A Simplied, General Approach to Simulating from Multivariate Copula Functions

This study investigates the sensitivity and uncertainty of hydrological droughts frequencies and severity in the Weihe Basin, China during 1960–2012, by using six commonly used univariate probability distributions and three Archimedean copulas to fit the marginal and joint distributions of drought characteristics. The Anderson-Darling method is used for testing the goodness-of-fit of the univariate model, and the Akaike information criterion (AIC) is applied to select the best distribution and copula functions. The results demonstrate that there is a very strong correlation between drought duration and drought severity in three stations. The drought return period varies depending on the selected marginal distributions and copula functions and, with an increase of the return period, the differences become larger. In addition, the estimated return periods (both co-occurrence and joint) from the best-fitted copulas are the closet to those from empirical distribution. Therefore, it is critical to select the appropriate marginal distribution and copula function to model the hydrological drought frequency and severity. The results of this study can not only help drought investigation to select a suitable probability distribution and copulas function, but are also useful for regional water resource management. However, a few limitations remain in this study, such as the assumption of stationary of runoff series.

A statistical analysis method for significant wave height and spectral peak frequency considering the random and time-varying effects based on copula function and Bayesian inference

Joint probability distribution of winds and waves from wave simulation of 20 years (1989-2008) in Bohai Bay

In this paper, a probabilistic static voltage stability assessment method is proposed to study the influence of correlated renewable energy and loads on static voltage stability of distribution network. Through non-parametric kernel density estimation, the probability distribution function of distributed renewable generation output power is obtained, and the joint probability distribution is established by Copula function. Based on the quasi-monte Carlo simulation, the samples of random input variables are gained, and then the distribution of load margin is calculated. The efficiency and accuracy of quasi-monte Carlo simulation is verified, and the impact of the correlation between distributed generation on the static voltage stability of distribution network is studied. Besides, the influence of distributed generation configuration on voltage stability is also discussed.

Contemporary financial risk management is significantly based on the analysis of time series of returns. One of the most significant errors frequently committed by analysts is the predominant use of normal distributions when it is clear that the returns are not normal. Copula models and models for non-normal multivariate distributions provide new tools to solve the problem because the obtained results are immediately applicable in portfolio management, option pricing and measuring risk without assuming normality. Therefore, both a theoretician and a practitioner are interested in multivariate models for returns and copula functions. The copula function models provide an effective and interesting technique of constructing multivariate distribution starting from marginal ones. Due to Sklar\'s result established in 1959, we can present any multivariate distribution with a help of corresponding marginal distributions and a selected copula function. In this work we present an application of copula function to construct multivariate conditional distributions of times series. In the last part of this paper dynamic models such as DCC-MVGARCH and conditional copula are analyzed. Moreover, we also present an application of bootstrap in the context of copula function. This work is appended by examples showing practical application of our work.

Urban lifeline Natech events are coupled systems composed of multiple risks and entities with complex dynamic transmission chains. Predicting risk evolution probabilities is the core task for achieving the safety management of urban lifeline Natech events. First, the risk evolution mechanism is analyzed, where urban lifeline Natech events exhibit spatial evolution characteristics, which involves dissecting the parallel and synergistic effects of risk evolution in spatial dimensions. Next, based on fitting marginal probability distribution functions for natural hazard and urban lifeline risk evolution, a Multi-dimensional Copula (MdC) function for the joint probability distribution of urban lifeline Natech event risk evolution is constructed. Building upon the MdC function, a Markov Chain Monte Carlo (MCMC) model for predicting risk evolution probabilities of urban lifeline Natech events is developed using the Metropolis–Hastings (M-H) algorithm and Gibbs sampling. Finally, taking the 2021 Zhengzhou ‘7·20’ catastrophic rainstorm as a case study, joint probability distribution functions for risk evolution under Rainfall-Wind speed scenarios are fitted for traffic, electric, communication, water supply, and drainage systems (including different risk transmission chains). Numerical simulations of joint probability distributions for risk evolution are conducted, and visualizations of joint probability predictions for risk evolution are generated.

In multivariate survival analysis, estimating the multivariate distribution functions and then measuring the association between survival times are of great interest. Copula functions, such as Archimedean Copulas, are commonly used to estimate the unknown bivariate distributions based on known marginal functions. In this paper the feasibility of using the idea of local dependence to identify the most efficient copula model, which is used to construct a bivariate Weibull distribution for bivariate Survival times, among some Archimedean copulas is explored. Furthermore, to evaluate the efficiency of the proposed procedure, a simulation study is implemented. It is shown that this approach is useful for practical situations and applicable for real datasets. Moreover, when the proposed procedure implemented on Diabetic Retinopathy Study (DRS) data, it is found that treated eyes have greater chance for non-blindness compared to untreated eyes.

Joint-life insurance pays a sum insured when the first death occurs. This insurance has a case based on the order of exit from the cohort, namely joint life and last survivor. The former means that one of the insured leaves the cohort, while the latter means the last member of the insured has left his or her cohort. For some reasons of simplicity, the insurance premium is usually calculated with the assumption that the husband and wife are mutually independent. However, this assumption is considered unrealistic. Couples are open to the same risks, hence explaining joint survival model should involve dependence structures between the distribution of spouse mortality. In line with this, to understand the dependence structure of multiple random variables, the approach used is Copula. In this context, Copula relates the marginal distribution function of these variables to the joint life distribution. One of the advantages from Copula is that the random variables do not have to come from the same distribution, hence Copula is considered good enough to explain the dependence of the mortality rate between husband and wife. This study aimed to develop a joint survival model for calculating joint life insurance premiums using the concept of Archimedean Copula to discover the minimum premium value by conducting the following steps: first, identifying the marginal distributions of mortality for genders using Indonesian Mortality Table IV (TMI/Tabel Mortalitas Indonesia IV); second, Archimedean copula function-based constructing survival models that captures the relationship between these variables; third, setting dependency parameter θ; fourth, calculating the joint life premium using Archimedean copula based survival modeled for each correlation dependency level; and carrying out optimization to find the minimum premium value. This can be achieved by formulating the problem as an optimization problem, considering an objective function that yields the lowest premium till satisfying the financial requirements of the insurance company.

‘NC‐Tinius’ soybean [ Glycine max (L.,) Merr.] (Reg. No. CV‐503, PI 664027) was cooperatively developed and released by North Carolina State University (NCSU) and the North Carolina Agricultural Research Service. It was first tested in North Carolina yield trials in 2003. NC‐Tinius is a determinate, maturity‐group‐V, conventional cultivar. It is a high‐yielding conventional line adapted to the northern and central regions of eastern North Carolina. It was derived from the cross of the low‐palmitate and low‐linolenate conventional line TN99‐76,077 to a high‐yielding conventional line V91‐3036. In the NCSU breeding trials (11 environments), NC‐Tinius averaged 3443 kg ha −1 , or 353 kg ha −1 more than ‘5601T’, under full‐season conditions. In eight average‐ to high‐yield environments in the North Carolina Official Variety Trials, NC‐Tinius produced significantly higher yields (4031 kg ha −1 ) than did the standard cultivar 5601T in narrow (19 cm) row spacing. Seed protein and oil contents were similar to those of 5601T. NC‐Tinius is resistant to Soybean mosaic virus (G1), frogeye leaf spot (caused by Cercospora sojina Hara), bacterial pustule [caused by Xanthomonas campestris pv. glycines (Nakano) Dye], and stem canker (caused by Diaporthe phaseolorum var. meridionalis ) and is moderately resistant to sudden death syndrome (caused by Fusarium solani f. sp. glycines ).

This article proposes the use of copula (copula function) for the purpose of two-dimensional analysis of the sums of precipitation as measured with a Hellman rain-gauge. The sums of precipitation are characterized by a two-dimensional random variable: the sum of uninterrupted sequence of rainfalls which were measured in Jelcz-Laskowice and the corresponding (coincident) sum of precipitation at the Botanical Garden in Wrocław. Several problems occur from the very start: debonding from time and lack of precipitation on one of stations. For the purpose of greater precision and correction it should be stated that in order to apply the two-dimensional copula functions we will use a random vector determining the sum of uninterrupted sequences of rainfalls at two simultaneous stations. In that way, this will not be a characteristics of the phenomenon, but rather the definition of two-dimensional random variable under analysis. Data for analysis has been derived from observational logs of the Institute of Meteorology and Water Management, branch in Wrocław. The results obtained in years 1980-2014 were subject to analysis. The aim of the work was to find the best two-dimensional probability distribution of a random variable (OpadJelcz, OpadOgród). The following were analysed from among the known copulas: the Archimedean copulas (the Gumbel copula, the Frank copula and the Clayton copula) and the Gaussian elliptical copula. The study of fitting of copulas to observed variables was carried out using the Spearmann's rank correlation coefficient and the best fitting was obtained for the Frank's copula.

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.

The paper in the first place reviews the information and the uncertainty measures of joint and marginal probability distributions of the sets and subsets of random events. Next it reminds on the relations of the unconditional and conditional entropy of joint and marginal distributions and their combinations. Then it elaborates the ways how these measures can be applied in the sea surface uncertainty evaluation, that is, how to gain the information from the wave properties visually observed and brought together in the global wave statistics. Furthermore, the examples offered demonstrate the computational procedures for the sea surface uncertainty evaluation and prospective usefulness of the information obtainable from the joint and marginal distributions of observed explicit and combined wave properties.

Estimation of distribution algorithms (EDAs) are a class of evolutionary optimisation algorithms based on probability distribution model. This article extends the basic EDAs for tackling multi-objective optimisation problems by incorporating Archimedean copulas for constructing probability distribution model, and using the concept of Pareto dominance. In the algorithm, the marginal distributions from the current population are firstly estimated by kernel estimation method and are used to estimate the parameter of the Archimedean copula function generator by using the maximum likelihood method. Afterwards, the multivariate Archimedean copula sample algorithm is used to generate current offsprings population by sampling the n-dimensional Laplace transform Archimedean copula. The population with the current population and current offsprings population is sorted based on non-domination, and the best individuals are selected to form the next population based on rank and the crowding distance. The proposed algorithm is tested to compare with NSGA-II, PAES and SPEA2 using a set of benchmark functions. Both convergence and diversity metrics are used to evaluate the performance of the algorithm. The experimental results show that the algorithm outperforms NSGA-II, PAES and SPEA2 in two metrics.

Groundwater is an essential water resource in the Yarkant River Basin Irrigation District, which is the largest oasis in Xinjiang, China. This study used a novel approach to analyze the relationship between groundwater depth and three driving factors by developing Copula Functions from monthly time series data collected from 16 monitoring wells. More specifically, marginal distribution and joint distribution functions were established, and the conditional probabilities for three data ranges were calculated using both two- and three-dimensional Copula Functions. The developed statistical models showed that groundwater exploitation, runoff, and surface water irrigation significantly affected groundwater depth. The most significant effect on water table declines was associated with groundwater exploitation and lagged 1-month behind the groundwater withdrawals. The amount of runoff and irrigation were both inversely related to water table depth due to groundwater recharge. Frank Copula Functions were found to produce the best fit to the joint distribution of the variables and were used herein, allowing for the establishment of a detailed probability distribution of the change in groundwater depth under the combined effect of multiple controlling factors. The results provided key insights into how irrigation and groundwater withdrawals can be adjusted to effectively manage groundwater resources.

Both Estimation of Distribution Algorithms (EDAs) and Copula Theory are hot topics in different research domains. The key of EDAs is modeling and sampling the probability distribution function which need much time in the available algorithms. Moreover, the modeled probability distribution function can not reflect the correct relationship between variables of the optimization target. Copula Theory provides a correlation between univariable marginal distribution functions and the joint probability distribution function. Therefore, Copula Theory could be used in EDAs. Because Archimedean copulas possess many nice properties, an EDA based on Archimedean copulas is presented in this paper. The experimental results show the effectiveness of the proposed algorithm.