Copula Families Research Articles

A Comparison of Information Criterion for Choosing Copula Models

The object of this paper is to analyse the ability of the Information Criterion in selecting the best copula model. For this study, we carry out a simulation exercise considering five one-parameter copula families: Normal, Student-t with ν degree freedom, Clayton, Gumbel and Frank. For each family copulas, three degrees of dependence and three size samples. The Information Criterion included in the comparison are AIC, BIC, HQIC, SIC. The results obtained are as follow; (i) we find that for a high dependence level (0.9) the reliability of the Information Criterion (IC) is quite good, but it reduces with the dependence level; (ii) the performance of the IC not only depends on the dependence degree but the size sample. In the case of considering negative dependence the reliability of the IC does not depend on the dependence degree but the size sample. As the size sample reduce the performed of the IC reduce. To last, in a comparison among the IC considered, we find that the BIC criterion is the most reliable follow by SIC. AIC and HQIC reaps similar results.

Coupling Bayesian Network and copula theory for water shortage assessment: A case study in source area of the South-to-North Water Division Project (SNWDP)

Updating the appropriate risk measures is vital for management practice in reservoir operation and offers new insights for decision-making. Water diversion may cause water shortage crises both in the source area and in the downstream flow. By focusing on the effect of water demand in the source area and reservoir operation, this study proposed a risk assessment framework with the Bayesian network and copula-based estimation of the parameters. The risk assessment framework comprised four parts: dependencies estimation, specifically established copula, a Bayesian network based on the copula-based estimation of the parameters, and scenario analysis. This framework was applied to the Danjiangkou Reservoir, part of the source area for the Middle Route Project of the South-to-North Water Division Project in China. The study results showed that the framework provided valuable decision-making information. The Clayton and Gumbel functions evidenced superior capabilities for simulating the dependence of hydrological variables using the copula family. Since it revealed the potential correlation of variables, the Bayesian network based on the copula-based estimation of the parameters was more robust and accurate compared with the classic Bayesian network. According to the results yielded in different scenarios, occurrences presenting the highest degree of disadvantage for the Danjiangkou Reservoir will be increased from 8.66% to 23.08% for upstream situations and 59.03% for downstream low-water situations. These results indicate the disadvantages and potential strategies for the successful operation of the reservoir, which will be helpful to both researchers and managers.

Evolution and copula modelling of drought duration and severity over Africa using CORDEX‐CORE regional climate models

AbstractThis study assessed the dependence of drought severity and duration across four subregions (Southern, Western, Eastern and Central) in Africa using copula modelling. The analysis was carried out for the reference period 1991–2020 and the future period 2071–2099. Simulated daily precipitation at a horizontal resolution of 0.22° were obtained from three regional climate models (RCMs) participating in the Coordinated Output for Regional Evaluations within the Coordinated Regional Downscaling Experiment (CORDEX‐CORE). The RCMs were downscaled by three global climate models and validated using three high‐resolution gridded daily precipitation products obtained from The Climate Hazards Group InfraRed Precipitation with Stations data (CHIRPS), Climate Prediction Center Africa Rainfall Climatology Version 2.0 (CPC‐ARC2) and Tropical Applications of Meteorology using SATellite data and ground‐based observations (TAMSAT). Comparison of precipitation from the RCMs with the three gridded reference observations shows relatively good performance across the different regions with median value within the range of 3–5 mm·day−1 in Central Africa, 1.5–4.5 mm·day−1 in Western Africa, 1.8–2.8 mm·day−1 in Eastern Africa and 0.5–2.2 mm·day−1 in Southern Africa. On the other hand, the correlation values of drought duration–severity from CORDEX‐CORE models in the different regions of Africa exhibit predominantly strong values greater than 0.8 for both historical and projected climate. The analysis also considered two families of copulas: Archimedean (Frank, Clayton, Gumbel) and Elliptical (Gaussian, Student's t). The performance of the copula functions were estimated using the Akaike information criteria (AIC). Generally, there is a good agreement in the distribution of observed precipitation among the three observational data products across the subregion of Africa, with slight differences attributed to the different processing algorithms of the products. Across West Africa, mean precipitation values from CPC, CHIRPS and TAMSAT was 2.86, 3.16 and 3.07 mm·day−1, respectively. The CCLM‐NCC underestimated the mean values of precipitation reported in the observation data, while CCLM‐MPI and CCLM‐HAD overestimated the mean values of precipitation in the West Africa region.

A multivariate frequency-severity framework for healthcare data breaches

Data breaches in healthcare have become a substantial concern in recent years and cause millions of dollars in financial losses each year. It is fundamental for government regulators, insurance companies, and stakeholders to understand the breach frequency and the number of affected individuals in each state, as these are directly related to the federal Health Insurance Portability and Accountability Act (HIPAA) and state data breach laws. However, an obstacle to studying data breaches in healthcare is the lack of suitable statistical approaches. We develop a novel multivariate frequency-severity framework to analyze breach frequency and the number of affected individuals at the state level. A mixed effects model is developed to model the square root transformed frequency, and the log-gamma distribution is proposed to capture the skewness and heavy tail exhibited by the distribution of numbers of affected individuals. We further discover a positive nonlinear dependence between the transformed frequency and the log-transformed numbers of affected individuals (i.e., severity). In particular, we propose to use a D-vine copula to capture the multivariate dependence among conditional severities, given frequencies due to its inherent temporal structure and rich bivariate copula families. The rejection sampling technique is developed to simulate the predictive distributions. Both the in-sample and out-of-sample studies show that the proposed multivariate frequency-severity model that accommodates nonlinear dependence has satisfactory fitting and prediction performances.

Copula-based projections of wind power: Ireland as a case study

Wind energy is a key element in the ongoing push to decarbonize the energy supply. The first step in the development of a wind farm at a specific site is to conduct feasibility studies including accurate long-term wind potential assessments and estimates of the annual energy production. However, as evidence of climate change becomes more apparent recently, concerns about the planning and utilization of wind resources in the face of these new conditions have increased. Accurate projections are needed to determine the frequency distribution of wind speeds in an area and, on this basis, estimate the energy production. The purpose of this study is to analyze the wind resource, to estimate its potential and to prepare zoning maps of wind energy production to determine the most suitable sites for wind farms in Ireland. For this purpose, wind data from ten Global Circulation Models and different climate-change scenarios were used during the historical and future period of 1981–2010 and 2021–2050, respectively. These data were evaluated in the study area and then a multi-criteria decision-making method was applied to choose the best representative climate models over the area. In order to post-process the outputs of the selected models, 17 statistical distributions and 26 Copula families were applied. Results showed that the average wind speed in the region during the historical period is expected to decrease in 2021–2050 by approximately 2–7% based on the climate scenarios. Additionally, wind power density maps were produced for the study area.

Copula-based bivariate drought severity and duration frequency analysis considering spatial–temporal variability in the Ceyhan Basin, Turkey

In recent times, copula families have been mostly employed in bivariate drought duration and severity. Archimedean copulas are mainly used conveniently among various potential families for hydrologic design and water resource management. In this study, drought events are defined by standard precipitation index (SPI3 time series) series for 24-gauge stations in Ceyhan Basin Turkey considering different elevation levels (high, average, and low). Both duration and severity are tested for their dependence with Mann–Kendall and Spearman Rho trend test at a 0.05 significant level. While Gamma and Weibull perform better than the other distributions at average elevation, Log-normal and Weibull distributions are found to fit the DD and DS series better for most stations located in low and high elevations, respectively. The best fit copula is found to construct a joint distribution function. The influence of upper tail dependencies is tested to select the most appropriate copula function. Gumbel and BB1 copulas are often regarded as the best copulas for the DD and DS series, respectively. After acquiring the best fit copula, the joint return periods are modeled for each station. Finally, considering drought risk categories (mild, moderate, severe, and extreme drought), spatial distributions of drought risk-return period are constructed. High risks are observed in the middle part of the basin based on TDS and T′DS under mild drought categories. Apart from using univariate frequency analysis, bivariate frequency analysis is performed to model the joint return period based on copula theory which is great of importance for hydrology design and water resource management.

An information ratio-based goodness-of-fit test for copula models on censored data.

Copula is a popular method for modeling the dependence among marginal distributions in multivariate censored data. As many copula models are available, it is essential to check if the chosen copula model fits the data well for analysis. Existing approaches to testing the fitness of copula models are mainly for complete or right-censored data. No formal goodness-of-fit (GOF) test exists for interval-censored or recurrent events data. We develop a general GOF test for copula-based survival models using the information ratio (IR) to address this research gap. It can be applied to any copula family with a parametric form, such as the frequently used Archimedean, Gaussian, and D-vine families. The test statistic is easy to calculate, and the test procedure is straightforward to implement. We establish the asymptotic properties of the test statistic. The simulation results show that the proposed test controls the type-I error well and achieves adequate power when the dependence strength is moderate to high. Finally, we apply our method to test various copula models in analyzing multiple real datasets. Our method consistently separates different copula models for all these datasets in terms of modelfitness.

Bivariate Copulas Based on Counter-Monotonic Shock Method

This paper explores the properties of a family of bivariate copulas based on a new approach using the counter-monotonic shock method. The resulting copula covers the full range of negative dependence induced by one parameter. Expressions for the copula and density are derived and many theoretical properties are examined thoroughly, including explicit expressions for prominent measures of dependence, namely Spearman’s rho, Kendall’s tau and Blomqvist’s beta. The convexity properties of this copula are presented, together with explicit expressions of the mixed moments. Estimation of the dependence parameter using the method of moments is considered, then a simulation study is carried out to evaluate the performance of the suggested estimator. Finally, an application of the proposed copula is illustrated by means of a real data set on air quality in New York City.

Total positivity of copulas from a Markov kernel perspective

The underlying dependence structure between two random variables can be described in manifold ways. This includes the examination of certain dependence properties such as lower tail decreasingness (LTD), stochastic increasingness (SI) or total positivity of order 2, the latter usually considered for a copula (TP2) or (if existent) its density (d-TP2). In the present paper we investigate total positivity of order 2 for a copula's Markov kernel (MK-TP2 for short), a positive dependence property that is stronger than TP2 and SI, weaker than d-TP2 but, unlike d-TP2, is not restricted to absolutely continuous copulas, making it presumably the strongest dependence property defined for any copula (including those with a singular part such as Marshall-Olkin copulas). We examine the MK-TP2 property for different copula families, among them the class of Archimedean copulas and the class of extreme value copulas. In particular we show that, within the class of Archimedean copulas, the dependence properties SI and MK-TP2 are equivalent.

Maximal asymmetry of bivariate copulas and consequences to measures of dependence

Abstract In this article, we focus on copulas underlying maximal non-exchangeable pairs ( X , Y ) \left(X,Y) of continuous random variables X , Y X,Y either in the sense of the uniform metric d ∞ {d}_{\infty } or the conditioning-based metrics D p {D}_{p} , and analyze their possible extent of dependence quantified by the recently introduced dependence measures ζ 1 {\zeta }_{1} and ξ \xi . Considering maximal d ∞ {d}_{\infty } -asymmetry we obtain ζ 1 ∈ 5 6 , 1 {\zeta }_{1}\in \left[\frac{5}{6},1\right] and ξ ∈ 2 3 , 1 \xi \in \left[\frac{2}{3},1\right] , and in the case of maximal D 1 {D}_{1} -asymmetry we obtain ζ 1 ∈ 3 4 , 1 {\zeta }_{1}\in \left[\frac{3}{4},1\right] and ξ ∈ 1 2 , 1 \xi \in \left(\frac{1}{2},1\right] , implying that maximal asymmetry implies a very high degree of dependence in both cases. Furthermore, we study various topological properties of the family of copulas with maximal D 1 {D}_{1} -asymmetry and derive some surprising properties for maximal D p {D}_{p} -asymmetric copulas.

Probabilistic Peak Demand Estimation Using Members of the Clayton Generalized Gamma Copula Family

Climate change impacts many aspects of life and requires innovative thinking on various issues. The electricity sector is affected in several ways, including changes in the production components and consumption patterns. One of the most important issues for Independent System Operators, a state-controlled organization responsible for ensuring the reliability, availability, and quality of electricity delivery in the country, is the response to climate change. This is reflected in the appropriate design of production units to cope with the increase in demand due to extreme heat and cold events and the development of models aimed at predicting the probability of such events. In our work, we address this challenge by proposing a novel probability model for peak demand as a function of wet temperature (henceforth simply temperature), which is a weighting of temperature and humidity. We study the relationship between peak demand and temperature using a new Archimedean copula family, shown to be effective for this purpose. This family, the Clayton generalized Gamma, is a multi-parameter copula function that comprises several members. Two new measures of fit, an economic measure and a conditional coverage measure, were introduced to select the most appropriate family member based on the empirical data of daily peak demand and minimum temperature in the winter. The Clayton Gamma copula showed the lowest cost measure and the best conditional coverage and was, therefore, proven to be the most appropriate member of the family.

Time to Kidneys Failure Modeling in the Patients at Adama Hospital Medical College: Application of Copula Model.

Kidney failure is a common public health problem around the world. The vast majority of kidney failure cases in Sub-Saharan African nations, including Ethiopia, go undetected and untreated, resulting in practically certain mortality cases. This study was aimed primarily to model the time to (right and left) kidneys failure in the patients at Adama Hospital Medical College using the copula model. A retrospective cohort study. The copula model was used to examine join time to the right and left kidneys failure in the patients by specifying the dependence between the failure times. We employed Weibull, Gompertz, and Log-logistic marginal baseline distributions with Clayton, Gumbel, and Joe Archimedean copula families. This research comprised a total of 431 patients, out of which, 170 (39.4%) of the total patients failed at least one kidney during the follow-up period. Factors such as sex, age, family history of kidney disease, diabetes mellitus, hypertension, and obesity were found to be the most predictive variables for kidney failure in the patients. There was a 41 percent correlation between the patients' time to the right and left kidneys failure. The patients' kidney failure risk factors included being a male, older adult, obese, hypertensive, diabetic and also having a family history of kidney disease. The dependence between the patient's time to the right and left kidneys failure was strong. The best statistical model for describing the kidney failure datasets was the log-logistic-Clayton Archimedean copula model.

A Copula-Based Simulation of Wave-Induced Pore Water Pressure Gradient and Local Acceleration Within Surf Zone for Natural and Laboratory Barred Beach Profiles

Wave-induced pressure gradients and local accelerations are important interconnected physical mechanisms involving several hydrodynamic and morphodynamic coastal phenomena. Therefore, to provide a reliable and realistic hydrodynamic and morphodynamic simulation, the dependencies among different parameters, such as water level, pressure gradient, local acceleration, and sediment concentration, should be considered. Herein, a copula-based simulation is presented for modeling multivariate parameters and maintaining their statistical characteristics within the surf zone. Archimedean and elliptical copula families are applied to investigate the dependency construction between the parameters in two case studies: one from a field site on the east coast of Japan, and another from a large-scale laboratory barred beach profile. The dependency between variables is evaluated using Kendall’s τ correlation coefficient. The water level, pressure gradient, and local acceleration are shown to be significantly correlated. The correlation coefficients between the variables for the natural beach are lower than the laboratory data. The marginal probabilistic distribution functions and their joint probabilities are estimated to simulate the variables using a copula approach. The performance of the simulations is evaluated via the goodness-of-fit test. The analysis shows that the laboratory data are comparable to the field measurements, implying that the laboratory simulation results can be applied universally to model multivariable joint distributions with similar hydrodynamic conditions.

On Some Mixing Properties of Copula-Based Markov Chains

This paper brings some insights of psi '-mixing, psi ^*-mixing and psi-mixing for copula-based Markov chains and the perturbations of their copulas. We provide new tools to check Markov chains for psi-mixing or psi '-mixing. We show that perturbations of psi '-mixing copula-based Markov chains are psi '-mixing while perturbations of psi-mixing Markov chains are not necessarily psi-mixing Markov chains, even when the perturbed copula generates psi-mixing. The Farlie–Gumbel–Morgenstern, gaussian and Ali-Mikhail-Haq copula families are considered among other examples. A statistical study is provided to emphasize the impact of perturbations on copula-based Markov chains in a simulation study. Moreover, we provide a correction to a statement made in Longla et al. (J Korean Stat Soc, 1–23, 2021) on psi-mixing.

Technical and allocative inefficiency in production systems: a vine copula approach

Abstract Modeling the error terms in stochastic frontier models of production systems requires multivariate distributions with certain characteristics. We argue that canonical vine copulas offer a natural way to model the pairwise dependence between the two main error types that arise in production systems with multiple inputs. We introduce a vine copula construction that permits dependence between the magnitude (but not the sign) of the errors. By using a recently proposed family of copulas, we show how to construct a simulated likelihood based on C-vines. We discuss issues that arise in the estimation of such models and outline why such models better reflect the dependencies that arise in practice. Monte Carlo simulations and a classic empirical application to electricity generation plants illustrate the utility of the proposed approach.

Formula omitted]-copula from the viewpoint of tail dependence matrices

The tail dependence coefficient is a bivariate measure of dependence in the tail, and the Tail Dependence Matrix (TDM) is a bidimensional array of these coefficients corresponding to a random vector. The TDM serves as a parsimonious measure of multivariate tail dependence akin to the correlation matrix in the context of dependence. The set of all TDMs corresponding to d-dimensional random vectors is a convex polytope with an intricate description known only for d up to six, with both its numbers of facets and vertices growing at least exponentially in d. We posit that the richness of its subset that a copula family can accommodate is a practically vital feature to be considered in its choice for modeling in the presence of tail dependence. In this paper, our focus is on the t-copula family, which is a popular choice for parametric modeling in risk management and financial econometrics. We discuss some geometric properties of the subset of TDMs supported by the t-copula family and provide an efficient algorithm to determine the t-copula that best captures the tail dependence specified by a target TDM.

Extreme generators of shock induced copulas

In a recent paper, extreme points (in the Krein-Milman sense) of the class of semilinear copulas were introduced, motivated by the lack of known extreme copulas such as shuffles of M. We propose an extension of this concept to the class of all bivariate shock induced copulas, the most well-known part of them being the Marshall-Olkin copulas. This class properly contains semilinear copulas. Our technique coincides with the existing notion on them and has some advantages. First, it is defined on a wider family of copulas, which is helpful in finding more examples of extreme copulas. Second, we show that they are dense in each class they belong to (including the class of semilinear copulas) in a stronger sense than in the Krein-Milman approach; actually, they are dense in a similar way as shuffles of M are dense in the set of all copulas. Third, this definition enables practitioners to give stochastic interpretation of extremality. Roughly speaking, a shock induced copula is extreme whenever the inducing shocks have pairwise disjoint supports.

Joint probability distribution of air temperature and global solar radiation for outdoor design conditions based on copula approach

In the common method of compiling outdoor design conditions by annual cumulative frequency of occurrence, the probability of simultaneous occurrence of the selected meteorological elements is not considered, which may lead to an overestimation of air-conditioning equipment capacity design. This paper aims at developing a novel approach for producing the outdoor design conditions, taking into consideration the dependency between the air temperature and global solar radiation, which have a strong influence on the HVAC capacity design. These two meteorological elements are analyzed for specific simultaneous occurrence probability based on the joint distribution. For this purpose, the copula approach which can describe the dependence between these two meteorological elements is used for modeling the joint distribution function. We investigated copula family selection for air temperature and global solar radiation at each hour of hourly weather data for 10 years from 2001 to 2010 in Osaka, Japan. The Joe-Frank copula has the best fitting results for the dependence between the air temperature and global solar radiation. According to the comparison of original exceeding probability and simultaneous occurrence probability, the results show that the design conditions based on the common method can significantly exceed that of the simultaneous occurrence probability method. Practical Application: Outdoor design conditions are used in the building design stage to estimate building performance and select the appropriate air conditioning equipment capacity. Excessive air-conditioning capacity may lead to a large amount of unnecessary energy wastage. Insufficient capacity may lead to not meeting the desired indoor environment conditions of utilization in extreme weather conditions, especially for large office buildings. Therefore, this paper provides a new approach to generate more realistic meteorological conditions for air-conditioning design to improve building energy efficiency.

Stochastic representation of FGM copulas using multivariate Bernoulli random variables

A one-to-one correspondence between Fréchet's class of multivariate Bernoulli distribution with symmetric marginals and the well-known family of Farlie-Gumbel-Morgenstern (FGM) copulas is established. A new stochastic representation of the family of d-variate FGM copulas is introduced. The representation is bijective: from any d-variate Bernoulli distribution, one may define a corresponding d-variate FGM copula; and for any d-variate FGM copula, one finds the corresponding d-variate Bernoulli distribution. The proposed stochastic representation has many advantages, notably establishing stochastic orders, constructing subclasses of FGM copulas and sampling. In particular, one may use the stochastic representation to develop computational methods to perform sampling from subclasses of FGM copulas, which scale well to large dimensions.

Analytical bias correction for two-step fixed effects models with copula-distributed errors

We derive an analytical bias correction for two-step fixed effects models with copula-distributed errors. We work out the approximate bias correction for the Gaussian copula and present a numerical computation exercise for three other copula families. The results are derived for both n and T large.