Copula-based Model 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.

Two-stage pseudo maximum likelihood estimation of semiparametric copula-based regression models for semi-competing risks data.

We propose a two-stage estimation procedure for a copula-based model with semi-competing risks data, where the non-terminal event is subject to dependent censoring by the terminal event, and both events are subject to independent censoring. With a copula-based model, the marginal survival functions of individual event times are specified by semiparametric transformation models, and the dependence between the bivariate event times is specified by a parametric copula function. For the estimation procedure, in the first stage, the parameters associated with the marginal of the terminal event are estimated using only the corresponding observed outcomes, and in the second stage, the marginal parameters for the non-terminal event time and the copula parameter are estimated together via maximizing a pseudo-likelihood function based on the joint distribution of the bivariate event times. We derived the asymptotic properties of the proposed estimator and provided an analytic variance estimator for inference. Through simulation studies, we showed that our approach leads to consistent estimates with less computational cost and more robustness than the one-stage procedure developed in Chen YH (Lifetime Data Anal 18:36-57, 2012), where all parameters were estimated simultaneously. In addition, our approach demonstrates more desirable finite-sample performances over another existing two-stage estimation method proposed in Zhu H et al., (Commu Statistics-Theory Methods 51(22):7830-7845, 2021) . An R package PMLE4SCR is developed to implement our proposed method.

Enhancing Drought Risk Assessment in the Punjab, Pakistan: A Copula-Based Modeling Approach for Future Projections

Abstract Drought is one of the most complicated and challenging natural hazards, which occurs nearly in every part of the world and poses recurring challenges to agriculture, food security, livestock, human health, and water management. Pakistan has a long history of drought; however, this study focuses on drought analysis and projection in the province of Punjab, Pakistan, as it provides around 60% of the country’s food product, significantly contributing to the national food supply and economy. This study utilized the previous 56 years (1962–2017) of climate data to calculate the reconnaissance drought index (RDI) and then extracted the drought variables of durations and severity for each meteorological station. The best-fit marginal probability distribution and copula models were chosen for the stations based on numerical as well as graphical evaluation. Lognormal and exponential probability distributions, as well as Gumbel, are selected as the best-fit probability distributions for both drought characteristics and bivariate copula model, respectively, for projections. From the projections, we can infer that the smaller return periods indicate high vulnerability while longer return periods with low vulnerability. The results suggest that Faisalabad, Bahawalpur, Bahawalnagar, and Multan stations have the lowest return periods, indicating high vulnerability, and may experience drought more frequently in the future. Mianwali, Khanpur, Lahore, and Sialkot stations may have an intermediate vulnerability to drought events. The stations of Jhelum, Murree, and Sargodha have larger return periods, implying lower susceptibility to drought events in the future. The projected results provide insights for policymakers and stakeholders to optimize the risk of droughts on agriculture production, livestock, water management, human health, and food security in Punjab, Pakistan.

Temporal changes in precipitation and correlation with large climate indicators in the Hengshao Drought Corridor, China

Understanding the lagged correlations between regional precipitation and large scale climate indicators (LCI) is crucial for water management and drought risk assessment. Copula functions are commonly used to quantify the relationship between precipitation and these factors. However, existing studies often neglect monsoon inland areas and lack quantification of the lagged correlations of precipitation at the monthly scale. To address these limitations, this study constructed a 3-dimensional vine copula-based model to assess the impact of large scale climatic change on regional precipitation. The analysis focused on the Hengshao Drought Corridor (HDC), the driest region in Hunan Province, China. The results indicate that precipitation in this region is influenced primarily by the Pacific Ocean’s circulation pattern, particularly the Niño1 + 2 and Niño3 indices, with a 2 ∼ 3 month lag. The R-Vine Copula provided the best fit when identifying lagged correlations, but due to its fitting center, it did not perform as well as the C-Vine Copula in generating stochastic simulated precipitation. The study revealed that the HDC receives more precipitation when the Niño1 + 2 and Niño3 are asynchronous, but less precipitation when they are synchronous, making the area more susceptible to drought events. These findings have implications for the HDC and other monsoon inland regions worldwide, providing insights to cope with climate extremes and develop appropriate strategies.

Spatial copula-based modeling of claim frequency and claim size in third-party car insurance: A Poisson-mixed approach for predictive analysis

The number and amount of claims, referred to as the sum of claims or the total claim/loss amounts in insurance literature, are crucial pieces of information for insurance companies. The analysis of these numerical values can provide essential insights for targeted planning. This study explores a spatial approach for jointly modeling claim frequency and claim size. We assume that the number of accidents follows a Poisson distribution with a variable mean, and this mean, in turn, has a distribution commonly known as a mixed distribution. The spatial dependence structure within the observations is then modeled using an appropriate copula. By estimating the parameters of the proposed model, we draw prediction maps for both claim frequencies and total claim size. These maps will contribute to the prediction of future claim dynamics, offering insurers the opportunity to refine their market strategies and enhance their overall risk management approach based on evolving spatial patterns.

Monitoring bivariate zero-inflated Poisson processes: an alternative to copula-based bivariate attribute control chart

Occurrences of two (or more) types of defects are not uncommon in high quality (or zero-inflated) processes. Bivariate (or multivariate) attribute control charts are required to be developed for monitoring such processes. Fatahi et al. (2012) introduced the idea of applying copula function to derive joint distribution of two correlated zero-inflated Poisson distributions, which can be used to develop bivariate attribute control chart (BACC) for monitoring bivariate zero-inflated Poisson (BZIP) processes. However, the copula-based model often fails to represent bivariate zero-inflated data, which we come across in real-life situations. Li et al. (1999) model for BZIP distribution is quite flexible. Thus, it can overcome the limitations of Fatahi et al. (2012) model. In this paper, BACC is developed based on Li et al. (1999) model. The performances of the proposed BACC, in terms of in-control and out-of-control average run lengths, are evaluated extensively using simulations. The results show that the proposed BACC are well capable of detecting shifts in the parameters of a BZIP process. Further, two case studies on BZIP process data obtained by past researchers reveal that the BACC developed based on Li et al. (1999) model is more effective than the BACC developed based on copula-based model.

Exploring the Relationship and Predictive Accuracy for the Tadawul All Share Index, Oil Prices, and Bitcoin Using Copulas and Machine Learning

Financial markets are increasingly interlinked. Therefore, this study explores the complex relationships between the Tadawul All Share Index (TASI), West Texas Intermediate (WTI) crude oil prices, and Bitcoin (BTC) returns, which are pivotal to informed investment and risk-management decisions. Using copula-based models, this study identified Student’s t copula as the most appropriate one for encapsulating the dependencies between TASI and BTC and between TASI and WTI prices, highlighting significant tail dependencies. For the BTC–WTI relationship, the Frank copula was found to have the best fit, indicating nonlinear correlation without tail dependence. The predictive power of the identified copulas were compared to that of Long Short-Term Memory (LSTM) networks. The LSTM models demonstrated markedly lower Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Scaled Error (MASE) across all assets, indicating higher predictive accuracy. The empirical findings of this research provide valuable insights for financial market participants and contribute to the literature on asset relationship modeling. By revealing the most effective copulas for different asset pairs and establishing the robust forecasting capabilities of LSTM networks, this paper sets the stage for future investigations of the predictive modeling of financial time-series data. The study highlights the potential of integrating machine-learning techniques with traditional econometric models to improve investment strategies and risk-management practices.

Dependence Modelling for Heavy-Tailed Multi-Peril Insurance Losses

The Danish fire loss dataset records commercial fire losses under three insurance coverages: building, contents, and profits. Existing research has primarily focused on the heavy-tail behaviour of the losses but ignored the relationship among different insurance coverages. In this paper, we aim to model the aggregate loss for all three coverages. To study the pairwise dependence of claims from all types of coverage, an independent model, a hierarchical model, and some copula-based models are proposed for the frequency component. Meanwhile, we applied composite distributions to capture the heavy-tailed severity component. It is shown that consideration of dependence for the multi-peril frequencies (i) significantly enhances model goodness-of-fit and (ii) provides more accurate risk measures of the aggregated losses for all types of coverage in total.

Advanced Copula-Based Models for Type II Censored Data: Applications in Industrial and Medical Settings

Copula models are increasingly recognized for their ability to capture complex dependencies among random variables. In this study, we introduce three innovative bivariate models utilizing copula functions: the XLindley (XL) distribution with Frank, Gumbel, and Clayton copulas. The results highlight the fundamental characteristics and effectiveness of these newly introduced bivariate models. Statistical inference for the distribution parameters is conducted using a Type II censored sampling design. This employs maximum likelihood and Bayesian estimation techniques. Asymptotic and credible confidence intervals are calculated, and numerical analysis is performed using the Markov Chain Monte Carlo method. The proposed methodology’s applicability is illustrated by analyzing several real-world datasets. The initial dataset examines burr formation occurrences and consists of two observation sets. Additionally, the second and third datasets contain medical information. The second dataset focuses on diabetic nephropathy, while the third dataset explores infection and recurrence time among kidney patients.

Utility in Time Description in Priority Best–Worst Discrete Choice Models: An Empirical Evaluation Using Flynn’s Data

Discrete choice models (DCMs) are applied in many fields and in the statistical modelling of consumer behavior. This paper focuses on a form of choice experiment, best–worst scaling in discrete choice experiments (DCEs), and the transition probability of a choice of a consumer over time. The analysis was conducted by using simulated data (choice pairs) based on data from Flynn’s (2007) ‘Quality of Life Experiment’. Most of the traditional approaches assume the choice alternatives are mutually exclusive over time, which is a questionable assumption. We introduced a new copula-based model (CO-CUB) for the transition probability, which can handle the dependent structure of best–worst choices while applying a very practical constraint. We used a conditional logit model to calculate the utility at consecutive time points and spread it to future time points under dynamic programming. We suggest that the CO-CUB transition probability algorithm is a novel way to analyze and predict choices in future time points by expressing human choice behavior. The numerical results inform decision making, help formulate strategy and learning algorithms under dynamic utility in time for best–worst DCEs.

A copula-based model to describe the uncertainty of overtopping variables on mound breakwaters

Rising sea levels caused by climate change are increasing the risk of overtopping on coastal structures. Moreover, there is a growing societal concern about the visual impact of these structures, which leads to the lowering of their crest freeboards. In previous studies, safety during overtopping events was assessed considering the overtopping layer thickness (hc), the overtopping flow velocity (uc) and the individual wave overtopping volume (V). Existing models in the literature to estimate hc, uc and V on mound breakwater crests are mainly deterministic, involve a chain of successive estimations leading to accumulated errors and/or do not account for the dependencies between hc, uc and V. This study proposes a model to describe the joint probability distribution of hc, uc and V based on bivariate copulas. Experimental data from small-scale 2D physical tests conducted on mound breakwaters with three armor layers (single-layer Cubipod®, and double-layer cubes and rocks) in depth-limited breaking wave conditions on two mild bottom slopes and dimensionless crest freeboards between 0.33 and 3.20 is used. Lognormal distribution functions are proposed for each variable and a multivariate dependence model is developed through a one-tree vine-copula. The parameters of this model are quantified directly using wave characteristics and the structure geometry minimizing the accumulated errors in the final predictions. The application of the model is illustrated by computing the probability of not fulfilling at least a tolerability limit for one of the studied variables (OR probability). The OR probability is computed both considering the dependence and assuming independence between the variables and a significant difference is obtained. It is concluded that by accounting for the multivariate dependence between the variables, it is possible to reduce the crest freeboard and, thus, achieve a more economic design within the required safety level.

Gaussian Copula Regression Modeling for Marker Classification Metrics with Competing Risk Outcomes

Decisions regarding competing risks are usually based on a continuous-valued marker toward predicting a cause-specific outcome. The classification power of a marker can be summarized using the time-dependent receiver operating characteristic curve and the corresponding area under the curve (AUC). This paper proposes a Gaussian copula-based model to represent the joint distribution of the continuous-valued marker, the overall survival time, and the cause-specific outcome. Then, it is used to characterize the time-varying ROC curve in the context of competing risks. Covariate effects are incorporated by linking linear components to the skewed normal distribution for the margin of the marker, a parametric proportional hazards model for the survival time, and a logit model for the cause of failure. Estimation is carried out using maximum likelihood, and a bootstrap technique is implemented to obtain confidence intervals for the AUC. Information-criteria strategies are employed to find a parsimonious model. The performance of the proposed model is evaluated in simulation studies, considering different sample sizes and censoring distributions. The methods are illustrated with the reanalysis of a prostate cancer clinical trial. The joint regression strategy produces a straightforward and flexible model of the time-dependent ROC curve in the presence of competing risks, enhancing the understanding of how covariates may affect the discrimination of a marker.

Application of Copula Methods in Financial Risk Management: Case of the Zimbabwe Stock Exchange and the Victoria Falls Stock Exchange.

In order to minimize risks and create a safe investing environment, financial risk management is becoming more and more crucial for individuals, financial organizations, and even entire nations. Accurately assessing financial risks and using that information to inform wise investment choices can give an investor a competitive edge as well as significant returns. In actuality, real-world financial variables limit the ability to estimate financial risks. On the other hand, a wealth of data indicates that financial variables typically have asymmetric dependency, skewness, and fat tails. In three ways, the conventional approaches to financial risk management based on normally distributed hypotheses are put to the test by these stylized characteristics of financial variables. First, the univariate normal distribution or other elliptical distributions are unable to adequately fit the distribution of the univariate variable. Second, despite their straightforward tractability, multivariate variables’ extra kurtosis and skewness are not captured by their normal distribution. As a result, the dependence risks associated with multivariate financial variables may be underestimated. Finally, when the joint distribution of various variables is non-elliptical, linear correlation which is typically employed to characterize the dependence of various variables in traditional portfolio risk management is likewise insufficient. This research uses a promising technique based on copulas in conjunction with GARCH and realized volatility models to examine the risks associated with multivariate financial variables in order to address these issues. The multivariate distributions are constructed using copulas in conjunction with GARCH and Realized Volatility models which are then utilized to evaluate portfolio risks in financial market. The findings demonstrate that copula-based models outperform conventional models in fitting financial data. Second, a variety of marginal models have a notable impact on the value at risk of the portfolio, including the GARCH and realized volatility models. Lastly, both the dependence structure and the marginal distribution exhibit notable skewness. Consequently, compared to the normal or Student-t distribution, the skewed Student-t distribution fits some datasets better.

A Comparative Perspective on Multivariate Modeling of Insurance Compensation Payments with Regression-Based and Copula-Based Models

A Comparative Perspective on Multivariate Modeling of Insurance Compensation Payments with Regression-Based and Copula-Based Models

Machine learning and copula-based analysis of past changes in global droughts and socioeconomic exposures

The intensification of the hydrological cycle has altered the spatiotemporal redistribution of terrestrial water storage (TWS). The long-term evolutionary mechanism controlling global TWS droughts and their bivariate risks (i.e., drought and severity) remain uncertain. Using machine learning-based TWS reconstructions, we explored the drivers, changes, and impacts of TWS droughts during 1940–2022 at the global scale. We developed a machine learning framework to detect the dominant climate/vegetation factors governing TWS. During 1940–1970, precipitation and vapor pressure deficit were the primary factors influencing TWS; however, the leaf area index was the dominant factor during 1992–2022. We evaluated past changes in drought frequency, duration, severity, and intensity, and found a substantial intensification tendency in most land areas. Subsequently, we evaluated the bivariate risks by combing a copula-based modeling approach and the most likely realization method, revealing a fivefold intensification over most regions. Changes in the marginal distributions of duration and severity accounted for 40–60% of the overall changes in bivariate drought risk, while the contribution from their dependence varied globally. Approximately 80–90% of the global population and gross domestic product were exposed to increasing bivariate drought risk, indicating the need to improve the adaptivity of society and ecosystems to climate change.

Copula-based modeling and simulation of 3D systems of curved fibers by isolating intrinsic fiber properties and external effects

In this paper we lay the foundation for data-driven 3D analysis of virtual fiber systems with respect to their microstructure and functionality. In particular, we develop a stochastic 3D model for systems of curved fibers similar to nonwovens, which is fitted to tomographic image data. By systematic variations of model parameters, efficient computer-based scenario analyses can be performed to get a deeper insight how effective properties of this type of functional materials depend on their 3D microstructure. In a first step, we consider single fibers as polygonal tracks which can be modeled by a third-order Markov chain. For constructing the transition function of the Markov chain, we formalize the intuitive notions of intrinsic fiber properties and external effects and build a copula-based transition function such that both aspects can be varied independently. Using this single-fiber model, in a second step we derive a model for the entire fiber system observed in a bounded sampling window and fit it to two different 3D datasets of nonwovens measured by CT imaging. Considering various geometric descriptors of the 3D microstructure related to effective properties of the pore space, we evaluate the goodness of model fit by comparing geometric descriptors of the 3D morphology of model realizations with those of tomographic image data.

A time-varying Copula-based approach to quantify the effects of antecedent drought on hot extremes

Hot extremes may adversely impact human health and agricultural production. Owing to anthropogenic and climate changes, the close and dynamic interaction between drought and hot extremes in most areas of China need to be revisited from the perspective of nonstationarity. This study therefore proposes a time-varying Copula-based model to describe the nonstationary dependence structure of summer extreme temperature (SET) and antecedent soil moisture condition to quantify the dynamic risk of hot extremes conditioned on dry/wet condition. A general statistical inspection procedure which was composed of maximum likelihood (ML)-based estimation, nonstationary Goodness-of-fit (GOF) tests, the log likelihood ratio (LR) tests and minimum corrected Akaike information criterion-based selection was promoted to select the best-fitted nonstationary models efficiently. This study proposed a new approach to identify the soil moisture driving law over extreme temperature from the point view of tail monotonicity and nonstationary risk assessment. Owing to the LTI-RTD (left tail increasing and right tail decreasing) tail monotonicity for dependence structure of these two extremes derived from most areas, two kinds of the driving laws of soil moisture over SET were detected. Because of the spatiotemporal divergence of sensitivity index derived from tail monotonicity (SITM), we can conclude that the spatial and temporal heterogeneity of response degree of ET over the variations of antecedent dry/wet conditions is evident. Incorporation of nonstationarity and tail monotonicity helps identify the changes of driving mechanism (laws) between soil moisture and hot extremes.

Multivariate probability analysis of wind-wave actions on offshore wind turbine via copula-based analysis

Floating structures in service encounter extreme natural hazards, such as correlated extreme wind-wave loads. This study aims to provide realistic multivariate models of environmental parameters to accurately describe the statistical characteristics of metoceanic conditions, which are essential for the reliability-based design of marine structures. The multiload analysis method includes three major steps: (1) to identify the best-fit univariate marginal distributions of the wind-wave parameters. Mixed models composed of a set of single-mode parametric distributions were found to be more suitable for the measured variables. (2) To construct joint probability distributions of multiple variables using copula-based models, the performance of various models namely, trivariate metaelliptical copulas, Archimedean copulas, and vine copula models. It was observed that joint distributions can be best modelled using vine copula models with considerable flexibility. The final process involves establishing environmental surfaces and contours based on the inverse first-order reliability method. The most unfavourable combination of extreme design conditions can be defined on the environmental surface which is useful for evaluating the response characteristics of offshore wind turbines. These expressions provide a preliminary assessment of wind-wave loads on offshore structures.

Dynamic factor copula-based modeling for market risk optimization with an application to the real industry in China

As finance returns to its fundamental purpose of serving the real economy, its connections with various industries are strengthening. Accurately depicting the interdependence among these industries and mitigating financial risks has become increasingly critical. The dependence among China's real industries is dynamic rather than static, which is particularly pronounced during the COVID-19 pandemic. In this paper, we propose a dynamic factor model to optimize the risk of high-dimensional portfolios. To describe the dependence structure, we employ the factor copula model, driven by a GAS (Generalized Autoregressive Score) model. By combining the dynamic factor model with a mean-ES (Expected Shortfall) model, we construct a dynamic factor copula-mean-ES model. Our empirical findings, based on an analysis of 24 industries in China, suggest that the dynamic heterogeneous factor copula model is the most suitable for describing portfolio risk. Furthermore, the mean-ES model ensures the lowest portfolio risk for a given expected return. Accurate return predictions enable leveraging market information to develop a "good knowledge" of dynamic copula and risk optimization. This "good knowledge" of dynamic copulas facilitates precise return prediction and effective risk optimization of portfolios, thereby addressing the relationship between risk prevention and sustainability. Moreover, it reveals the internal connection between China's real industry and the risk landscape of the financial market.

Prediction of Rapid Early Progression and Survival Risk with Pre-Radiation MRI in WHO Grade 4 Glioma Patients.

Recent clinical research describes a subset of glioblastoma patients that exhibit REP prior to the start of radiation therapy. Current literature has thus far described this population using clinicopathologic features. To our knowledge, this study is the first to investigate the potential of conventional radiomics, sophisticated multi-resolution fractal texture features, and different molecular features (MGMT, IDH mutations) as a diagnostic and prognostic tool for prediction of REP from non-REP cases using computational and statistical modeling methods. The radiation-planning T1 post-contrast (T1C) MRI sequences of 70 patients are analyzed. An ensemble method with 5-fold cross-validation over 1000 iterations offers an AUC of 0.793 ± 0.082 for REP versus non-REP classification. In addition, copula-based modeling under dependent censoring (where a subset of the patients may not be followed up with until death) identifies significant features (p-value < 0.05) for survival probability and prognostic grouping of patient cases. The prediction of survival for the patients' cohort produces a precision of 0.881 ± 0.056. The prognostic index (PI) calculated using the fused features shows that 84.62% of REP cases fall under the bad prognostic group, suggesting the potential of fused features for predicting a higher percentage of REP cases. The experimental results further show that multi-resolution fractal texture features perform better than conventional radiomics features for prediction of REP and survival outcomes.