Copula Construction Research Articles

Copula-Based Regression with Mixed Covariates

In this paper, we focused on developing copula-based modeling procedures that effectively capture the dependence between response and explanatory variables. Building upon the work of Noh et al. (J. Am. Stat. Assoc. 2013, 108, 676–688) we extended copula-based regression to accommodate both continuous and discrete covariates. Specifically, we explored the construction of copulas to estimate the conditional mean of the response variable given the covariates, elucidating the relationship between copula structures and marginal distributions. We considered various estimation methods for copulas and distribution functions, presenting a diverse array of estimators for the conditional mean function. These estimators range from non-parametric to semi-parametric and fully parametric, offering flexibility in modeling regression relationships. An adapted algorithm is applied to construct copulas and simulations are carried out to replicate datasets, estimate prediction model parameters, and compare with the OLS method. The practicality and efficacy of our proposed methodologies, grounded in the principles of copula-based regression, are substantiated through methodical simulation studies.

Innovative Methods of Constructing Strict and Strong Fuzzy Negations, Fuzzy Implications and New Classes of Copulas

This paper presents new classes of strong fuzzy negations, fuzzy implications and Copulas. It begins by presenting two theorems with function classes involving the construction of strong fuzzy negations. These classes are based on a well-known equilibrium point theorem. After that, a construction of fuzzy implication is presented, which is not based on any negation. Finally, moving on to the area concerning copulas, we present proof about the third property of copulas. To conclude, we will present two original constructions of copulas. All the above constructions are motivated by a specific formula. For some specific conditions of the variables x, y and other conditions for the function f(x), the formula presented produces strict and strong fuzzy negations, fuzzy implications and copulas.

Construction of copulas for bivariate failure rates

This paper aims to develop a method to construct an asymmetric copula, based on which a closed form of the cumulative bivariate failure rate can be obtained. The construction method differs from existing ones. This new method can facilitate the derivation of some results such as the estimation of the expected number of occurrences for a system whose failure process is modelled by a bivariate stochastic process or the expected cost in optimisation of maintenance policies.

Construction of New Bivariate Semilinear Copulas Based on the Ali–Mikhail–Haq, Farlie–Gumbel–Morgenstern and Plackett Families

Copulas are quite a popular, flexible, and useful tool for multivariate modeling in many fields of applications, including finance and insurance, actuarial sciences, biomedical studies, geostatistics, and hydrology. In this paper, we use the concept of diagonal function to build new bivariate copulas based on the well known and important Ali–Mikhail–Haq (AMH), Farlie–Gumbel–Morgenstern (FGM) and Plackett families. We also study some dependence properties of the newly constructed semilinear copulas, namely, the Spearman’s [Formula: see text] coefficient. In particular, we find that the range of this rank correlation coefficient varies between weak and moderate values for the transformed AMH and FGM families, although it varies between moderate and strong values for the transformed Plackett families. Moreover, in all cases, the Spearman’s [Formula: see text] coefficient reaches negative and positive values. Finally, we present a synthetic data generation algorithm, as well as a classical method of estimation of the copula association parameter [Formula: see text] for each of the six transformed bivariate families.

Vine Copula Model: Application to Chemical Elements in Water Samples

Copula can link the bivariate distribution function with marginal distribution functions without requiring specific information about the interdependence among random variables. There are several types of copulas, such as elliptical copulas, Archimedean copulas, and extreme value copulas. However, in multivariate modeling, each type of copula has limitations in modeling complex dependence structures in terms of symmetry and tail dependence properties. The class of vine copulas overcomes these limitations by constructing multivariate models using bivariate copulas in a tree-like structure. The bivariate copulas used in this study include the Clayton, Gumbel, Frank, Gaussian, and Student's t copula families. This study discusses the construction of vine copula models, parameter estimation, and their applications. The construction of vine copulas is done through the decomposition of conditional probability density functions and substituting bivariate copula density functions into the decomposition results. The data used in the study is the logarithm of the concentration of chemical elements in water samples in Colorado. The parameter estimation method used is pseudo-maximum likelihood with sequential estimation. Model selection is then performed using the Akaike information criterion (AIC) to determine the most suitable model. The results indicate that Caesium and Titanium have a dependency relationship with Scandium. Moreover, Scandium and Titanium exhibit the strongest dependence compared to other variable pairs.

Versatile copulas and their stance-marking uses in conversational Odia, an Indo-Aryan language

Recent typological studies have identified some major functions associated with copulas, including tense-aspect-mood (TAM) and agreement (AGR) marking. In this paper, using conversational data from Odia, an Indo-Aryan language, we examine the speaker stance functions in copula constructions. More specifically, in addition to focus marking as a highlighting strategy, we show how copulas help to anchor the speaker’s subjective (attitudinal and affective) and intersubjective (socio-interactional) stance. Our analysis reveals that these (inter)subjective stances can be realized epistemically through the tense-mood marking functions of the copula, socio-interactionally through its agreement marking functions, and emphatically through its focus marking functions. The findings from this study contribute to our understanding of the relationship between copulas and stance-taking/stance-marking in spontaneous conversations in Indo-Aryan languages, and potentially other languages as well.

Curvilinear patchwork constructions of (quasi-)copulas with given curvilinear sections

In this paper, we focus on constructions of (quasi-)copulas with a given curvilinear section by the curvilinear patchwork operation. It is shown that the curvilinear patchwork of any two quasi-copulas with a common curvilinear section is always a quasi-copula, and some sufficient conditions for the curvilinear patchwork of two copulas with a common curvilinear section to be a copula are also provided. The density, the singular component and tail dependence coefficients of copulas constructed by the curvilinear patchwork operation are discussed. The concept of a simple curvilinear section is introduced, and an elementary way of creating copulas with simple curvilinear sections is developed by the curvilinear patchwork construction. Best-possible bounds for the set of copulas with a common simple curvilinear section are given. By the curvilinear patchwork construction, two generalized bound copulas with given curvilinear sections and a new extension of the Farlie-Gumbel-Morgenstern family of copulas are also proposed.

Copula Construction in Kathariya Tharu

Kathariya Tharu is one of the least studied Indo-Aryan languages spoken by the Kathariya Tharu community living in Kailali district of Nepal and Gorakhpur, Gonda, Lakhimpur, and Khiri districts of India. This is a preliminary study which analyses the morphosyntactic structure of copula construction in this language. The linguistic data for this study were collected from Ghodaghodi municipality, ward no. 9, Sisaiya, ward no.12, Kota and Kailari rural municipality, ward no. 8, Lausa and were analyzed with the help of the Toolbox software. R. M. W. Dixon’s Basic Linguistic Theory has been employed as analytical tool, though insights have also been taken from other linguists. Kathariya Tharu employs the verbal copula construction strategy with three copula verbs- ho, ba, and rəh. Unlike other NIA languages in which the relation between the copula subject and copula complement is prominent, in Kathariya Tharu, it is the copula subject that determines the selection of the copula verb. However, the person, relation and function distinctions between the copula subject and copula complement disappears in the past tense and so, the same copula rəh is used with all the subjects.

Development of a non-Gaussian copula Bayesian network for safety assessment of metro tunnel maintenance

The operation and maintenance of metro systems play a crucial role in urban development, with a focus on ensuring the serviceability and safety of metro tunnels. Accurate evaluation of the condition of these tunnel states requires investigating the complex interaction of multiple factors that impact their safety state. This study developed a hybrid model that integrates pair copula constructions (PCCs) and Bayesian networks (BN) to assess the safety state of metro tunnels, considering complex dependencies among these factors. First, key performance indicators (KPIs) were selected to assess tunnel safety, based on six failure modes. Then, an improved copula-based PC algorithm was employed to learn the non-Gaussian bayesian network model, which eliminated the normality assumption in the marginal densities of the KPIs. Finally, a combination of forward reasoning and GM(1,1) was utilized to predict the safety state of the metro tunnel in time series, facilitating the formulation of effective operation and maintenance plans. Furthermore, the proposed approach was applied to a real case study of the Wuhan metro system to demonstrate its effectiveness and applicability. The results highlighted the significant influence of some KPIs, such as convergence deformation, dislocation displacement, and stripping area, on metro tunnel safety. These KPIs emerged as key factors requiring focused attention in the operation and maintenance of metro tunnels.

Construction of bivariate symmetric and asymmetric copulas and its relationship to ratios of conditional hazard rate functions

ABSTRACT A new method for the construction of both symmetric and asymmetric, absolutely continuous bivariate copulas is introduced. This research was inspired by the well-known cross ratio (CR), studied by Clayton (1978), which is a local dependence measure describing the association between the components of a bivariate random vector, as well as by an alternative risk ratio (RR) proposed recently by Abrams et al. (2022), which unravels the local relation between the two components of the vector. The practical meaning of RR is illustrated in two real data sets by the authors. In this article, various properties of the proposed parametric families of copulas are studied, for example quadrant dependence ordering, tail dependence coefficients and several copula-based measures of dependence. Some relationships of the copulas to RR are discussed and illustrative scatterplots are presented.

Transformed Pair Copula Construction of Pareto Copula and Applications

The study introduced the transformed copula models and a D-vine structure for three variables. The numerical applications of the models are evaluated using two different sets of real-life data that exhibit nearly no dependence, highly dependent, over, and under dispersed characteristics. We examined only the volatilities of the first data using the exponentiated weighted moving average (EWMA). The parameter estimates of the models were obtained based on the maximum likelihood estimation method for the bivariate copula models and the Dissmann algorithm for sequential top-down estimation for the D-vine structure. The results showed that the introduced copula models outperformed some existing copula models in terms of their fit statistics for both real-life and simulated data sets. In addition, the Gaussian copula model gave a better fit to the D-vine structure than some existing copula models and could be recommended for modeling a D-vine structure comprising of variables that are positively weak correlated and highly correlated.

An Analysis of the Part of Speech of “shi(是)” in the “shi(是)” Copula Construction in Guanzi(『管子』)

The origin and formative causes of Chinese copula shi(是) were the important issues in Chinese studies. The particular contents of Guanzi (『管子』) established its essential position in ancient books of the Pre-Qin Dynasty, which was the important document for studies of ancient Chinese as well as the valuable corpus for studying the part of speech of the early copula shi(是). This research analyzed the copula construction of shi(是) in Guanzi(『管子』) from different aspects including grammatical components and semantic relationships for exploring the part of speech of shi(是). The results indicated that the part of speech of shi(是) in Guanzi(『管子』) had both referential and copulative features, as well as intense judgement. Due to the referential features, shi(是) was not a copula at the present stage, but belonged to the predecessor of the copula shi(是).

Linguistic factors modulating gender assignment in Spanish–English bilingual speech

AbstractDrawing on naturally-occurring bilingual speech from a well-defined codeswitching community in Southern Arizona, this study examined the influence of semantic gender (a.k.a. biological gender), analogical gender, and other-language phonemic cues in modulating gender assignment in Spanish–English codeswitched speech. Thirty-four Spanish–English early bilinguals completed a forced-choice elicitation task involving two codeswitching environments: Spanish determiner–English noun switches (Task 1) and English–Spanish switched copula constructions (Task 2). The results revealed that for human-denoting nouns, bilinguals assigned grammatical gender based on the presupposed sex of a noun's referent in both syntactic environments tested. As for inanimate nouns, bilinguals were more likely to assign masculine over feminine gender to such nouns in determiner–noun switches, but not in switched copula constructions. Other-language phonemic cues did not influence the assignment mechanism. A methodological implication is that the study replicated the codeswitching patterns observed in naturally-occurring bilingual speech from the same bilingual community.

A Novel Construction Method of (OP) Polynomial and Rational Fuzzy Implications

Abstract: In this article, we develop new constructed methods with specific conditions. The first method is a generalization of convex combination using n fuzzy implications. The second method is a parameterization of Lukasiewicz implication in an Ordering Property (OP) fuzzy implication form. The innovation in this work is the presentation of three new constructed methods of (OP) polynomial and (OP) rational fuzzy implications. We investigate some families of Ordering Property (OP) and Ordering Property (OP) Rational fuzzy implications. To these methods, we give some coefficient conditions in order to satisfy basic properties like ordering property (OP), identity property (IP) and contrapositive symmetry (CP). Background: Fuzzy implication functions are one of the main operations in fuzzy logic. They generalize the classical implication, which takes values in the set {0, 1}, to fuzzy logic, where the truth values belong to the unit interval [0, 1]. The study of this class of operations has been extensively developed in the literature in the last 30 years from both theoretical and applicational points of view. Introduction: In this paper, we develop five new methods for constructing fuzzy implications with specific properties. The paper starts by presenting the first fuzzy implication construction machine that uses n fuzzy implications with specific conditions. Next, we parameterize Lukasiewicz implication and create new families of (OP) polynomial and (OP) rational implications. For each method we investigate which conditions are satisfied and we give some examples. Methods: The first constructed method uses n fuzzy implications in a linear product representation. The second method is an (OP) polynomial implication a parameterized Lukasiewicz implication. The third method is a rational implication with five parameters. In the fourth method we give a general form in the previous method by changing variables x and y with increasing functions. Finally, the last method is another (OP) rational implication with three parameters. Results: In each method we present the properties that are satisfied. We generalize the (OP) polynomial and rational by replacing the variables with monotonic functions or add powers on them. Finally, we generalize and we give examples of new produced fuzzy implications. Conclusion: As a future work, we can create new families of rational implications by changing the polynomials of the numerator and denominator so that they satisfy more properties. Finally, the new methods we presented can contribute in the construction of uninorms and copulas under certain conditions.

Development of a Joint Probabilistic Rainfall‐Runoff Model for High‐to‐Extreme Flow Projections Under Changing Climatic Conditions

AbstractMachine learning (ML) models have been widely used for hydrological simulation. Previous studies have reported that conventional ML models fail to accurately simulate extreme flows which are crucial for design flood estimation and associated risk analysis in the context of climate change. Therefore, this study proposes a joint probabilistic rainfall‐runoff model (JPRR) for improving high‐to‐extreme flow projection. With the aid of paired copula constructions, bootstrap aggregation, and multi‐model ensemble approaches, the proposed model is able to effectively characterize the dependence relationships of predictors (i.e., precipitation time series with different moving sums) with various probability distributions. JPRR has been applied to four pristine basins in China, representing different climate zones and landscapes. The results reveal that JPRR significantly outperforms three well‐known ML models (i.e., random forest, artificial neural networks, and long short‐term memory) in high‐to‐extreme flow simulations. In JPRR, the copulas exhibiting the right tail dependence play a more important role in streamflow simulations at mountainous basins. Moreover, a significant difference in streamflow projections (from 2030 to 2099) derived from JPRR and benchmark models imply that flood risks from conventional ML models may be underestimated under changing climatic conditions.

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.

Copula Regression for Compound Distributions with Endogenous Covariates with Applications in Insurance Deductible Pricing

This article concerns deductible pricing in nonlife insurance contracts. The primary interest of insurers is the effect of the contract deductible on a policyholder’s aggregate loss that is determined by a compound distribution where the sum of individual claim amount is stopped by the number of claims. Policyholders choose the deductible level based on their hidden risks, which makes deductible endogenous in the regressions for both claim frequency and claim severity. To address the endogeneity in the regression for the compound aggregate loss, we introduce a novel approach using pair copula constructions to jointly model the policyholder’s deductible, number of claims, and individual claim amounts, in the context of compound distributions. The proposed method provides insurers an empirical tool to uncover the underlying risk distribution of the potential customers. In the application we consider an insurance portfolio from the property insurance program that provides property coverage for building and contents of local government entities of the Wisconsin. Using the historical data on policyholder and insurance claims, we first provide empirical evidence of the endogeneity of the deductible. Interestingly, we find that the policyholder’s deductible is negatively associated with the claim frequency but positively associated with the claim severity. For the portfolio of policyholders, the endogenous deductible model provides superior prediction for 65% and 71% of policyholders for claim frequency and severity, respectively. The endogeneity of deductible shows significant managerial implications on insurance operations. In particular, the risk score suggested by the proposed method allows the insurer to identify additional profitable underwriting strategies which are quantified by the Gini indices of 0.22 and 0.13 when switching from the exogenous deductible premium and the insurer’s contract premium, respectively. Supplementary materials for this article are available online.

Bayesian Computation in Dynamic Latent Factor Models

Bayesian computation for filtering and forecasting analysis is developed for a broad class of dynamic models. The ability to scale-up such analyses in non-Gaussian, nonlinear multivariate time series models is advanced through the introduction of a novel copula construction in sequential filtering of coupled sets of dynamic generalized linear models. The new copula approach is integrated into recently introduced multiscale models in which univariate time series are coupled via nonlinear forms involving dynamic latent factors representing cross-series relationships. The resulting methodology offers dramatic speed-up in online Bayesian computations for sequential filtering and forecasting in this broad, flexible class of multivariate models. Two examples in nonlinear models for very heterogeneous time series of nonnegative counts demonstrate massive computational efficiencies relative to existing, simulation-based methods, while defining similar filtering and forecasting outcomes.

A Non-stationary and Probabilistic Approach for Drought Characterization Using Trivariate and Pairwise Copula Construction (PCC) Model

Under variable climatic conditions, the conventional Standardized Precipitation Index (SPI) and Reconnaissance Drought Index (RDI) are inadequate for predicting extreme drought characteristics. Non-stationary Standardized Precipitation Index (NSPI) and Non-stationary Reconnaissance Drought Index (NRDI) are, therefore, developed by fitting non-stationary distributions. The Generalized Additive Model in Location, Scale and Shape (GAMLSS) framework, with time varying location parameters considering the external covariates, is used to fit the non-stationary distributions. Multivariate ENSO Index (MEI), Southern Oscillation Index (SOI), Sea Surface Temperature (SST), and Indian Ocean Dipole (IOD) are considered as external covariates for the non-stationary drought assessment. The performances of stationary and non-stationary models are compared. The study also concentrated on the trivariate and the Pairwise Copula Construction (PCC) models to estimate the drought return periods. The comparison of two copula models revealed that the PCC model performed better than the trivariate Student’s t copula model. The recurrence intervals arrived at for the drought events are different for trivariate copula model and PCC model. This study showed that non-stationary drought indices will be helpful in the accurate estimate of the drought characteristics under the changing climatic scenario.

Data‐driven look‐ahead voltage stability assessment of power system with correlated variables

Probabilistic voltage stability assessment (VSA) of distribution systems has recently gained significant research attention due to the proliferation of renewable energy sources (RESs). This study presents a nonparametric data-driven VSA in a look-ahead framework by considering correlated variables. First, an accurate but relatively fast correlation model is developed by considering stochastic interdependency with unknown dependency structure. Then, the probabilistic density function (PDF) of the voltage stability index is estimated for near-future minutes. The proposed method employs Bernstein vine copula (BVC) for estimating the unknown underlying dependence structure among different variables. This non-parametric method imposes no assumption on the dependence structure and can handle the problems where no information is available for judging about dependence structure and its belongingness to a specific type of copula function. Additionally, global sensitivity analysis (GSA), with an embedded surrogate model based on the extreme learning machine (ELM), is utilized to rank the loads and RESs according to their contribution to voltage stability margin. Furthermore, a parallel implementation of vine copula construction and dimension reduction is developed based on the GSA to speed up the proposed VSA. The efficiency of the proposed method is examined by simulating the results of modified IEEE 33-node and 123-node systems.