Archimedean Copulas Research Articles

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.

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.

Extremal Dependence Modelling of Global Horizontal Irradiance with Temperature and Humidity: An Application Using South African Data

The interaction between global horizontal irradiance (GHI) and temperature helps determine the maximum amount of solar power generated. As temperature increases, GHI increases up to the point that it increases at a decreasing rate and then decreases. Therefore, system operators need to know the maximum possible solar power which can be generated. Using the multivariate adaptive regression splines, extreme value theory and copula models, the present paper seeks to determine the maximum temperature that will result in the generation of the maximum GHI ceteris paribus. The paper also discusses extremal dependence modelling of GHI with temperature and relative humidity (RH) at one radiometric station using South African data from 16 November 2015 to 16 November 2021. Empirical results show that the marginal increases of GHI converge to 0.12 W/m2 when temperature converges to 44.26 °C and the marginal increases of GHI converge to −0.1 W/m2 when RH converges to 103.26%. Conditioning on GHI, the study found that temperature and RH variables have a negative extremal dependence on large values of GHI. Due to the nonlinearity and different structure of the dependence on GHI against temperature and RH, unlike previous literature, we use three Archimedean copula functions: Clayton, Frank and Gumbel, to model the dependence structure. The modelling approach discussed in this paper could be useful to system operators in power utilities who must optimally integrate highly intermittent renewable energies on the grid.

Semiparametric spatial frailty modeling for survival data based on copulas

In this study, we describe a frailty model for spatially correlated survival data based on the class of Archimedean copulas in the spatial regression, assuming that counties are allowed to have varying sizes of spatial structure. In the spatial frailty model, a Bayesian nonparametric prior for a baseline survival function is provided to improve model flexibility. The proposed approach was evaluated by analyzing survival data for acute myeloid leukemia in the North West of England.

Identification and Risk Characteristics of Agricultural Drought Disaster Events Based on the Copula Function in Northeast China

Accurate feature identification of drought disaster events is required for proper risk management in agriculture. This study improved the crop water deficit index (CWDI) by including the daily meteorological, crop development stage, soil moisture content, and yield data for 1981–2020 in northeastern China. Two drought characteristic variables (drought duration and intensity) were extracted using the theory of runs to produce the improved crop water deficit index (CWDIwp). Thresholds for the bivariate indicators were also determined for agricultural drought events of varying severity. A joint distribution model for drought variables was constructed based on five types of Archimedean copulas. The joint probability and the joint recurrence period for agricultural drought events were analyzed for drought events with varying intensities in northeast China. The results suggest that the CWDIwp can reliably identify the onset, duration, and intensity of drought events over the study area and can be used to monitor agricultural drought events. The conditional probability of drought intensity (duration) decreased as the drought duration (intensity) threshold increased, whereas the drought recurrence period increased as the threshold for drought duration and intensity rose. In the period (1981–2020), drought intensity in the three Northeastern provinces showed an increasing trend in the order Jilin Province > Liaoning Province > Heilongjiang Province. The spatial distribution of the joint probability and the joint recurrence period was obvious, and the joint probability showed a decreasing distribution trend from west to east. The distribution trend for the joint probability was opposite to that of the joint recurrence period. Furthermore, the areas with high drought probability values corresponded to the areas with low values for the recurrence period, indicating that the drought occurrence probability was higher, and the recurrence period value was lower in the drought-prone areas. The high-risk drought areas (60–87%) were in western Liaoning and western Jilin, with a recurrence period of 1–3 years, whereas the low-risk areas (<40%) were located in the mountainous areas of eastern Liaoning and eastern Jilin. The joint probability and joint recurrence period for agricultural drought events of varying severity were quite different, with the probability following the order light drought > moderate drought > severe drought > extreme drought. The order for the recurrence period was light drought < moderate drought < severe drought < extreme drought. The results provide technical support for disaster prevention and mitigation in drought risk management.

Usual stochastic ordering of the sample maxima from dependent distribution‐free random variables

In this paper, we discuss stochastic comparison of the largest order statistics arising from two sets of dependent distribution‐free random variables with respect to multivariate chain majorization, where the dependency structure can be defined by Archimedean copulas. When a distribution‐free model with possibly two parameter vectors has its matrix of parameters changing to another matrix of parameters in a certain mathematical sense, we obtain the first sample maxima is larger than the second sample maxima with respect to the usual stochastic order, based on certain conditions. Applications of our results for scale proportional reverse hazards model, exponentiated gamma distribution, Gompertz–Makeham distribution, and location‐scale model, are also given. Meanwhile, we provide two numerical examples to illustrate the results established here.

Modeling Bivariate Dependency in Insurance Data via Copula: A Brief Study

Copulas are a quite flexible and useful tool for modeling the dependence structure between two or more variables or components of bivariate and multivariate vectors, in particular, to predict losses in insurance and finance. In this article, we use the VineCopula package in R to study the dependence structure of some well-known real-life insurance data and identify the best bivariate copula in each case. Associated structural properties of these bivariate copulas are also discussed with a major focus on their tail dependence structure. This study shows that certain types of Archimedean copula with the heavy tail dependence property are a reasonable framework to start in terms modeling insurance claim data both in the bivariate as well as in the case of multivariate domains as appropriate.

Parametric Vine Copula Framework in the Trivariate Probability Analysis of Compound Flooding Events

The interaction between oceanographic, meteorological, and hydrological factors can result in an extreme flooding scenario in the low-lying coastal area, called compound flooding (CF) events. For instance, rainfall and storm surge (or high river discharge) can be driven by the same meteorological forcing mechanisms, tropical or extra-tropical cyclones, resulting in a CF phenomenon. The trivariate distributional framework can significantly explain compound events’ statistical behaviour reducing the associated high-impact flood risk. Resolving heterogenous dependency of the multidimensional CF events by incorporating traditional 3D symmetric or fully nested Archimedean copula is quite complex. The main challenge is to preserve all lower-level dependencies. An approach based on decomposing the full multivariate density into simple local building blocks via conditional independence called vine or pair-copulas is a much more comprehensive way of approximating the trivariate flood dependence structure. In this study, a parametric vine copula of a drawable (D-vine) structure is introduced in the trivariate modelling of flooding events with 46 years of observations of the west coast of Canada. This trivariate framework searches dependency by combining the joint impact of annual maximum 24-h rainfall and the highest storm surge and river discharge observed within the time ±1 day of the highest rainfall event. The D-vine structures are constructed in three alternative ways by permutation of the conditioning variables. The most appropriate D-vine structure is selected using the fitness test statistics and estimating trivariate joint and conditional joint return periods. The investigation confirms that the D-vine copula can effectively define the compound phenomenon’s dependency. The failure probability (FP) method is also adopted in assessing the trivariate hydrologic risk. It is observed that hydrologic events defined in the trivariate case produce higher FP than in the bivariate (or univariate) case. It is also concluded that hydrologic risk increases (i) with an increase in the service design life of the hydraulic facilities and (ii) with a decrease in return periods.

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.

Stochastic comparisons on extremes of burr type XII samples associated with Archimedean copula and heterogeneous shape parameters

This study deals with random variables from Burr Type XII samples having heterogeneous shape parameters and homogeneous frailty parameters. For samples whose survival copula (whose copula) is Archimedean copula, sufficient conditions based on the heterogeneity of shape parameter vector for the stochastic tail ordering and usual stochastic ordering between sample minimums (sample maximums) are provided. As for independent samples, the convex transform order and tail hazard rate order (tail reversed hazard rate order) between sample minimums (maximums) are discussed. Several numerical examples are also presented to illustrate the theoretical findings.

On reliability analysis in k-out-of-n systems under Archimedean copula dependence

In this paper, we investigate k-out-of-n system with n independent parallel subsystems comprising dependent components drawn from a heterogeneous population consisting of m different subpopulations. The components within each subpopulation are assumed to be dependent, while the subpopulations are independent of each other. We also assume that the subpopulations have different Archimedean copulas for their dependence. Under this setup, we discuss the k-out-of-n systems reliability by comparing two selection probabilities or two allocation policies in the sense of majorization order. In addition, we present some numerical examples to illustrate all the results established here. Finally, some concluding remarks are made.

Bivariate Survival Copula Analysis of Glaucoma Patients during Blindness: Glaucoma Cases at Alert Hospital in Addis Ababa City of Ethiopia

Glaucoma is a worldwide problem that causes vision loss and even blindness, with a prevalence rate ranging from 1.9% to 15%. In Ethiopia, glaucoma is the fifth cause of blindness. This study aimed to explore the dependence between blindness of the right and the left eyes of glaucoma patients and assess the effects of the covariates under the dependence structure. A retrospective cohort study. The study population included the glaucoma patients at Alert hospital from January 1, 2018, to December 30, 2021. The copula model was used to estimate the time to the blindness of the right and the left eyes of the glaucoma patients by specifying the dependence between the event times. Out of 537 glaucoma patients, 224 (41.71%) became blind at least in one eye during the follow-up period. The results of the Clayton copula model revealed that factors, such as age, residence, diabetes mellitus, stage of glaucoma, and hypertension are considered the most prognostic factors for blindness in glaucoma patients. The findings also revealed that there was a strong dependence between the time to the blindness of the right and the left eyes in the glaucoma patients (τ = 0.43). Based on the obtained results, high age, urban residence, hypertension, diabetes mellitus, and higher stage of glaucoma were factors associated with time to the blindness in the glaucoma patients. There was also a dependence between the right and the left eyes of the glaucoma patients. The results revealed that the Clayton Archimedean copula model was the best statistical model for accurate description of glaucoma patients' datasets.

CHARACTERISTICS OF JOINT DISTRIBUTION MODELS AND RETURN PERIOD OF EXTREME RAINFALL VOLUME AND INTENSITY

This article highlights a new approach to finding a copula model that most closely matches extreme rainfall characteristics in South Sulawesi, Indonesia. In climatology research, data that does not meet normal assumptions are often found due to extreme observations; therefore, a method is needed to overcome the dependency of variables that do not meet the normality assumption. The copula approach is a method that can determine the relationship between such variables. Copula can describe the dependency structure between variables with different margins and model its tail dependencies. This study discusses the application of Archimedean copula in modeling the dependency structure of both variables, namely the intensity of the 75th percentile (I75) and volume of the 75th percentile (P75). For explaining the dependencies of both variables, the best copula model was selected using the empirical copula method. The results showed that 16 stations followed the Clayton copula model, 23 stations followed the Gumbel copula model, and 14 stations followed the Frank copula model. After knowing the distribution model characteristics and the intensity and volume of extreme rainfall, the return period value on the intensity and volume of extreme rainfall was calculated. The return period value can be useful as information and evaluation valuable material for preparing natural hazard mitigation programs.

Portfolio credit risk with Archimedean copulas: asymptotic analysis and efficient simulation

Portfolio credit risk with Archimedean copulas: asymptotic analysis and efficient simulation

Co-occurrence probability of water balance elements in a mountain catchment on the example of the upper Nysa Kłodzka River

Conditions of the formation of key elements of the water balance, such as precipitation and runoff, and relations between them in the mountain catchment area are very complicated, conditioned both by the climatic factor and the physiographic characteristics of the catchment area. The aim of the study is to determine relations between precipitation and runoff in the Kłodzka Valley (KV) located in mountain areas of south-western Poland. Analyzes were based on precipitation in KV and discharges of the Nysa Kłodzka River and its tributaries, recorded in hydrological years 1974–2013. The bivariate Archimedean copulas were used to describe the degree of synchronicity between these variables. The study area shows a considerable variability in the conditions of transformation of precipitation into runoff. It is conditioned both by the pluvial regime and the physical-geographical characteristics of the catchment area. As a result, sub-catchments with diversified hydrological activity and their role in the formation of water resources of the entire KV were identified. Among them, the Biała Lądecka River sub-catchment was found to be the most hydrologically active, and the sub-catchment of Bystrzyca Dusznicka River the most inert, despite e.g. quite similar synchronicity of precipitation compared to the average precipitation in KV. At the same time, the KV rivers are characterized by different types of runoff regime and characteristic of the water balance structure. The methodology presented can be useful in determining dependencies between selected elements of the water balance and evaluation of water resources availability in source areas of mountain rivers.

Archimedean Copulas and Goodness of Fit Test

The main endeavor of this paper is to compare the result of parameter estimation for Archimedean copulas by using Kendal coefficient and Goodness of fit test. It is seen that on modeling dependency structure of data by the GOF method, at the same time, we are able to estimate parameters and also test the compatibility of copulas to data.

Analysis of Hydrologic Drought Frequency Using Multivariate Copulas in Shaying River Basin

Droughts, considered one of the most dangerous and costly water cycle expressions, always occurs over a certain region, lasting several weeks or months, and involving multiple variables. In this work, a multivariate approach was used for the statistical characterization of hydrological droughts in Shaying River Basin with data from 1959–2008. The standard runoff index (SRI) and the run theory were employed to defined hydrological drought character variables (duration, severity, and intensity peak). Then, a multivariate joint probability analysis with four symmetric and corresponding asymmetric Archimedean Copulas was presented; and the multivariate frequency analysis with the joint return periods (Tand and Tor) were estimated. The results showed that the hydrological droughts have a severity of 4.79 and 5.09, and the drought intensity peak is of 1.35 and 1.50 in Zhoukou station and Luohe station, respectively; the rank correlation coefficients τ are more than 0.5, which means multivariate copulas can effectively describe the joint frequency distributions among multivariate variables. Drought risk shows a spatial variation: the downstream observed at Zhoukou station is characterized by a higher multivariate drought risk. In general, multivariate copulas provide a reliable method when constructing a comprehensive drought index and evaluating multivariate drought characteristics. Thus, this paper can provide useful indications for the multi-dimensional droughts’ risks assessment in Shaying River Basin.

Ordering fail-safe systems having dependent components with Archimedean copula and exponentiated location-scale distributions

In this paper, we study stochastic comparisons of two fail-safe systems with dependent components having an Archimedean copula for the joint survival function and general exponentiated location-scale lifetime distributions. We first present sufficient conditions for the usual stochastic order between fail-safe systems when the components have different exponentiated location-scale distributions but with the same Archimedean copula. Next, the hazard rate ordering of fail-safe systems is discussed when the components in the two systems are independent and follow multiple-outlier exponentiated location-scale models. Several examples are then presented to illustrate the theoretical results established here.

A Mixture of Clayton, Gumbel, and Frank Copulas: A Complete Dependence Model

Knowledge of the dependence between random variables is necessary in the area of risk assessment and evaluation. Some of the existing Archimedean copulas, namely the Clayton and the Gumbel copulas, allow for higher correlations on the extreme left and right, respectively. In this study, we use the idea of convex combinations to build a hybrid Clayton–Gumbel–Frank copula that provides all dependence scenarios from existing Archimedean copulas. The corresponding density and conditional distribution functions of the derived models for two random variables, as well as an estimator for the proportion parameter associated with the proposed model, are also derived. The results show that the proposed model is able to show any case of dependence by providing coefficients for the upper tail and lower tail dependence.