choices of appropriate distances and nonparametric estimators of copula are discussed, and the proposed methods are compared with testing procedures based on bootstrap and multiplier techniques. The consistency of the testing procedures is established. In a simulation study the authors investigate the finite sample size and power performances of three types of test statistics, Kolmogorov-Smirnov, Cramér-von-Mises, and Anderson-Darling statistics, together with several nonparametric estimators of a copula, including recently developed kernel type estimators. Finally, they apply the testing procedures on some real data. The Canadian Journal of Statistics 38: 555-581; 2010 © 2010 Statistical Society of Canada [ABSTRACT FROM AUTHOR]

Abstract: Simultaneous recordings from groups of neurons request to improve models. To switch from single unit description to multivariate models describing the coding activity of two or more neurons, we propose to use the copula notion. This mathematical object catches the coupling properties and allows a mathematical description of the dependencies between two or more random variables. Its use is here illustrated by means of toy examples and further applications are discussed. [Copyright &y& Elsevier]

This paper presents a probabilistic analysis framework for the reliability evaluation of turbine disc considering the correlation of multi-failure modes. A system-level zone division method is first applied to decompose the whole structure into different serial zones. Due to the same input random variables, there is correlation between failure modes. Thus, a mathematical copula function method is introduced to quantify the correlation between failure modes after reliability calculation of separate zones, during which process, dependent random variables are transformed to independent ones using Nataf transformation method. Meanwhile, to guarantee the accuracy and efficiency of calculation, adaptive surrogate model based on local radial point interpolation method (LRPIM) is established in each zone. Two main failure modes, i.e., low cycle fatigue and creep-fatigue are considered during the reliability analysis on a turbine disc. The results reveal that the reliability of the turbine disc changes with the correlation between failure modes. Also, sensitivity analysis shows that rotating speed and maximum temperature are two dominant factors affecting the turbined disc's reliability. Finally, the comparisons among three methods including the proposed method, the zone-based method without considering correlation and Monte Carlo (MC) method based on physics of failure (POF) of correlation are conducted. It is demonstrated that the proposed method in this study is more efficient and accurate for evaluating structural reliability with multi-failure modes coupling. Moreover, the proposed method provides an available prospect for reliability-based design optimization of multiple failure structure, contributing to enhance reliability in mechanical design. [ABSTRACT FROM AUTHOR]