Copula Function Research Articles

Technical and environmental inefficiency measurement in agriculture using a flexible by‐production stochastic frontier model

AbstractIn light of the urgent need for farms to mitigate greenhouse gas emissions while maintaining economic viability, this paper analyses technical and environmental inefficiencies and their determinants based on a flexible multi‐equation by‐production stochastic frontier model, which accounts for the stochastic dependence between good and bad outputs via a copula function. An empirical application to Dutch dairy farms illustrates the distortions in the inefficiency scores and in the estimates of their determinants that occur when the dependence between good and bad outputs is ignored. The empirical results indicate a strong positive dependence between the good output (milk) and the bad output (methane emissions), which is particularly pronounced in the upper tail of the distribution. Notably, farms exhibit high efficiency in maximising their good output and minimising their bad output. Subsidies are negatively related to good output inefficiency but positively related to bad output inefficiency, while stock density exhibits a negative association with inefficiencies in both outputs. Disregarding output dependence leads to distortions in inefficiency estimates, particularly affecting the estimates for their determinants.

Parametric and nonparametric estimation stress strength model based on copula function under first-failure progressively unified hybrid censoring schemes

Parametric and nonparametric estimation stress strength model based on copula function under first-failure progressively unified hybrid censoring schemes

Drought disaster risk assessment and yield insurance pricing of maize: a case study of Liaoning Province, China

Starting from the ‘four-factors’ theory of natural disaster risk formation, the study first selects indices that represent hazard, vulnerability, exposure, and emergency response and recovery capability to construct the drought risk assessment index system. Then the natural disaster risk index method is employed to evaluate the risk of maize drought disasters in Liaoning. Finally, the pure premium rate for maize yield insurance is determined based on the risk assessment results and the Copula function. The results show that: (1) The hazard, vulnerability, and exposure index in northwestern Liaoning are high, while emergency response and recovery capability is relatively low. Therefore, northwestern Liaoning is identified as a high-risk area for drought risk. (2) There exists a negative correlation between the drought risk index and maize yield in most areas of Liaoning; that is, higher drought risk is associated with lower maize yield. (3) Under the same coverage level, pure rates are higher in the northwestern and eastern Liaoning and relatively lower in the southwestern and southeastern Liaoning. Under different coverage levels, higher coverage corresponds to increased pure rates. The results can offer valuable guidance and benchmarks for relevant departments in Liaoning Province to assess the impact of drought risk on agriculture.

A Copula Function–Monte Carlo Method-Based Assessment of the Risk of Agricultural Water Demand in Xinjiang, China

A Copula Function–Monte Carlo Method-Based Assessment of the Risk of Agricultural Water Demand in Xinjiang, China

Variation Characteristics of Actual Evapotranspiration and Uncertainty Analysis of Its Response to Local Climate Change in Arid Inland Region of China

Exploring the variation characteristics of actual evapotranspiration (ETa) and its response to climate change in the arid inland region of China is of great significance for strengthening regional water resources management and maintaining ecological environment security and stability. Taking the Dulan River Basin as the research area, based on the meteorological data from the Wulan Station and hydrological data from the Shanggaba Station from 1981 to 2020, the variation characteristics of ETa at the annual scale were analyzed. The ETa estimation model and joint distribution model of P and potential evapotranspiration (ET0) was constructed based on climate factors, and the uncertainty of ETa response to climate change was explored with the water balance method, multiple linear regression, marginal distribution function, Copula function, and Monte Carlo algorithm. The results showed that the multi-year mean value of ETa in the study area was 261.6 mm, and the interannual process showed an insignificant upward trend, and had no abrupt change during the period. There were two obvious main cycles, which were 19-year periodic changes on the 30-year time scale and 6-year periodic changes on the 9-year time scale. The ETa estimation model based on precipitation (P) and ET0 had good simulation accuracy. The optimal marginal distributions of P and ET0 were Pearson-III (P-III) distribution and Generalized Extreme Value (GEV) distribution, respectively. The Copula joint distribution probability density of P and ET0 was a symmetric saddle-shaped distribution. ETa showed an inverted ‘S’ distribution with the change in joint guarantee rate of P and ET0, ranging from 116.9 mm to 498.6 mm. ETa was an interval range under a certain joint guarantee rate. The research results can provide support for the assessment of ETa, and help to further understand the driving mechanism of climate change on ETa in the arid inland region of China.

The Impact of Drought on Vegetation at Basin Scale: A Case Study of the Wei River Basin, China

Droughts in the Weihe River Basin are occurring more frequently and are becoming more intense. These events negatively affect industrial production, economic development, and ecosystems. Studying how vegetation changes in response to them is of practical significance. We report temporal and spatial trends in vegetation cover, use a copula function to analyze relationships between drought and vegetation cover, and assess the probability of vegetation loss in different drought scenarios. A vegetation index trends upwards from north to south in this basin; from 2001 to 2017, vegetation cover also trends upward in most areas, although it decreases in areas with high vegetation cover. An escalated susceptibility to drought has been observed in the southern and eastern sectors, where proximity to the riverbank correlates with heightened drought sensitivity, particularly in zones of intensified vegetation density. The probability of vegetation loss at the same vegetation loss preset point gradually increases with increased drought severity. These results will facilitate the formulation of countermeasures to prevent and combat the effects of drought on vegetation and land management.

Risk Assessment of Urban Water and Energy Supply Using Copula Function: A Water–Energy Nexus Approach in an Arid City

Planning for the future of water and energy supply systems in urban areas requires a thorough assessment of associated risks. In this study, monthly water and energy demand data from 2011 to 2022 in an arid city was used to predict the corresponding demands from 2023 to 2032 using the seasonal auto-regressive integrated moving average (SARIMA) method. The aim is to estimate future water and energy supply risks both individually and jointly, using cumulative distribution functions (CDFs) derived from historical data. The main focus is to calculate the combined risk of water and energy, referred to as the water–energy nexus (WEN) risk. Based on the interdependent relationship between water and energy, the Copula function was utilized to model the bivariate distribution between these two variables. Pearson correlation analysis indicated a strong correlation between water and energy supplies. Among the distributions fitted to the data, the log-normal and gamma distributions were the best fit for water supply and energy supply systems, respectively, with the lowest Akaike information criterion (AIC) values. The Gumbel Copula, with a parameter of 1.66, was identified as the most suitable for modeling the joint distribution, yielding the lowest AIC value. The results indicate that the risks associated with energy supply, water supply, and their joint dependency could exceed 0.8% in the future, highlighting a potentially critical situation for the city. The trend analysis revealed that forecasted water and energy demands and their corresponding risks and the WEN risk are expected to have a significant upward trend in the future. Finally, local authorities need to explore alternative sources to supply water and energy in the future to address the ever-growing water and energy demands.

Two-stage pseudo maximum likelihood estimation of semiparametric copula-based regression models for semi-competing risks data.

We propose a two-stage estimation procedure for a copula-based model with semi-competing risks data, where the non-terminal event is subject to dependent censoring by the terminal event, and both events are subject to independent censoring. With a copula-based model, the marginal survival functions of individual event times are specified by semiparametric transformation models, and the dependence between the bivariate event times is specified by a parametric copula function. For the estimation procedure, in the first stage, the parameters associated with the marginal of the terminal event are estimated using only the corresponding observed outcomes, and in the second stage, the marginal parameters for the non-terminal event time and the copula parameter are estimated together via maximizing a pseudo-likelihood function based on the joint distribution of the bivariate event times. We derived the asymptotic properties of the proposed estimator and provided an analytic variance estimator for inference. Through simulation studies, we showed that our approach leads to consistent estimates with less computational cost and more robustness than the one-stage procedure developed in Chen YH (Lifetime Data Anal 18:36-57, 2012), where all parameters were estimated simultaneously. In addition, our approach demonstrates more desirable finite-sample performances over another existing two-stage estimation method proposed in Zhu H et al., (Commu Statistics-Theory Methods 51(22):7830-7845, 2021) . An R package PMLE4SCR is developed to implement our proposed method.

An efficient surrogate model for system reliability with multiple failure modes

AbstractThe failure modes of complex systems are often highly dependent. Aiming at the reliability analysis of engineering systems with multiple failure modes, an efficient surrogate model for system reliability with multiple failure modes is proposed, namely, the Copula Adaptive Kriging surrogate model (CAKSM). First, the initial Kriging model is established based on the initial samples. In order to select more effective added samples to update the Kriging model, a new learning function, namely, Copula Unites with Expected Improvement Function (CUEIF), is established. The function can make a reasonable sample selection by considering the failure correlation. Then, based on the optimal Copula function, the reliability data of each failure mode are coupled and analyzed. In order to adapt to various computing scales, regularization is introduced to improve the prediction speed of the surrogate model, and the probability density function of each performance function is obtained by introducing nonparametric kernel density estimation. Finally, the effectiveness of the CAKSM method and the learning function CUEIF is verified through four examples.

Introducing Copula Functions to Estimate the Reliability of Dependent Mechanical Systems

This paper addresses the challenge of assessing the reliability of complex mechanical systems where components are inherently correlated in their failure modes. Traditionally, the assumption of independence among these components has been employed, but it often fails to capture the real-world complexities. To overcome this limitation, copula functions are introduced as a robust methodology for modeling the dependent relationships between correlated variables within mechanical systems. This paper aims to demonstrate the utility of copulas in estimating system reliability while accounting for these dependencies. The results reveal that the Clayton copula emerges as the most suitable model for representing dependence in such systems. Importantly, the reliability estimates obtained through copula-based methods not only reflect the complex interdependencies accurately but also align with the principles of the boundary theory of reliability. This research underscores the potential of copula-based reliability estimation as a valuable alternative, offering a more comprehensive and precise assessment of reliability in complex mechanical systems and holding significant promise for practical engineering applications. This framework allows the consideration of dependence among the observed variables that is usually overlooked in engineering practice.

Probabilistic modeling of multivariate extreme non-Gaussian wind loads and its applications in envelope engineering

Probabilistic modeling for non-Gaussian wind load fields under extreme winds is crucial to the quantitative risk and reliability analysis of envelope structures. Based on the multi-point synchronous wind load data during typhoons, a probabilistic model is proposed to characterize multivariate non-Gaussian wind loads to improve the accuracy of the risk analysis considering data dimension reduction. In the proposed model, the Pareto distribution function is utilized to fit the marginal distribution, and the copula function is employed to construct the dependent structure of wind loads from different locations. In the marginal fitted results, the extreme wind load value in the edge region is about 0.10–0.15 higher than those from inner measurement points at quantile p = 95 %. In addition, the variation gradient from the different variable directions is approximately equal, and a sharp variation occurs at the point (0.10, 0.10) for bivariate joint distributions. Engineering applications given in this article are composed of three parts: calculating regional exceeding probability, estimating wind load vector under a specific regional exceeding probability, and determining the location of joint extreme wind loads using the joint gradient value. Finally, risk analysis of existing roof coverings is carried out, the edge corner region is weaker, and two engineering recommendations are given. For determining the regional design wind load, a comprehensive calculation framework considering the temporal dependence of different locations is established, and the reduction rates are in the range (5 %, 25 %).

Quantile and expectile copula-based hidden Markov regression models for the analysis of the cryptocurrency market

The role of cryptocurrencies within the financial systems has been expanding rapidly in recent years among investors and institutions. It is therefore crucial to investigate this phenomenon and develop statistical methods able to capture their interrelationships, the links with other global systems and, at the same time, the serial heterogeneity. Here we introduce hidden Markov regression models for jointly estimating quantiles and expectiles of cryptocurrency returns using regime-switching copulas. The proposed approach allows us to focus on extreme returns and describe their temporal evolution by introducing time-dependent coefficients, evolving according to a latent Markov chain. Moreover to model their time-varying dependence structure, we consider elliptical copula functions defined by state-specific parameters. Maximum likelihood estimates are obtained via an expectation-maximization algorithm. The empirical analysis investigates the relationship between daily returns of five cryptocurrencies and major world market indices.

Optimal configuration of dynamic VAR compensators considering uncertainty and correlation

AbstractThe transient voltage stability of modern power systems is challenged by the increasing uncertainty on source‐load dual sides. To cope with it, a robust optimal configuration method for dynamic VAR compensators (DVCs) is proposed. First, a series of power‐flow scenarios are generated with uncertainty and correlation considered, where nonparametric kernel density estimation is used to predict the marginal distribution of wind speeds and combined Copula function is employed to characterize correlations between nearby wind farms. Then, the variation‐of‐information indicator is introduced to assess the similarity among power‐flow (PF) scenarios, and representative PF (RPF) scenarios are screened with the help of Spectral clustering algorithm. Finally, locating and sizing of DVCs for RPF scenarios are achieved. By synthesizing the compensation schemes for RPF scenarios, the robust configuration scheme is suggested. Simulation studies in the modified New England 10‐machine 39‐bus system show the proposed method can ensure transient voltage security of the studied power system under source‐load uncertainty.

Joint frequency analysis of streamflow and sediment amount with copula functions in the Kızlırmak Basin, Turkey

Joint frequency analysis of streamflow and sediment amount with copula functions in the Kızlırmak Basin, Turkey

A method for configuring hybrid electrolyzers based on joint wind and photovoltaic power generation modeling using copula functions

Considering the specific wind and photovoltaic power characteristics of a certain region, this study investigates the optimal ratio of Alkaline Electrolysis Cells (AEL) to Proton Exchange Membrane (PEM) electrolyzers in a hybrid electrolysis system for hydrogen production. A flexible model for configuring the hybrid electrolysis system is proposed, based on a copula function for joint wind and solar power modeling. This model generates wind and photovoltaic power generation scenarios using the copula function, incorporating a selection mechanism to ensure that the output scenarios are more representative of the actual data characteristics of wind and photovoltaic power output. Consequently, considering both the fluctuation and amplitude, the wind and photovoltaic power data are decomposed using the Ensemble empirical mode decomposition method. The decomposed components are then allocated to the two types of electrolyzers. Furthermore, the optimal configuration of the hybrid electrolysis system is determined by minimizing the costs associated with wasted power, electricity purchases, and other expenses. Finally, a case study of a 100 MW wind farm and a 50 MW photovoltaic power station in Northwest China is presented, concluding that the optimal configuration ratio of AEL to PEM electrolyzers is 2:1. In a Matlab/Simulink platform, the performance metrics of the hybrid electrolysis system were validated. It was found that the hydrogen production rate of the hybrid electrolyzer is comparable to that of the PEM electrolyzer, but with a lower required cost. Additionally, the hydrogen production rate and volume of the optimal configuration for the hybrid electrolyzer determined by the model proposed in this paper are higher than those obtained through the optimization algorithm's optimal configuration.

Pricing Model Construction and Empirical Research on Catastrophe Bond Pricing for Three Major Crops in China

Agricultural disaster risks are significant for the sector's development, and existing risk management tools have limitations.This paper introduces a nationwide catastrophe bond for wheat, maize, and rice , offering broad coverage for China's agriculture. The paper utilizes marginal distribution and Copula function fitting to model the disaster losses of the three major crops from 1990 to 2022. It employs Monte Carlo simulation to generate a 10,000-year event set and rank correlation to align simulated data with real data. The bond price is then calculated using the Wang model, addressing the challenge of data scarcity and providing an innovative approach to catastrophe bond design. The study finds that bond prices increase with the coverage period and risk level, with interest-risk bonds offering higher returns and risks than principal-and-interest risk bonds. Catastrophe bonds present a more stable risk diversification method compared to reinsurance, both in terms of returns and potential losses.To further the development of agricultural catastrophe bonds, the paper recommends optimizing the issuance environment and establishing a robust market infrastructure.

Research on Credit Risk Contagion of Commercial Banks Based on Copula Model

Research on the credit risk contagion effect of commercial banks is a key issue in credit risk management of commercial banks. This paper constructs a measurement framework for the credit risk contagion effect of commercial banks based on the Copula function theory. On the basis of selecting the appropriate Copula function and measuring specific parameters, taking Bank of China and Industrial and Commercial Bank of China as examples, the tail correlation coefficient was used to measure the credit risk contagion effect of the two banks, and relevant conclusions were drawn.

Reliability analysis of an aircraft load mechanism considering multi-site damage dependency

Aircraft load mechanisms are the important actuators of the aircraft rudder control systems. By sharing the same operation circumstances and working conditions, the damage processes at multiple sites are dependent. In this paper, we firstly conduct a wear experiment and a stress relaxation experiment of an aircraft load mechanism. To consider the individual variability of the damage processes, the stochastic processes are adopted. Then, vine copula function is introduced to analyze the multivariate dependence of the damage processes. Furthermore, a time-varying dependent model is formed to model the dynamic damage interaction. The reliability of the aircraft load mechanism is assessed by using the Rosenblatt transformation and Monte Carlo method. The obtained results show that overlooking the dependency will lead to a conservative reliability result. Finally, the effect of the damage processes on the mechanism is analyzed, which provides effective guidance for the maintenance of the mechanism.

Detection of flood trends and drivers in the Taihu Basin, China

Study regionTaihu Basin, China Study focusFloods threaten humans, the environment, economic activity, and infrastructure. In this study, a new trend test and flood-frequency methods were adopted to detect extreme floods and their distributions based on flood-event identification. To fully understand the phased process of the influence of human activities on extreme hydrological processes, 12 copula functions were employed creatively in combined static and dynamic time-varying correlation aspects between extreme precipitation and floods. New hydrological insights for the regionAlthough both significant and insignificant increasing trends of the annual maximum water level in all three hydrological districts were examined, the periodic oscillations of all the stations were similar. Thus, it was significant to fully detect the periodical variation of floods. Extreme floods occurred mainly in the 1990s, as measured by frequency estimates. Generally, the nonstationary response relationship between heavy rain and an extreme water level was gradually strengthened; that is, a certain magnitude precipitation seemed to induce a greater-intensity flood event as time passed. Through the identification of historical flood events and the analysis of the rise and fall processes of floods, we found that the main reason for variation in the response relationship was the increase in the water level before the rising stage, rather than the water level rising in the Taihu Basin. Our study findings further existing knowledge on the regional flood-control design standard and can ensure the coexistence of humans and water systems in the future.

A bivariate dependent degradation model based on artificial neural network supported stochastic process and Copula function

AbstractIn order to use the high ability of the artificial neural network (ANN) in data fitting, this paper introduces an ANN in stochastic process to describe the mean function for degradation modeling. Due to the fact that the existing method cannot handle the bivariate dependent degradation conditions, a bivariate dependent degradation model based on Copula function and ANN‐supported stochastic processes is proposed. Considering the random effects caused by individual difference, it is assumed that the unknown parameters in the stochastic processes and Copula functions are randomly distributed. Based on the maximum likelihood and moment estimation methods, a related statistical inference method for ANN training and parameter estimation is developed to use the bivariate dependent degradation model. An actual fatigue crack dataset is used to demonstrate the validity of the proposed method. The obtained results show that the dependent relationship between two degradation indicators should not be neglected, and it can be efficiently handled by the proposed method. Furthermore, the proposed degradation model can provide reliability and degradation intervals with enough precision due to the fact that it considers the random effects caused by individual difference.