Copulas
for which
are called
copulas, of which well-studied examples are idempotent copulas and complete dependence copulas. As such, we shall work mainly with the topology induced by the modified ...Sobolev norm, with respect to which the class
of pre-idempotent copulas is closed and the class of factorizable copulas is a dense subset of
. Identifying copulas with Markov operators on
, the one-to-one correspondence between pre-idempotent copulas and partial isometries is one of our main tools. In the same spirit as Darsow and Olsen’s work on idempotent copulas, we obtain an explicit characterization of pre-idempotent copulas, which is split into cases according to the atomicity of its associated
-algebras, where the nonatomic case gives all factorizable copulas and the totally atomic case yields conjugates of ordinal sums of copies of the product copula.
This paper develops a new method for interactive multi-criteria group decision-making (MCGDM) with probabilistic linguistic information and applies to the emergency assistance area selection of ...COVID-19 for Wuhan. First, a new possibility degree for PLTSs is defined and a new possibility degree algorithm is devised to rank a series of probabilistic linguistic term sets (PLTSs). Second, some new operational laws of PLTSs based on the Archimedean copulas and co-copulas are defined. A generalized probabilistic linguistic Choquet (GPLC) operator and a generalized probabilistic linguistic hybrid Choquet (GPLHC) operator are developed and their desirable properties are discussed in details. Third, a tri-objective nonlinear programming model is constructed to determine the weights of DMs. This model is transformed into a linear programming model to solve. The fuzzy measures of criterion subsets are derived objectively by establishing a goal programming model. Fourth, using the probabilistic linguistic Gumbel weighted average (PLGWA) operator, the collective normalized decision matrix is obtained by aggregating all individual normalized decision matrices. The overall evaluation values of alternatives are derived by the probabilistic linguistic Gumbel hybrid Choquet (PLGHC) operator. The ranking order of alternatives is generated. Finally, an emergency assistance example is illustrated to validate the proposed method of this paper.
•A new possibility degree of PLTSs is defined.•A new similarity degree of PLTSs is defined.•Define new operational laws of PLTSs based on Archimedean copulas and co-copulas.•Construct programming models to determine DMs’ weights and fuzzy measures of criteria.•A new method for the interactive MCGDM with PLTSs is put forward.
Coupling of multiple environmental factors in the engineering sector, in particular, joint distribution modeling based on a high-dimensional environmental dataset containing circular data is very ...topical. We propose a modeling framework applicable to the 3D joint distribution of circular-linear-linear (C-L-L) dataset consisting of a parametric model based on copulas and a nonparametric kernel density estimation model. In the parametric model, the pair-copula decomposition concept of vine copulas represents the C-L-L dependence structure as a combination of C-L and L-L ones modeled by C-L and L-L copulas, respectively. This allows one to solve the circular variable's cyclicity problem in the trivariate joint distribution and assess C-L and L-L dependencies between paired variables in the C-L-L dataset. A case study is used to establish the joint distribution model based on annual observations of wind direction, wind speed, and air temperature via the structural health monitoring system of the Jiangyin Bridge, China. The modeling framework validity is proved by the case study, which reveals significant differences in the conditional univariate distribution features of wind speed and air temperature, and conditional joint distribution features of two variables under different wind directions. Therefore, in engineering problems sensitive to the wind direction, the latter's effect cannot be neglected.
•A modeling framework applicable to joint distribution of circular-linear-linear (C-L-L) dataset is introduced.•The C-L-L dependence structure is decomposed as a combination of C-L and L-L ones via vine copulas.•The cyclicity problem of the circular variable in the trivariate joint distribution model is solved.•The C-L and L-L dependencies between paired variables in the C-L-L dataset are considered.•A kernel density estimation model for C-L-L dataset is proposed for judging the accuracy of copula model.
We propose an approach to construct a new family of generalized Farlie–Gumbel–Morgenstern (GFGM) copulas that naturally scales to high dimensions. A GFGM copula can model moderate positive and ...negative dependence, cover different types of asymmetries, and admits exact expressions for many quantities of interest such as measures of association or risk measures in actuarial science or quantitative risk management. More importantly, this paper presents a new method to construct high-dimensional copulas based on mixtures of power functions and may be adapted to more general contexts to construct broader families of copulas. We construct a family of copulas through a stochastic representation based on multivariate Bernoulli distributions and Coxian-2 distributions. This paper will cover the construction of a GFGM copula and study its measures of multivariate association and dependence properties. We explain how to sample random vectors from the new family of copulas in high dimensions. Then, we study the bivariate case in detail and find that our construction leads to an asymmetric modified Huang–Kotz FGM copula. Finally, we study the exchangeable case and provide insights into the most negative dependence structure within this new class of high-dimensional copulas.
The (sub)copulas play a central role in statistics for capturing the stochastic dependence structure of random variables. A subcopula is a map from the Cartesian product of two closed subsets of the ...unit closed interval containing zero and one to the unit closed interval, which is grounded, 2-increasing and has uniform margins. A copula is a subcopula whose domain is the Cartesian product of the unit closed interval and itself. The uniform metric on the set of all copulas can be extended to a metric on the set of all subcopulas. In this work, we investigate topological structures of these spaces by using tools from infinite-dimensional topology. Our main result is as follows: there exists a homeomorphism from the space of subcopulas onto the Hilbert cube which sends the space of all copulas onto an end of the Hilbert cube and sends the space of exchangeable copulas onto an end of the end of the Hilbert cube. As a consequence, we obtain that the spaces of subcopulas, copulas and exchangeable copulas are homeomorphic to the Hilbert cube and hence have the fixed point property.
We study systemic risk and dependence between oil and renewable energy markets using copulas to characterize the dependence structure and to compute the conditional value-at-risk as a measure of ...systemic risk. We found significant time-varying average and symmetric tail dependence between oil returns and several global and sectoral renewable energy indices. Our evidence on systemic risk indicates that oil price dynamics significantly contributes around 30% to downside and upside risk of renewable energy companies. These results have important implications for risk management and renewable energy policies.
•We study systemic risk and dependence between oil and renewable energy markets.•Dependence and conditional value-at-risk is obtained through copulas.•Oil and renewable energy displayed time-varying average and symmetric tail dependence.•Oil price contribution to the downside and upside risks of renewable energy companies was around 30%.
This paper brings some insights of
ψ
′
-mixing,
ψ
∗
-mixing and
ψ
-mixing for copula-based Markov chains and the perturbations of their copulas. We provide new tools to check Markov chains for
ψ
...-mixing or
ψ
′
-mixing. We show that perturbations of
ψ
′
-mixing copula-based Markov chains are
ψ
′
-mixing while perturbations of
ψ
-mixing Markov chains are not necessarily
ψ
-mixing Markov chains, even when the perturbed copula generates
ψ
-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
ψ
-mixing.
•A reliable nonlinear multivariate drought index (NMDI) was constructed.•Copulas can better reflect the nonlinear relationship among multiple single drought indices.•Drought event including three ...drought characteristics was redefined with the NMDI.•Multivariate drought risk was fully assessed.•Partitions where had higher drought risk were found.
It is vital to identify drought events and to evaluate multivariate drought characteristics based on a composite drought index for better drought risk assessment and sustainable development of water resources. However, most composite drought indices are constructed by the linear combination, principal component analysis and entropy weight method assuming a linear relationship among different drought indices. In this study, the multidimensional copulas function was applied to construct a nonlinear multivariate drought index (NMDI) to solve the complicated and nonlinear relationship due to its dependence structure and flexibility. The NMDI was constructed by combining meteorological, hydrological, and agricultural variables (precipitation, runoff, and soil moisture) to better reflect the multivariate variables simultaneously. Based on the constructed NMDI and runs theory, drought events for a particular area regarding three drought characteristics: duration, peak, and severity were identified. Finally, multivariate drought risk was analyzed as a tool for providing reliable support in drought decision-making. The results indicate that: (1) multidimensional copulas can effectively solve the complicated and nonlinear relationship among multivariate variables; (2) compared with single and other composite drought indices, the NMDI is slightly more sensitive in capturing recorded drought events; and (3) drought risk shows a spatial variation; out of the five partitions studied, the Jing River Basin as well as the upstream and midstream of the Wei River Basin are characterized by a higher multivariate drought risk. In general, multidimensional copulas provides a reliable way to solve the nonlinear relationship when constructing a comprehensive drought index and evaluating multivariate drought characteristics.
•Bivariate Archimedean copulas NN13, NN19 and NN20 performed well for modeling drought frequency variations in China.•Bivariate copula-based models are derived for seven climatic sub-regions and ...entire mainland China.•Drought probabilities in sub-region III, V, VI and VII, are significantly greater than other sub-regions.•Regions with high probability are often associated with short return periods.
Probabilistic modelling of drought events is a significant aspect of water resources management and planning. In this study, popularly applied and several relatively new bivariate Archimedean copulas were employed to derive regional and spatial based copula models to appraise drought risk in mainland China over 1961–2013. Drought duration (Dd), severity (Ds), and peak (Dp), as indicated by Standardized Precipitation Evapotranspiration Index (SPEI), were extracted according to the run theory and fitted with suitable marginal distributions. The maximum likelihood estimation (MLE) and curve fitting method (CFM) were used to estimate the copula parameters of nineteen bivariate Archimedean copulas. Drought probabilities and return periods were analysed based on appropriate bivariate copula in sub-region I–VII and entire mainland China. The goodness-of-fit tests as indicated by the CFM showed that copula NN19 in sub-regions III, IV, V, VI and mainland China, NN20 in sub-region I and NN13 in sub-region VII are the best for modeling drought variables. Bivariate drought probability across mainland China is relatively high, and the highest drought probabilities are found mainly in the Northwestern and Southwestern China. Besides, the result also showed that different sub-regions might suffer varying drought risks. The drought risks as observed in Sub-region III, VI and VII, are significantly greater than other sub-regions. Higher probability of droughts of longer durations in the sub-regions also corresponds to shorter return periods with greater drought severity. These results may imply tremendous challenges for the water resources management in different sub-regions, particularly the Northwestern and Southwestern China.
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