Mixed Copula Research Articles

Joint probability distribution of coastal winds and waves using a log-transformed kernel density estimation and mixed copula approach

The planning, designing, and risk assessment of coastal structures heavily rely on realistic statistical analyses of coastal winds and waves (CWW). In this study, we propose a novel bivariate joint probabilistic approach to describe the complex dependence and model the joint probability distribution (JPD) for CWW. A log-transformed kernel density estimation (KDE) is applied to fit the univariate marginal cumulative distribution (MCD) of the CWW measured at ShiDao station. As a connectivity function between the univariate MCD and JPD, a mixed copula is developed using a linear combination of Gumbel, Clayton, and Frank copula. The analytical results indicate that the optimum MCD for the ocean parameters can be effectively obtained using the log-transform KDE when compared with the Kernel-Pareto and GEV. Compared with five commonly used single copulas (i.e., Gaussian, t, Gumbel, Clayton, and Frank copula), the mixed copula is more suitable for describing the bivariate complex dependency structure between the ocean parameters and can yield more realistic JPD estimations. The return period and conditional probability calculated by the mixed copula were discussed and compared with the best fit single copula, which concluded that the log-transformed KDE and mixed copula approach can improve the precision of statistical analyses for CWW.

Wind direction fluctuation analysis for wind turbines

Fluctuations are a key characteristic of the wind resource. It is important to quantitatively analyze wind direction fluctuation due to its influence on the optimization of wind turbine yaw control. Based on wind resource data available from SCADA systems, a method is proposed to describe wind direction fluctuations in terms of fluctuation amplitude A and fluctuation duration T. A Weibull distribution is employed to fit the marginal probability density of both these two measures of wind direction fluctuations, and a mixed Copula used to connect the marginal distributions, establishing the joint probability density function. This representation has been verified through comparison with the real operating SCADA data. A set of indicators are extracted from the probability distribution which can accurately quantify the local wind direction fluctuation characteristics of a wind turbine. These indicators can be helpful in the optimization of the yaw control system parameters, facilitating an improvement in the power generating performance of the wind turbine. • Wind direction fluctuations are studied with statistic method and SCADA data. • Weibull distribution and copula are used to model the wind direction fluctuations. • A set of indicators are extracted to quantify the wind direction fluctuations.

Based on M-Copula Reliability Analysis of Random Load Correlation

Load is one of the main causes of structural failure, and the correlation among loads would affect the evaluation results of structural performance. The purpose of this paper is to analyze the influence of the correlation among multiple loads on the structural reliability. In this paper, the nonparametric kernel density estimation (NKDE) method is used to estimate the probability density function (PDF) of related loads. In addition, the mixed copula (M-Copula) model is proposed, which combines Gumbel copula, Frank copula, Clayton copula, and weight coefficient, and the model parameters are fitted by MATLAB software to get the correlation of related loads. The reliability based on the related load combination is calculated according to the constructed model. After analyzing three numerical cases, the results show that the probability characteristics of NKDE estimation are very close to the actual conditions, and the reliability calculated by the M-Copula model is larger than those calculated by JCSS, Turkstra, and Gong methods. Using the M-Copula model for load correlation would avoid underestimating the reliability of the structure, which is conducive to structural economic development.

Asymmetric dependence structures between emission allowances and energy markets: new evidence from China's emissions trading scheme pilots.

This article investigates the dependence structure between China's emissions allowances and energy (coal, oil, and natural gas) markets using autoregressive conditional heteroskedasticity (GARCH) and mixed Copula functions. The dependence structures between China's eight emissions allowances and energy markets demonstrate significant regional heterogeneity; an increase (decrease) in coal, crude oil, and natural gas prices induces an increase (decrease) in eight regional emissions allowance prices and confirms regional divergence and market heterogeneity. The Beijing, Guangdong, Hubei, and Chongqing emissions allowances and coal markets demonstrate significant left-tail dependence structures, while the Shanghai, Tianjin, Shenzhen, and Fujian emissions allowances and coal markets demonstrate complex mixed dependence structures. Seven of China's emissions allowances and crude oil markets confirm the existence of complex asymmetric dependence structures, except for the symmetric dependence structure between the Fujian emissions allowances and crude oil markets. However, eight of China's emissions allowances and natural gas markets demonstrate complex asymmetric dependence structures.

An Efficient Prediction Framework for Multi-parametric Yield Based on Uncertainty of Performance-Relevant Structure

Due to the aggressive shrinking of integrated circuit (IC) fabrication technology, variability in design parameters leads to significant yield loss. Therefore, efficient parametric yield prediction has been a critical task for today’s ICs, especially when multiple performances are considered simultaneously. However, most of the previous yield prediction works have failed to take into account uncertainty of performance-relevant structures. Neglecting the uncertainty effects will tend to result in predictive accuracy loss. To avoid the issue, this paper proposes a multi-parametric yield prediction framework based on the uncertainty of performance-relevant structures. In the proposed framework, saddle point estimation and mixed copula function are applied to maintain the uncertainty-relevant structures among performance metrics and predict the multi-parametric yield accurately. First, the framework constructs a general statistical model for performance metrics to illustrate the uncertainty of performance-relevant structures. The general model is explicitly expressed in terms of underlying process, voltage, and temperature (PVT) parameters. Then, based on the general performance model and taking total leakage current and gate delay for example, their marginal distributions are estimated by assuming the PVT parameters as normal random variables. Finally, a mixed copula-based yield prediction method is suggested to solve the uncertainty-relevant structure problem. Experimental results demonstrate that the proposed yield prediction framework is capable of predicting multi-parametric yield under arbitrary performance-relevant structures. Compared to Monte Carlo (MC) analysis, the relative errors of yield prediction are less than 5%, which means the framework has good accuracy and efficiency.

Multidimensional portfolio risk measurement: a mixed copula approach

It is an increasingly challenging task to explore the risk measurement for multidimensional portfolios with nonlinear correlative assets. A risk measurement scheme based on the mixed copula theory is proposed in this paper, where the mixed copula is constructed by the linear combination of three single Archimedean copulas, embodying greater flexibility than single copula in connecting different types of marginal distributions. In the scenario, ARMA-EGARCH model with t innovation is employed to fit marginal distributions, and the parameter values of the mixed copulas are inferred by maximum likelihood estimation (MLE) method, and interior point algorithm is used to calculate the extreme values of the MLE, VaR and CVaR, corresponding to the optimal portfolio with the minimum risk. Finally, an empirical study on five international stock market indexes in Europe is performed to verify the feasibility and effectiveness of the scheme.

Multidimensional portfolio risk measurement: a mixed copula approach

It is an increasingly challenging task to explore the risk measurement for multidimensional portfolios with nonlinear correlative assets. A risk measurement scheme based on the mixed copula theory is proposed in this paper, where the mixed copula is constructed by the linear combination of three single Archimedean copulas, embodying greater flexibility than single copula in connecting different types of marginal distributions. In the scenario, ARMA-EGARCH model with t innovation is employed to fit marginal distributions, and the parameter values of the mixed copulas are inferred by maximum likelihood estimation (MLE) method, and interior point algorithm is used to calculate the extreme values of the MLE, VaR and CVaR, corresponding to the optimal portfolio with the minimum risk. Finally, an empirical study on five international stock market indexes in Europe is performed to verify the feasibility and effectiveness of the scheme.

Probabilistic optimal power flow of AC/DC system with multiport current flow controller

To evaluate the impact of the randomness and correlation of photovoltaic (PV) and load on AC/DC systems with a multiport current flow controller (M-CFC), this paper proposes a probabilistic optimal power flow calculation for AC/DC systems based on a nonparametric kernel density estimation. First, according to the M-CFC model, the DC power flow calculation method with M-CFC was deduced, and its influence on line loss was analyzed. Second, a nonparametric kernel density estimation with an adaptive bandwidth was used to accurately describe the probability distribution of the PV and load, and correlation samples of the PV and load were obtained by the mixed copula function. Then an optimization model that considers system loss and static security was established, and a fast nondominated sorting genetic algorithm based on the elite strategy (NSGA-II) was used to calculate the multi-objective probability optimal power flow of the AC/DC system. Finally, a case study was performed on the modified IEEE39 bus system using measured PV and load data. We verified the nonparametric kernel density estimation with an adaptive bandwidth can better adapt to random component uncertainty, and M-CFC can improve the static security of the system.

Reliability and sensitivity analysis for bearings considering the correlation of multiple failure modes by mixed Copula function

The correlation between contact fatigue failure and wear failure of rolling bearing is analyzed, and the reliability model of rolling bearing based on multi-correlation failure mode is established. Based on the improved linear fatigue cumulative damage theory and wear theory, the limit state equations of two failure modes are established, and the correlation of state functions of contact fatigue and wear failure modes is described by the mixed Copula function. The random sample data are generated by Monte Carlo method, and the unknown parameters are estimated by genetic algorithm based on the minimum deviation square criterion, then the dynamic reliability model of rolling bearing is established. The quadratic polynomial without cross-term is used as the implicit functional proxy model, then the Latin hypercube sampling method and least squares theory are employed to estimate parameters of mixed Copula function and finally, the reliability sensitivity analysis of multi-failure modes of bearing is presented by fourth-order moment estimation method. Taking a certain type of spherical roller bearing as an example, the bearing reliability analysis considering the correlation of failure modes is carried out. The results show that compared with the bearing reliability model based on the assumption of independent failure modes and the weakest link theory, the bearing reliability model based on multi-correlation failure is more consistent with the practical application.

Time Series Generation and Complex Correlation Assessment for Multiple Wind Farms

Analysis and calculation of power system often requires a large number of medium and long-term wind power output series as data foundation. However, for most wind farms that have not been built or put into operation for a long time, the current output data is limited and it is difficult to support related research. It is necessary to use a limited amount of measured data to generate lots of wind power output time series that are similar to the actual data. Considering the concentrated development mode of wind power, this paper proposes a time series generation method for multiple wind farms based on the Markov Chain and Monte Carlo (MCMC) method and combined with high-dimensianal Markov process. The mixed Copula function fitting model is used to describe the spatial correlation of wind farms accurately. The measured wind power ouput is used to verify the results. It shows that the proposed method can simulate the output characteristics of single wind farm while maintaining the spatial correlation among different wind farms with geographical proximity

Reliability analysis of stochastic structure with multi-failure modes based on mixed Copula

In view of that there exist complicated correlations among failure modes of the stochastic structure because of the same uncertain variables that are contained in different failure modes, many reliability analysis methods based on empirical probability model have been developed to calculate and assess the structural reliability. However, the empirical distributions of uncertain variables are not always available or not accurate. Thus, based on the available sample data of random variables of the stochastic structural system to be analyzed, this paper presents a new reliability analysis method in which the mixed Copula is constructed to describe the correlations among multi-failure modes. First, the Gumbel Copula and Clayton Copula, which have the upper and lower tail dependence separately, are utilized to construct a mixed Copula so as to describe the correlations among multi-failure modes. Then, the sample data of the random variables is substituted into the performance functions of failure modes to calculate the sequence values thereof, and the probability density function (PDF) and cumulative distribution function (CDF) of each performance function of the structure are estimated by using the nonparametric kernel density estimation (NKDE) technique. Subsequently, the Canonical Maximum Likelihood (CML) method is applied to estimate the correlation parameters in the mixed Copula. Finally, the main girder structure of an overhead traveling crane is investigated by using the proposed method, and the results indicate that the new reliability analysis method is effective and valid. Compared with traditional reliability analysis methods, the new method, in which the available sample data of random variables is directly used to calculate the values of performance functions so as to avoid the errors caused by empirical distributions, provides some important theories and engineering contributions for the reliability design analyses and safety assessments of actual engineering structures, especially these with complex correlations among multi-failure modes.

Multiple failure mode reliability modeling and analysis in failure crack propagation based on time-varying copula

The fatigue failure of crack structure is influenced by fracture failure and static strength failure in the process of its growth. And the correlation characteristics between them change with the crack growth, which brings difficulties to safety assessment. To solve this problem, we propose a reliability model of failure crack propagation (FCP) based on time-varying copula. Meanwhile, a dynamic parameter estimation is put forward to build the time-varying copula. First, a flexible mixed copula model was constructed, by means of linear weighted model, to describe the nonlinear dependence among failure modes. Then the time-varying correlation parameters of copula were obtained by maximum likelihood estimation and least square polynomial fitting. Therefore, the dynamic parameter estimation of copula was realized, and then we used the built time-varying copula model to solve the reliability of joint failure. Finally, a numerical example was provided to illustrate the feasibility and application of the built time-varying copula model.

Copula-based interpolation methods for air temperature data using collocated covariates

This paper introduces two copula-based interpolation methods to produce air temperature maps in a data-scarce area: a spatial copula interpolator including covariates, and a mixed copula interpolator. The methods allow a construction of the conditional distribution of air temperature given the collocated covariates. Our study compared the new methods with the spatial copula interpolator, the ordinary kriging predictor and the cokriging predictor. Daily mean air temperature was used from weather stations and ERA_Interim reanalysis weather data at 174 locations in the Qazvin Plain, Iran. Spatial copula interpolator including covariates resulted in more precise predictions as shown by leave-two-out cross-validation. Visual inspection of air temperature maps demonstrated that the new methods well represented spatial variability of air temperature at a 1 km spatial resolution. The results showed an improved performance of the new methods to describe both spatial variability and co-variability between variables. The methods are potentially useful for other sparsely and irregularly distributed weather data.

Construction and Application of Mixed Copula Model

The Copula theory has been widely used in many scientific fields. In this study, we discuss model selection for the Copula theory. We derive the mixed Copula model based on the Kendall rank correlation coefficient. This new method is applied to forecast the value at risk of the portfolio in the foreign exchange market. The results show that the proposed mixed Copula model proposed in this paper is better than the other, commonly used, Copula model.

Wind farm’s clustering equivalent method based on mixed-copula function considering wind speed correlation of complex terrain

Large-scale grid connected wind farms are the main development trend of wind power in China. In the planning and design stage of wind farms, a simplified equivalent model is often used in wind farm equivalents. It makes the wind farm simply equivalent to a single wind turbine, neglecting the change of wind speed correlation caused by the terrain effect. This makes a large error in estimating wind power. In order to solve this problem, a method based on wind speed correlation is proposed to cluster the wind farm of complex terrain. By using the mixed Copula function with tail characteristics, a wind field correlation model with complex terrain features is built. Based on that, EM function is used to identify the Copula function parameters, and the equivalent clustering method of wind farms is built. Finally, the simulation analysis shows that the complex terrain wind farm clustering equivalent method considering wind speed correlation is verified.

Reliability Modeling for Gear Door Lock System With Dependent Failures Based on Copula

Abstract The gear door lock system (GDLS) is a hydraulic and mechatronic system with high degree of complexity and uncertainty, making the performance assessment of the system especially intractable. We develop copula models to estimate the reliability of GDLS with dependent failure modes. Based on the working principle of the GDLS, kinematic and dynamic model with imperfect joints is built in which Latin hypercube sampling (LHS) and kernel smoothing density are utilized to obtain the marginal failure probabilities. Then, copula models are utilized to describe the dependence between the two function failure modes. Furthermore, to be more accurate, mixed copula models are developed. The squared Euclidean distance is adopted to estimate the parameters of the above reliability models. Finally, the Monte Carlo simulation is conducted to evaluate different reliability models.

Research on Risk and Early Warning System of Inter‐Basin Multi‐Reservoir Water Transfer‐Supply Operation with Consideration of Uncertain Factors

AbstractAccording to traditional risk assessment methods it is impossible to have a quantitative analysis of the nonlinear relationship between uncertain factors of inter‐basin reservoir regulation. So, fuzzy mathematics and a mixed Copula function are used in this paper to determine existing water supply indexes. The joint probability distribution of runoff forecasting and water demand is simulated and it also makes a quantitative analysis of future flows. The information entropy principle is applied to determine early risk warning index and signals, and emergency water shortage measures are established for an inter‐basin water transfer‐supply operation. Then the early warning decision system of reservoir optimization water supply dispatching is built up. A practical example computation shows that the information of water supply and water shortage can be described effectively with such an early risk warning system regarding donor and recipient reservoirs. The early risk warning method is applied for the first time, which simultaneously takes into account water inflow, water utilization and water shortage regarding donor and recipient reservoirs. Copyright © 2018 John Wiley & Sons, Ltd.

Research of Wind Power Correlation With Three Different Data Types Based on Mixed Copula

With the increasing integration of large-scale wind farms, the stochastic characteristics of wind speed and the coupling relationship of geographically distributed wind farms become ineligible. Correlation investigation of wind farms based on copula theory is able to lay good foundation for further optimization and schedule in power systems. In this paper, three data types, including wind speed, calculated wind power, and actual wind power, are applied to explore the correlation of two geographically close wind farms. After graphical analysis and comparison of the time series, the structural and compositional characteristics of correlation by different data types are investigated based on mixed copula. Moreover, a two-stage filtration method is proposed to evaluate different types of copulas. Finally, taking into account the practical conditions after careful examination of actual wind power dataset, the practical conditions are applied to the calculated wind power dataset. Correlation research based on the adjusted calculated wind power dataset is further explored and revealed that it is more practical and prospective to provide reference for further power system operation with high penetration of wind farms.

Performance of Pairs Trading on the S&P 500: Distance and Mixed Copula Models

We propose an alternative investment strategy for pairs trading using Archimedean copulas in order to cover a wider range of tail dependence patterns and apply it to the S&P 500 stocks from 1990 to 2015. Empirical results show that our mixed copula approach generates higher average and risk adjusted excess returns and lower drawdown risk than the traditional distance method under different weighting structures when we control for trading costs. Particularly, the mixed copula and distance methods show a mean annualized value-weighted excess returns after costs on committed capital of 3.68% and 2.30% for top 5 pairs, with annual Sharpe ratios of 0.58 and 0.28, respectively. In addition, the mixed copula method shows a higher probability of yielding positive returns than the distance approach in different scenarios. The high returns can only partially be explained by common sources of systematic risk. The proposed mixture copulas have been found to be a superior fitting model over different market states, especially during subperiods of a stronger joint tail dependence.

Dynamic return-volatility dependence and risk measure of CoVaR in the oil market: A time-varying mixed copula model

This study investigates the risk level in the oil market measured by Value-at-Risk (VaR) and conditional VaR (CoVaR), as well as the dynamic and asymmetric dependence between WTI returns and crude oil volatility index (OVX), by constructing six time-varying mixed copula models. Results show that mixed copula between t copula and the 270-degree rotated Clayton copula is the optimal fitting copula to measure dynamic dependence. The estimated time-varying Kendall coefficients indicate that WTI returns and OVX present negative dependence most of the time. There exists a structural change point of dependence between WTI returns and OVX changes on April 17, 2009, while the dependence characteristics within the subsamples are similar to that in the whole sample, indicating the rationality of our time-varying mixed copula models. Finally, the tests show significant risk spillover from OVX to WTI returns and also asymmetric effects for CoVaRs in response to different upside and downside extreme OVX movements.