Copula Model Research Articles

Copulas and hydro-economic models for assessing the impacts of climate change in irrigated agriculture

Climate change significantly affects water resources through alterations in rainfall patterns, reduced rainfall quantities, rising temperatures, and prolonged drought cycles. Consequently, the escalating demand for water coupled with diminishing water availability is anticipated to lead to a critical water scarcity issue in the future unless proactive and innovative strategies are implemented.The impact of climate change on water resources is globally recognized as an increasingly vital concern, given its intricate interconnections with various sectors, notably agriculture, energy, and the provision of drinking water. Consequently, the scientific community has dedicated substantial efforts to devise optimal water resource management strategies in response to the challenges posed by climate change. Over the past four decades, hydro-economic models (HEM) have been instrumental in proposing solutions to adapt to evolving climatic conditions.This paper presents a newhydro-economic optimization model accounting for climatic uncertainties. The problem is formulated as a chance-constrained program in which the dependence structure between hydrologic and meteorologic variables is modeled using copula theory. The novelty of the approach lies in the capacity of the model to optimize the water resources, taking into account the dependence between agronomic, socio-economic, and hydrologic systems and climatic uncertainties.

An Improved Copula‐Based Framework for Efficient Global Sensitivity Analysis

AbstractGlobal sensitivity analysis (GSA) enhances our understanding of computational models and simplifies model parameter estimation. VarIance‐based Sensitivity analysis using COpUlaS (VISCOUS) is a variance‐based GSA framework. The advantage of VISCOUS is that it can use existing model input‐output data (e.g., water model parameters‐responses) to estimate the first‐ and total‐order Sobol’ sensitivity indices. This study improves VISCOUS by refining its handling of marginal densities of the Gaussian mixture copula model (GMCM). We then evaluate VISCOUS using three types of generic functions relevant to water system models. We observe that its performance depends on function dimension, input‐output data size, and non‐identifiability. Function dimension refers to the number of uncertain input factors analyzed in GSA, and non‐identifiability refers to the inability to estimate GMCM parameters. VISCOUS proves powerful in estimating first‐order sensitivity with a small amount of input‐output data (e.g., 200 in this study), regardless of function dimension. It always ranks input factors correctly in both first‐ and total‐order terms. For estimating total‐order sensitivity, it is recommended to use VISCOUS when the function dimension is not very high (e.g., less than 20) due to the challenge of producing sufficient input‐output data for accurate GMCM inferences (e.g., more than 10,000 data). In cases where all input factors are equally important (a rarity in practice), VISCOUS faces non‐identifiability issues that impact its performance. We provide a didactic example and an open‐source Python code, pyVISCOUS, for broader user adoption.

Copula Variational LSTM for High-Dimensional Cross-Market Multivariate Dependence Modeling.

We address a challenging problem-modeling high-dimensional, long-range dependencies between nonnormal multivariates, which is important for demanding applications such as cross-market modeling (CMM). With heterogeneous indicators and markets, CMM aims to capture between-market financial couplings and influence over time and within-market interactions between financial variables. We make the first attempt to integrate deep variational sequential learning with copula-based statistical dependence modeling and characterize both temporal dependence degrees and structures between hidden variables representing nonnormal multivariates. Our copula variational learning network weighted partial regular vine copula-based variational long short-term memory (WPVC-VLSTM) integrates variational long short-term memory (LSTM) networks and regular vine copula to model variational sequential dependence degrees and structures. The regular vine copula models nonnormal distributional dependence degrees and structures. VLSTM captures variational long-range dependencies coupling high-dimensional dynamic hidden variables without strong hypotheses and multivariate constraints. WPVC-VLSTM outperforms benchmarks, including linear models, stochastic volatility models, deep neural networks, and variational recurrent networks in terms of both technical significance and portfolio forecasting performance. WPVC-VLSTM shows a step-forward for CMM and deep variational learning.

Spatial variability characterization and modelling of 2.5D woven SiO2f/SiO2 composites

2.5D woven SiO2f/SiO2 composites possess geometric variabilities across scales, such as the spatial fluctuations of fibers and yarns or the random distributions of defects and pores formed in the manufacturing process, which induces a complex stress and strain distribution in material testing. In this study, a multivariate cross-correlated non-Gaussian random field based on Vine Copula is proposed to characterize the spatial variability of mechanical properties. A database of material mechanical properties is acquired by stochastic representative volume element (SRVE) simulation results which contains realistic meso-geometry characteristics and spatially random voids. Based on this database, the cross- and auto-correlations of the mechanical properties are quantified by Vine Copula and non-Gaussian random field model. In a comparison of different numerical models and experimental results, the proposed method is verified, which shows great advantages in characterizing the spatial variability and big potential abilities in damage or reliability analysis.

Efficiency gains in value-at-risk and expected shortfall estimation by using copulas and full maximum likelihood

We provide Monte Carlo evidence on the efficiency gains obtained in GARCH-based estimations of value-at-risk (VaR) and expected shortfall (ES) by incorporating dependence information through copulas and subsequently using full maximum likelihood (FML) estimates. First, we consider an individual returns series; in this case, the efficiency gain stems from exploiting the relationship with another returns series using a copula model. Second, we consider a portfolio returns series obtained as a linear combination of returns series related through a copula model; in this case, the efficiency gain stems from using FML estimates instead of two-stage ML estimates. We consider three copulas models in order to analyze the effect of the type and degree of tail dependence on the results. Our results show that, in these situations, using copula models and FML leads to a substantial reduction in the mean squared error of the VaR and ES estimates when there is a relatively high degree of dependence between returns (up to 70% in the presence of lower-tail dependence) and a notable improvement in the performance of backtesting procedures.

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.

A New Family of Archimedean Copulas: The Half-Logistic Family of Copulas

In this research, we introduce a truncation of the half-logistic distribution function as a multiplicative Archimedean generator. The corresponding Archimedean copula is obtained, namely the half-logistic family. The dependency structure of this copula is distinct from other well-known ones. Kendall’s tau correlation coefficient is obtained in exact form and found to cover the entire positive dependence range (i.e., [0, 1]). We have proven that this copula is positively ordered and has no tail dependencies. The density of this copula is shown to be totally positive of order two. An extension of this copula is also introduced by adding a second parameter. This extension allows for a negative correlation and connects the famous Frank copula to the half-logistic copula. Two datasets were used to compare the half-logistic copula with some other known copula models.

Risk Spillover from World Energy Markets to Pakistan Agricultural Commodity Markets. An application of Dependence switching Copula model

Risk Spillover from World Energy Markets to Pakistan Agricultural Commodity Markets. An application of Dependence switching Copula model

The Effects of Oil Price Volatility on South African Stock Market Returns

The aim of this study is to assess the response of the South African stock market returns to oil price volatility, based on the daily South African stock market index, using the GARCH-Copula modelling technique. The results of the analysis show evidence of an asymmetric impact of fluctuations in oil prices on South African stock market returns, using a copula model specification, particularly the bivariate symmetrized Joe-Clayton (SJC) copula. The results also revealed that the EGARCH process is the best univariate model to capture oil price volatility. Interestingly, this study also revealed that the tourism industry is most dependent on oil price fluctuations, due to its heavy reliance on transportation costs. The economic implications of this study also suggest that sectors affected by oil price fluctuations need specific long-term and short-term monetary policy strategies. It is recommended that in the short term, expansionary monetary policy could assist in mitigating the impact of higher oil prices, while in the long-term, policies aimed at reducing the volatility in oil prices would be of great help in alleviating its harmful effect on stock market returns.

Vine copula approach to the intra-sectoral dependence analysis in the technology industry

This study explores the dependence within the technology industry post-global financial crisis. Employing vine copula modeling, we focus on stocks of the biggest U.S. technology leaders. Our results reveal moderate to strong positive dependence, best modeled by the C-Vine structure, with Microsoft as a central variable exerting influence on the remaining companies. We also identify a significant heterogeneity in dependence specifications within the studied sample, including variations in its shape, symmetry and tails presence. This motivates the adoption of a wide range of copula families, ranging from elliptical to extreme value models, to capture the dependence characteristics.

Default dependence in the insurance and banking sectors: A copula approach

We employ a time-varying asymmetric copula model that combines the generalized autoregressive score model with the generalized hyperbolic skewed t copula to capture the dynamics and asymmetry of default dependence between insurers and banks. We identify the term structure of default dependence between these two sectors. The short-term and long-term dependence of default risk rise and converge during financial crises. We explore the determinants of the time-series variation in default dependence. While traditional macro variables can explain only a small portion of the variation in default dependence, we find a significant negative correlation between default dependence and global geopolitical risk.

Some systemic risk indicators

This paper aims to introduce novel systemic risk indicators based on risk allocation methods employed in actuarial science. We present diverse general approaches for constructing these indicators and utilize them to derive indicators based on commonly used risk measures such as Value at Risk, Tail Value at Risk, and Expectiles. Furthermore, we analyze the influence of the dependence structure on the behavior of these indicators using a range of copula models. To support our findings, we provide numerical illustrations.

Factorial survival analysis for treatment effects under dependent censoring.

Factorial analyses offer a powerful nonparametric means to detect main or interaction effects among multiple treatments. For survival outcomes, for example, from clinical trials, such techniques can be adopted for comparing reasonable quantifications of treatment effects. The key difficulty to solve in survival analysis concerns the proper handling of censoring. So far, all existing factorial analyses for survival data have been developed under the independent censoring assumption, which is too strong for many applications. As a solution, the central aim of this article is to develop new methods for factorial survival analyses under quite general dependent censoring regimes. This will be accomplished by combining existing nonparametric methods for factorial survival analyses with techniques developed for survival copula models. As a result, we will present an appealing F-test that exhibits sound performance in our simulation study. The new methods are illustrated in a real data analysis. We implement the proposed method in an R function surv.factorial(.) in the R package compound.Cox.

Bayes analysis of one-shot device testing data with correlated failure modes using copula models

Copula models are capable of modeling the dependence structure among the random variables, a phenomenon that is often required in the statistical analysis. Such models are the flexible substitutes of multivariate distributions because they model both the marginal distributions and the joint dependence structure distinctly. Because of such important features, the models are recognized as popular tools in a variety of situations including reliability engineering and survival analysis. The present paper studies a Bayesian approach using three Archimedean copulas, namely, the Gumbel Hougaard copula, the Frank copula and the Joe copula for analyzing one-shot device testing data with two correlated failure modes collected from a constant stress accelerated life test. One-shot devices are units that can be used only once and destroyed immediately after the use. Obviously, one obtains either left or right censored data on the failure times instead of actual failure times of the devices. Finally, all the considered copula models are compared using the Bayesian model selection tools. A real dataset is analyzed as an illustration of the proposed Bayesian methodology.

Assessing dependence between soil ecosystem services as a function of weather and soil: Application of vine copula modeling

Soils are natural ecosystems that provide ecosystem services, whose provision depends on multiple soil properties, climate conditions and human management. Dependence among soil ecosystem services (SESs) must therefore be considered to reliably assess risks of improving SES, as a function of weather conditions or soil properties. The present study described dependence among regulating and provisioning SESs predicted by a biophysical soil and crop model, based on a dataset of soils in France. We applied vine copula modeling as a statistical method that can model joint distribution functions of three SESs and enabled us to estimate probabilities of exceeding a level of one SES as a function of another SES. Trade-offs may need to be made between them to manage soil and water resources and achieve a given yield. By highlighting the degree of dependence among multiple SESs, copula models thus provide information that may improve understanding or management of ESs.

Portfolio Optimization Based on Artificial Neural Network and GARCH-EVT-Copula Models

Forecasting volatility is an essential task in the financial market, especially in portfolio optimization. To improve the prediction accuracy of the volatilities of assets we use a hybrid ANN-EGARCH model then combining with extreme value theory and Copula models to perform out-of-sample forecasting returns for six indices in Asia stock markets then we simulate one-day-ahead returns of these indices. We use EGARCH model to capture the leverage of return shocks due to COVID-19. Based on ANN-EGARCH-EVT-Copula models, we solve our portfolio optimization consisting of these six indices with different copula models. Using different performance measures to evaluate the efficiency of the models we show that under the Sharpe ratio and Sortino ratio the Gumbel copula gives better performance whereas with Average Drawdown and Max Drawdown measures, the Gaussian copula model is a best model for optimizing the portfolio.

The weighted characteristic function of the multivariate PIT: Independence and goodness-of-fit tests

Many authors have exploited the fact that the distribution of the multivariate probability integral transformation (PIT) of a continuous random vector X∈Rd with cumulative distribution function FX is free of the marginal distributions. While most of these methods are based on the cdf of W=FX(X), this paper introduces the weighted characteristic function (WCf) of W. A sample version of the WCf of W based on pseudo-observations is proposed and its weak limit in a space of complex functions is formally established. This result can be used to define test statistics for multivariate independence and goodness-of-fit in copula models, whose asymptotic behaviour comes from the weak convergence of the empirical WCf process. Simulations show the good sampling properties of these new tests, and an illustration is given on the multivariate Cook and Johnson dataset.

Pair programming with ChatGPT for sampling and estimation of copulas

Without writing a single line of code by a human, an example Monte Carlo simulation-based application for stochastic dependence modeling with copulas is developed through pair programming involving a human partner and a large language model (LLM) fine-tuned for conversations. This process encompasses interacting with ChatGPT using both natural language and mathematical formalism. Under the careful supervision of a human expert, this interaction facilitated the creation of functioning code in MATLAB, Python, and R. The code performs a variety of tasks including sampling from a given copula model, evaluating the model’s density, conducting maximum likelihood estimation, optimizing for parallel computing on CPUs and GPUs, and visualizing the computed results. In contrast to other emerging studies that assess the accuracy of LLMs like ChatGPT on tasks from a selected area, this work rather investigates ways how to achieve a successful solution of a standard statistical task in a collaboration of a human expert and artificial intelligence (AI). Particularly, through careful prompt engineering, we separate successful solutions generated by ChatGPT from unsuccessful ones, resulting in a comprehensive list of related pros and cons. It is demonstrated that if the typical pitfalls are avoided, we can substantially benefit from collaborating with an AI partner. For example, we show that if ChatGPT is not able to provide a correct solution due to a lack of or incorrect knowledge, the human-expert can feed it with the correct knowledge, e.g., in the form of mathematical theorems and formulas, and make it to apply the gained knowledge in order to provide a correct solution. Such ability presents an attractive opportunity to achieve a programmed solution even for users with rather limited knowledge of programming techniques.

A flexible Clayton-like spatial copula with application to bounded support data

The Gaussian copula is a powerful tool that has been widely used to model spatial and/or temporal correlated data with arbitrary marginal distributions. However, this kind of model can potentially be too restrictive since it expresses a reflection symmetric dependence. In this paper, we propose a new spatial copula model that makes it possible to obtain random fields with arbitrary marginal distributions with a type of dependence that can be reflection symmetric or not.Particularly, we propose a new random field with uniform marginal distributions that can be viewed as a spatial generalization of the classical Clayton copula model. It is obtained through a power transformation of a specific instance of a beta random field which in turn is obtained using a transformation of two independent Gamma random fields.For the proposed random field, we study the second-order properties and we provide analytic expressions for the bivariate distribution and its correlation. Finally, in the reflection symmetric case, we study the associated geometrical properties. As an application of the proposed model we focus on spatial modeling of data with bounded support. Specifically, we focus on spatial regression models with marginal distribution of the beta type. In a simulation study, we investigate the use of the weighted pairwise composite likelihood method for the estimation of this model. Finally, the effectiveness of our methodology is illustrated by analyzing point-referenced vegetation index data using the Gaussian copula as benchmark. Our developments have been implemented in an open-source package for the R statistical environment.

Asymmetric Dependence in Hydrological Extremes

AbstractExtremal dependence describes the strength of correlation between the largest observations of two variables. It is usually measured with symmetric dependence coefficients that do not depend on the order of the variables. In many cases, there is a natural asymmetry between extreme observations that cannot be captured by such coefficients. An example for such asymmetry is large discharges at an upstream and a downstream stations on a river network: an extreme discharge at the upstream station will directly influence the discharge at the downstream station, but not vice versa. Simple measures for asymmetric dependence in extreme events have not yet been investigated. We propose the asymmetric tail Kendall's τ as a measure for extremal dependence that is sensitive to asymmetric behavior in the largest observations. It essentially computes the classical Kendall's τ but conditioned on the extreme observations of one of the two variables. We show theoretical properties of this new coefficient and derive a formula to compute it for existing copula models. We further study its effectiveness and connections to causality in simulation experiments. We apply our methodology to a case study on river networks in the United Kingdom to illustrate the importance of measuring asymmetric extremal dependence in hydrology. Our results show that there is important structural information in the asymmetry that would have been missed by a symmetric measure. Our methodology is an easy but effective tool that can be applied in exploratory analysis for understanding the connections among variables and to detect possible asymmetric dependencies.