Archimedean Copulas Research Articles

Bivariate Frequency Analysis of Hydrological Drought Using Copula: A Case Study of Northern Iraq

In this research work, copula-based methodology is adopted to analyze the hydrological drought frequency. Standardized Runoff Index SRI was calculated using monthly-streamflow data for 50 years of two gauging stations in the northern region of Iraq. The drought duration and severity were extracted using run theory. Three Archimedean family and Gaussian copulas were used and compared to select the most appropriate copula model for bivariate frequency analysis of hydrological-drought characteristics. The dependence between drought duration and drought severity was estimated by Pearson's, Spearman's rho and Kendall's tau correlations. Various probability distributions were utilized to determine the best fit marginal distributions for drought characteristic variables based on the Kolmogorov-Smirnov and Chi-squared statistics. Uni-variate and joint return periods were estimated and compared. Generally, the results indicate that Archimedean copulas performed better than the Gaussian copulas. Exponential and Weibull distributions are the best fit for drought duration and severity, respectively, except for drought severity in case of the 9-month time scale at Lesser Zab region, where lognormal distribution was chosen. The current study can give helpful information for drought-risk assessment and water-resource management under climate change. KEYWORDS: Hydrological drought, Copula, SRI, Greater Zab river, Lesser Zab river, Bivariate return periods.

Snow Disaster Hazard Assessment on the Tibetan Plateau Based on Copula Function

In the context of global climate change, the Tibetan Plateau is particularly susceptible to meteorological disasters, including snow disasters. This study utilized daily temperature and precipitation data from 44 meteorological stations on the Tibetan Plateau spanning from 1960 to 2018 to construct a snow event dataset. Optimal marginal distribution and the copula function were chosen to calculate the joint return period and joint probability, which effectively assess the hazard of snow disasters in the region. Additionally, the study analyzed the comprehensive risk of snow disasters under various return periods by integrating social and economic data. The results indicate the following: (1) Based on the five different Archimedean copula functions, the joint return period of an error rate of each station was calculated to be less than 36%, which is significantly lower than the recurrence interval of univariate analysis; (2) High-hazard areas are predominantly concentrated in the northwest region of the Tanggula Mountains and the eastern foothills of the Bayankara Mountains. As the return period increases, the spatial distribution of snow disaster hazard probability shifts gradually from “double-core” to continuous distribution; and (3) the northwestern Karakorum Mountains and Bayankara Mountains are two distinct high-risk areas for snow disasters. The range of high-risk areas in the region expands with an increase in the return period.

On convergence and mass distributions of multivariate Archimedean copulas and their interplay with the Williamson transform

Motivated by a recently established result saying that within the class of bivariate Archimedean copulas standard pointwise convergence implies weak convergence of almost all conditional distributions this contribution studies the class Card of all d-dimensional Archimedean copulas with d≥3 and proves the afore-mentioned implication with respect to conditioning on the first d−1 coordinates. Several properties equivalent to pointwise convergence in Card are established and - as by-product of working with conditional distributions (Markov kernels) - alternative simple proofs for the well-known formulas for the level set masses μC(Lt) and the Kendall distribution function FKd as well as a novel geometrical interpretation of the latter are provided. Viewing normalized generators ψ of d-dimensional Archimedean copulas from the perspective of their so-called Williamson measures γ on (0,∞) is then shown to allow not only to derive surprisingly simple expressions for μC(Lt) and FKd in terms of γ and to characterize pointwise convergence in Card by weak convergence of the Williamson measures but also to prove that regularity/singularity properties of γ directly carry over to the corresponding copula Cγ∈Card. These results are finally used to prove the fact that the family of all absolutely continuous and the family of all singular d-dimensional copulas is dense in Card and to underline that despite of their simple algebraic structure Archimedean copulas may exhibit surprisingly singular behavior in the sense of irregularity of their conditional distribution functions.

Tail Dependence as a Measure of Teleconnected Warm and Cold Extremes of North American Wintertime Temperatures

Abstract Current models for spatial extremes are concerned with the joint upper (or lower) tail of the distribution at two or more locations. Such models cannot account for teleconnection patterns of 2-m surface air temperature (T2m) in North America, where very low temperatures in the contiguous United States may coincide with very high temperatures in Alaska in the wintertime. This dependence between warm and cold extremes motivates the need for a model with opposite-tail dependence in spatial extremes. This work develops a statistical modeling framework that has flexible behavior in all four pairings of high and low extremes at pairs of locations. In particular, we use a mixture of rotations of common Archimedean copulas to capture various combinations of four-corner tail dependence. We study teleconnected T2m extremes using ERA5 of daily average 2-m temperature during the boreal winter. The estimated mixture model quantifies the strength of opposite-tail dependence between warm temperatures in Alaska and cold temperatures in the midlatitudes of North America, as well as the reverse pattern. These dependence patterns are shown to correspond to blocked and zonal patterns of midtropospheric flow. This analysis extends the classical notion of correlation-based teleconnections to considering dependence in higher quantiles.

Remarks on a copula‐based conditional value at risk for the portfolio problem

SummaryWe deal with a multivariate conditional value at risk. Compared with the usual notion for the single random variable, a multivariate value at risk is concerned with several variables, and thus, the relation between each risk factor should be considered. We here introduce a new definition of copula‐based conditional value at risk, which is real valued and ready to be computed. Copulas are known to provide a flexible method for handling a possible nonlinear structure; therefore, copulas may be naturally involved in the theory of value at risk. We derive a formula of our copula‐based conditional value at risk in the case of Archimedean copulas, whose effectiveness is shown by examples. Numerical studies are also carried out with real data, which can be verified with analytical results.

Risk measurement of China's foreign energy investment portfolio based on Copula-VaR

Energy is an important resource for the development of the country, and investment in energy can promote the development of the national economy. Many scholars are currently using Copula models to predict the risk of energy investments to improve investment efficiency. However, most studies are not systematic enough and focus on countries outside of China. We use the Copula-VaR method with the Archimedean Copula function and the Copula-VaR method with the introduction of tail correlation to calculate the energy futures risk. The risk of six different percentages of China's foreign energy portfolio for three futures on natural gas, oil, and coal between January 3, 2015 and December 30, 2021 is calculated and compared to the traditional method. The results show that the risk values calculated using the improved Copula-VaR model are 0.00836, 0.00922, 0.00217, 0.00635, 0.00612 and 0.00827 higher under the 0.98 confidence level than under the 0.96 confidence level. It has a high accuracy compared with the traditional method. The research in this paper provides an idea for the design of energy investment programs in China

Ordering and ageing properties of developed sequential order statistics governed by the Archimedean copula

AbstractDeveloped sequential order statistics (DSOS) are very useful in modeling the lifetimes of systems with dependent components, where the failure of one component affects the performance of remaining surviving components. We study some stochastic comparison results for DSOS in both one-sample and two-sample scenarios. Furthermore, we study various ageing properties of DSOS. We state many useful results for generalized order statistics as well as ordinary order statistics with dependent random variables. At the end, some numerical examples are given to illustrate the proposed results.

The analysis of semi‐competing risks data using Archimedean copula models

In this paper, we derive the copula‐graphic estimator (Zheng and Klein) for marginal survival functions using Archimedean copula models based on competing risks data subject to univariate right censoring and prove its uniform consistency and asymptotic properties. We then propose a novel parameter estimation method based on the semi‐competing risks data using Archimedean copula models. Based on our estimation strategy, we propose a new model selection procedure. We also describe an easy way to accommodate possible covariates in data analysis using our strategies. Simulation studies have shown that our parameter estimate outperforms the estimator proposed by Lakhal, Rivest and Abdous for the Hougaard model and the model selection procedure works quite well. We fit a leukemia dataset using our model and end our paper with some discussion.

Reliability analysis for system with dependent components based on survival signature and copula theory

Recently, survival signature theory has been widely used in system reliability analysis. However, there are few studies that address the extension of survival signature theory to component-dependent systems. This study proposes a copula-based survival signature method to obtain an analytical solution for system reliability considering component dependence. It consists of two steps: constructing the dependence structure and utilizing the copula function to compute the analytical probability structure. The effectiveness of the Archimedean copula function in describing component dependence and its compliance with the exchangeability assumption in survival signature theory are first derived. Then, using Sklar's theorem, the analytical solution for the probability structure is determined based on the inclusion-exclusion principle. Finally, the hierarchical Archimedean copula function is employed to build a more sophisticated dependent structure. It is demonstrated that the satisfaction of the exchangeability assumption under the hierarchical Archimedean copula structures depends only on whether the joint distribution probability function of the components in the same type is constructed by a unique Archimedean copula function. Three separate systems with imprecise dependent information are employed in the reliability analysis in order to show the validity of the proposed method. The findings show that given a straightforward series-parallel system with a single type of component, the system reliability initially drops and then rises as the dependence parameters in the copula function grow at each time point. Additionally, the approach is more precise and effective than the simulation method when it comes to the study of complex systems.

Geostatistical Evaluation of a Porphyry Copper Deposit Using Copulas

Kriging has some problems such as ignoring sample values in giving weights to them, reducing dependence structure to a single covariance function, and facing negative confidence bounds. In view to these problems of kriging in this study to estimate Cu in the Iju porphyry Cu deposit in Iran, we used a convex linear combination of Archimedean copulas. To delineate the spatial dependence structure of Cu, the best Frank, Gumbel, and Clayton copula models were determined at different lags to fit with higher-order polynomials. The resulting Archimedean copulas were able to describe all kinds of spatial dependence structures, including asymmetric lower and upper tails. The copula and kriging methods were compared through a split-sample cross-validation test whereby the drill-hole data were divided into modeling and validation sets. The cross-validation showed better results for geostatistical estimation through copula than through kriging in terms of accuracy and precision. The mean of the validation set, which was 0.1218%, was estimated as 0.1278% and 0.1369% by the copula and kriging methods, respectively. The correlation coefficient between the estimated and measured values was higher for the copula method than for the kriging method. With 0.0143%2 and 0.0162%2 values, the mean square error was substantially smaller for copula than for kriging. A boxplot of the results demonstrated that the copula method was better in reproducing the Cu distribution and had fewer smoothing problems.

Stochastic comparisons of series systems with heterogeneous Gompertz-G components

This study compares two different series systems under various scenarios for which the components are assumed to be heterogeneous and follow Gompertz-G distributions. In the first scheme, the components of the systems are supposed to be independently distributed. In the second, we compared two series systems in the case that the independent components also experience random shocks. In the last scenario, we considered a case where the components of the systems have dependent structure sharing Archimedean copula. However, in all scenarios, the comparisons are performed based on the concepts of usual stochastic, the hazard rate, and the likelihood ratio orders through the majorization of the Gompertz-G parameters.

Stochastic comparison on active redundancy allocation to K-out-of-N systems with statistically dependent component and redundancy lifetimes

AbstractIn the literature on active redundancy allocation, the redundancy lifetimes are usually postulated to be independent of the component lifetimes for the sake of technical convenience. However, this unrealistic assumption leads to a risk of inaccurately evaluating system reliability, because it overlooks the statistical dependence of lifetimes due to common stresses. In this study, for k-out-of-n:F systems with component and redundancy lifetimes linked by the Archimedean copula, we show that (i) allocating more homogeneous redundancies to the less reliable components tends to produce a redundant system with stochastically larger lifetime, (ii) the reliability of the redundant system can be uniformly maximized through balancing the allocation of homogeneous redundancies in the context of homogeneous components, and (iii) allocating a more reliable matched redundancy to a less reliable component produces a more reliable system. These novel results on k-out-of-n:F systems in which component and redundancy lifetimes are statistically dependent are more applicable to the complicated engineering systems that arise in real practice. Some numerical examples are also presented to illustrate these findings.

Joint chance-constrained multi-objective multi-commodity minimum cost network flow problem with copula theory

This study focuses on a specific problem in network optimization, namely the minimum cost multi-commodity network flow (MCNF) problem. The problem is complicated by the presence of uncertain parameters, including various types of costs associated with each arc in the network. The study presents a multi-objective approach to solving this problem, where the coefficients of the capacity constraints are modelled as random variables with a normal distribution, and the dependence between them is modelled using an Archimedean copula. The capacity constraints are presented as joint chance constraints, and a multi-objective problem is formulated to deal with the uncertainty. This uncertain multi-objective problem is then converted into a certain multi-objective problem using fuzzy programming. The resulting certain multi-objective MCNF problem is converted to a certain single objective problem using second-order cone programming (SOCP), which is solved using either piecewise tangent approximation or piecewise linear approximation methods. The proposed approaches are tested using numerical examples and experimental tests to demonstrate their effectiveness in solving large-scale network problems efficiently. The results show that the proposed approaches are a useful tool for solving uncertain multi-objective MCNF problems in real-world applications.

Tourists’ travel mode and length of stay: Application of a fully nested Archimedean copula structure

Tourism decisions are interdependent of one another. This study aims to explore the interrelations among three pertinent tourism choices, i.e., travel season, transport mode and length of stay, from a data-driven perspective. To achieve a joint model of three travel behavioural decisions, this paper adopts a Frank-type fully nested Archimedean copula function, which is capable of efficiently incorporating a logistic regression model, a multinomial logit model and a lognormal accelerated failure time model. Then, we empirically evaluate the proposed model framework using single-destination Sydney-based trip-level data collected in 2017–2018. Gumbel, Clayton and Frank copula results are compared. To illustrate its application, the modelling results are used to build a simulated COVID-19 pandemic scenario according to the social distancing restrictions within New South Wales, Australia, and a three-dimensional elasticity analysis for trip destinations is performed. The findings suggest the model provides nuanced insights into simulating tourist behaviours and transport policy appraisal aimed at tourism recovery or/and development.

Bivariate socioeconomic drought assessment based on a hybrid framework and impact of human activities

Socioeconomic drought generally occurs when water supply cannot satisfy water demand. This study proposed a hybrid framework for socioeconomic drought assessment, involving the construction of standardized water supply-demand reliability index (SWSDRI), truncation of drought events, and classification of bivariate drought levels. Using the run theory, socioeconomic drought events and their properties, including duration and severity, were characterized by the SWSDRI series. Then, bivariate socioeconomic drought levels, based on the Kendall frequency of drought duration and severity, were classified using the copula function. Impacts of human activities on socioeconomic drought, including mainly the regulation of reservoirs and water consumption were quantified in the Dongjiang River basin, located in the south of China. Results showed that the framework, based on SWSDRI, facilitated the investigation of socioeconomic drought properties. The joint distributions of drought duration and severity, based on naturalized streamflow and under the individual and total impacts of human activities, were fitted well by the Archimedean copulas. Both drought duration and severity clearly exhibited the slight, moderate, severe, and extreme droughts classified by bivariate drought levels. For naturalized streamflow, socioeconomic drought occurred approximately once a year in the Dongjiang River basin. Design values of the combination of drought duration and severity accounted for 3.4 months and 8.0 hundred million cubic meters of the boundary between slight and moderate droughts, 4.9 months and 13.6 hundred million cubic meters of the boundary between moderate and severe droughts, and 6.9 months and 20.0 hundred million cubic meters of the boundary between moderate and severe droughts, respectively. Under the impact of human activities, in particular the regulation of reservoirs, socioeconomic drought was remarkably alleviated with a decrease of 45.8% in quantity. If the impact of water consumption was considered, drought events in quantitive terms remained almost unchanged, but more extreme and severe droughts occurred, because some slight, moderate, and severe droughts were aggravated by one level. Among the main reservoirs, the impact of the Xinfengjiang reservoir in the period of its operation was remarkable with a decrease in drought events by 47.7%. Under the impact of the Fengshuba reservoirs, drought events changed little in quantitive terms, but more slight droughts occurred because some moderate, severe, and extreme droughts were alleviated.

Hybridizations of Archimedean copula and generalized MSM operators and their applications in interactive decision-making with q-rung probabilistic dual hesitant fuzzy environment

The q-rung probabilistic dual hesitant fuzzy sets (qRPDHFSs), which outperform dual hesitant fuzzy sets, probabilistic dual hesitant fuzzy sets, and probabilistic dual hesitant Pythagorean fuzzy sets, are used in this research to develop an interactive group decision-making approach. We first suggest the Archimedean Copula-based operations on q-rung probabilistic dual hesitant fuzzy (qRPDHF) components and investigate their key features before constructing the approach. We then create some new aggregation operators (AOs) in light of these operations, including the qRPDHF generalized Maclaurin symmetric mean (MSM) operator, qRPDHF geometric generalized MSM operator, qRPDHF weighted generalized MSM operator, and qRPDHF weighted generalized geometric generalized MSM operator. These aggregation operators are better than current operators on qRPDHF because they can take into account the interactions between a large number of criteria and probability distributions. The evaluation findings are distorted since the present methodologies do not take expert involvement into account in order to achieve the required consistency level. We employ the idea of interaction, consistency, resemblance, and consensus-building among the decision-makers in our method to get around this. We create an optimization model based on the cross-entropy of the qRPDHF components to estimate the weights of the criterion. We provide contextual research on the choice of open-source software LMS in order to demonstrate the relevance of the recommended AOs. Likewise, we ran a sensitivity test on the weights of the criterion to make sure that our model is consistent. The comparison investigation has demonstrated that the suggested approach can overcome the challenges of previous works.

HR and RHR orderings of extremes of dependent variables under Archimedean copula

In this article, we discuss sufficient conditions for the hazard rate order on smallest order statistics for variables with Archimedean survival copula and having modified proportional hazard rates (modified scale) model. We also establish the reversed hazard rate order on largest order statistics from variables with Archimedean copula and having modified proportional reversed hazard rates model. The results established here extend the recent results of Li and Li (2019).

Multi‐attribute group decision‐making based on linguistic Pythagorean fuzzy copula extended power average operator

AbstractThis paper aims to propose a novel multi‐attribute group decision‐making (MAGDM) method based on linguistic Pythagorean fuzzy copula extended power average operator. Existing researches under linguistic Pythagorean fuzzy environment lack of the ability to handle with extreme values and the flexible operational rules. To fill these two gaps, this paper first provides the definition of Archimedean copula and co‐copula operational rules under linguistic Pythagorean fuzzy environment, which can reflect the connection among arguments and provide more choices for experts to express their preferences. Then, we gather the extended power average (EPA) operator to present some new aggregation operators, which can reduce the negative influence of extreme evaluation values. To show the application of the proposed method to MAGDM problems, we apply it to handle a case of takeout O2O platform assessment problem. The numerical case and comparative analysis with other existing methods illustrate that our proposed method is more scientific and flexible.

Investigation into the International Financial Reporting Standards 9 model flaws exposed during the pandemic

Orientation: Market events during the coronavirus disease 2019 (COVID-19) pandemic exposed flaws in the econometric models used to derive International Financial Reporting Standards (IFRS) 9 impairments. Models were unable to capture the level of government intervention or predict the economic recovery process because of the unprecedented nature of the pandemic.Research purpose: This study examines the causes of the challenges experienced with the IFRS 9 models during the pandemic and approaches to minimise this risk in the future.Motivation for the study: Structural correlation breaks forced banks to replace the IFRS 9 models with expert overlays or rapidly rebuild the models to reduce impairment volatility and manage the impact on earnings. Expert judgement may lead to biased outcomes.Research approach/design and method: Behavioural finance theory suggests that emotion and cognitive biases often lead to irrational investment decisions with disastrous consequences to the market. The link between market sentiment and economic outcomes is tested with natural language processing. Archimedean copulas are used to compare the dependence structures of market variables between different stress periods.Main findings: Market sentiment is closely related to the trends observed in major macroeconomic indicators. The nature of the dependence structures differs between stress periods.Practical/managerial implications: Sentiment may be a valuable exogenous variable to incorporate into economic forecast models. Learnings from one stress period are not necessarily applicable to another.Contribution/value-add: Government intervention and market sentiment had a significant impact on the economic outcomes and correlation breaks observed during the pandemic. Developing bespoke models for the different phases of the economic cycle may not necessarily lead to improved outcomes.

Stochastic comparisons of second-order statistics from dependent and heterogenous modified proportional hazard rate observations

This manuscript studies the stochastic comparisons of the second-order statistics from dependent or independent and heterogeneous modified proportional hazard rate observations. Some sufficient conditions on the usual stochastic order of the second-order statistics from dependent and heterogeneous observations are established under Archimedean copula. Some sufficient conditions are also provided in the hazard rate order of the second-order statistics arising from two sets of independent and heterogeneous or multiple-outlier modified proportional hazard rate observations. Some numerical examples are given to illustrate the theoretical findings.