Copula-based Model Research Articles

On copula-based collective risk models: from elliptical copulas to vine copulas

Several collective risk models have recently been proposed by relaxing the widely used but controversial assumption of independence between claim frequency and severity. Approaches include the bivariate copula model, random effect model, and two-part frequency-severity model. This study focuses on the copula approach to develop collective risk models that allow a flexible dependence structure for frequency and severity. We first revisit the bivariate copula method for frequency and average severity. After examining the inherent difficulties of the bivariate copula model, we alternatively propose modeling the dependence of frequency and individual severities using multivariate Gaussian and t-copula functions. We also explain how to generalize those copulas in the format of a vine copula. The proposed copula models have computational advantages and provide intuitive interpretations for the dependence structure. Our analytical findings are illustrated by analyzing automobile insurance data.

CODC: a Copula-based model to identify differential coexpression

Differential coexpression has recently emerged as a new way to establish a fundamental difference in expression pattern among a group of genes between two populations. Earlier methods used some scoring techniques to detect changes in correlation patterns of a gene pair in two conditions. However, modeling differential coexpression by means of finding differences in the dependence structure of the gene pair has hitherto not been carried out. We exploit a copula-based framework to model differential coexpression between gene pairs in two different conditions. The Copula is used to model the dependency between expression profiles of a gene pair. For a gene pair, the distance between two joint distributions produced by copula is served as differential coexpression. We used five pan-cancer TCGA RNA-Seq data to evaluate the model that outperforms the existing state of the art. Moreover, the proposed model can detect a mild change in the coexpression pattern across two conditions. For noisy expression data, the proposed method performs well because of the popular scale-invariant property of copula. In addition, we have identified differentially coexpressed modules by applying hierarchical clustering on the distance matrix. The identified modules are analyzed through Gene Ontology terms and KEGG pathway enrichment analysis.

Prediction of risk: Decoding the Serial Dependence of Stock Return Volatility with Copula

In this article, a new approach based on the copula theory is employed in the analysis and forecasting of hospitality and tourism-related stock return volatility (HTSRV). The application of copula-based models for univariate time series is state-of the-art methodologies with new perspectives for economic analysis in tourism and hospitality. This flexible method provides numerous functions for serial dependence specification of volatility series. Eight hospitality–tourism stocks are analyzed for their serial dependence structures to provide insight for forecasting stock return volatility. While the forecasting performance between our chosen copulas and benchmark models is inconclusive, the empirical results show that copulas well specify both linear and nonlinear serial dependence structures, which lead to forecasting results as good as or even better than those of the benchmark models. This property allows us to use copulas for HTSRV forecasting without the concern of model misspecification.

Measurement of Systemic Risk in Global Financial Markets and Its Application in Forecasting Trading Decisions

The global financial crisis in 2008 spurred the need to study systemic risk in financial markets, which is of interest to both academics and practitioners alike. We first aimed to measure and forecast systemic risk in global financial markets and then to construct a trade decision model for investors and financial institutions to assist them in forecasting risk and potential returns based on the results of the analysis of systemic risk. The factor copula-generalized autoregressive conditional heteroskedasticity (GARCH) models and component expected shortfall (CES) were combined for the first time in this study to measure systemic risk and the contribution of individual countries to global systemic risk in global financial markets. The use of factor copula-based models enabled the estimation of joint models in stages, thereby considerably reducing computational burden. A high-dimensional dataset of daily stock market indices of 43 countries covering the period 2003 to 2019 was used to represent global financial markets. The CES portfolios developed in this study, based on the forecasting results of systemic risk, not only allow spreading of systemic risk but may also enable investors and financial institutions to make profits. The main policy implication of our study is that forecasting systemic risk of global financial markets and developing portfolios can provide valuable insights for financial institutions and policy makers to diversify portfolios and spread risk for future investments and trade.

Probabilistic prediction of solar power supply to distribution networks, using forecasts of global horizontal irradiation

This paper presents a mathematical model for the prediction of the probabilities of reverse power flow exceeding predefined critical thresholds at feed-in points of a distribution network. The parametric prediction model is based on hourly forecasts of global horizontal irradiation and uses copulas, a tool for modeling the joint probability distribution of two or more strongly correlated random variables with non-Gaussian (marginal) distributions. The model is used for determining the joint distribution of forecasts of global horizontal irradiation and measured solar power supply at given feed-in points, where respective sample datasets were provided by Deutscher Wetterdienst and the N-ERGIE Netz GmbH. It is shown that the fitted model replicates important characteristics of the data such as the corresponding marginal densities. The validation results highlight strong performance of the proposed model. The copula-based model enables to predict the distribution of solar power supply conditioned on the forecasts of global horizontal irradiation, thus anticipating great fluctuations in the distribution network.

Gaussian copula joint models to analysis mixed correlated longitudinal count and continuous responses

Gaussian Copula joint models for analyzing mixed correlated longitudinal continuous and count responses with random effects are presented where the count response is defined by a latent variable approach, and its distribution is a member of the power series family of distributions. A copula-based joint model is proposed that accounts for associations between count and continuous responses. A full likelihood-based inference method for estimation is used by which maximum likelihood estimation of parameters is obtained. To illustrate the utility of the models, some simulation studies are performed. Finally, the proposed models are motivated by analyzing a medical data set where the correlated responses are the number of joint damaged obtained from the severity of osteoporosis (count response) and Body Mass Index (continuous response). Effects of some covariates on responses are investigated simultaneously. Besides, identifiability of parameters, detection of outlines by examining residuals and sensitivity analyses are fully investigated.

Fully and empirical Bayes approaches to estimating copula-based models for bivariate mixed outcomes using Hamiltonian Monte Carlo

We provide a fully Bayesian approach to conduct estimation and inference for a copula model to jointly analyze bivariate mixed outcomes. To obtain posterior samples, we use Hamiltonian Monte Carlo, which avoids the random walk behavior of Metropolis and Gibbs sampling algorithms. We also provide an empirical Bayes approach to estimate the copula parameter, which is useful when prior specification on that parameter is difficult. We further propose the use of Bayesian model selection criteria to select the most appropriate copula family. We conduct simulation studies to compare the two approaches and to examine copula selection performance and illustrate the application of the fully Bayesian approach on a burn injury data set.

A diagnostic test for specification of copulas under censorship

We propose a copula specification test for copula-based multivariate survival models under censorship. This flexible test is applicable to both Archimedean and non-Archimedean copulas and provides useful pointers for alternative copula construction when a null distribution is rejected. It is shown to be consistent and asymptotically distribution free. We demonstrate its good finite sample performance via Monte Carlo simulations. We apply this test to an insurance dataset on losses and expenses. Our test rejects the hypothesis of Gaussian copula. Furthermore, its diagnostic information suggests an alternative copula specification that captures the extreme-value dependence exhibited in the data.

A factorial Bayesian copula framework for partitioning uncertainties in multivariate risk inference

In this study, a factorial Bayesian copula (FBC) method is proposed to quantify parameter uncertainties in copula-based models and then reveal their impacts on hydrologic risk inferences within a multivariate context. In detail, Bayesian inference and factorial analysis are integrated into copula-based multivariate risk models to (1) quantify parameter uncertainties, (ii) reveal their individual and interactive effects, and (iii) identify their detailed contributions to uncertain risk inferences. Streamflow observations at Xiangxi and Wei River basins is China are used to illustrate the applicability of FBC. The results indicate that imprecise parameters in marginal distributions and the dependence structure would lead to extensive uncertainties in predictive joint return periods and failure probabilities. Also, individual and interactive effects of parameters are well revealed through multilevel factorial analysis, and the detailed contributions of one parameter to different failure probabilities under different service time scenarios are identified.

A Bayesian hierarchical copula model

Dependent data with hierarchical structures arises commonly from a variety of application, and analysis of such data is often challenging due to the complexity in modeling dependence structures and the computation intensity. In this paper, we propose a Bayesian hierarchical copula model (BHCM) to accommodate hierarchical structures of dependent data, where the subject-level dependence is modeled by the copula-based model and the hierarchical structure is described using random dependence parameters. We introduce a layer-by-layer sampling scheme for conducting Bayesian inferences. Our proposed BHCM enjoys the flexibility of modeling various complex association structures, while retaining manageable computation. Extensive simulation studies show that our proposed estimators outperform conventional likelihood-based estimators in a variety of finite sample settings. We apply the BHCM to analyze the Vertebral Column dataset arising from UCI Machine Learning Repository.

Multivariate Type G Matérn Stochastic Partial Differential Equation Random Fields

SummaryFor many applications with multivariate data, random-field models capturing departures from Gaussianity within realizations are appropriate. For this reason, we formulate a new class of multivariate non-Gaussian models based on systems of stochastic partial differential equations with additive type G noise whose marginal covariance functions are of Matérn type. We consider four increasingly flexible constructions of the noise, where the first two are similar to existing copula-based models. In contrast with these, the last two constructions can model non-Gaussian spatial data without replicates. Computationally efficient methods for likelihood-based parameter estimation and probabilistic prediction are proposed, and the flexibility of the models suggested is illustrated by numerical examples and two statistical applications.

Parametric and semiparametric copula-based models for the regression analysis of competing risks

Copula-based constructions of the joint distribution of the overall survival time and the cause-specific failure are developed for the analysis of competing risks data. Covariate effects are incorporated by characterizing the margins with parametric and semiparametric proportional hazards models for the survival times, and a logistic regression model for the cause of failure. Appealing aspects of the methods are the use of single-outcome techniques for the adequate modeling of the proportional hazards component, the ability to find parsimonious models using information-criterion metrics, and the inherent simplicity and usefulness of the interpretation of the parameters. Estimation is carried out using maximum likelihood and two-stage pseudolikelihood procedures for the parametric and semiparametric models, respectively. The performance of the estimators is evaluated in simulation studies. The methods are illustrated with the reanalysis of a follicular cell lymphoma dataset.

Copula-based modeling of earthen levee breach due to overtopping

The level of protection offered by an earthen levee is typically described in terms of flood water level that the levee is capable of containing. If a larger flood occurs, floodwaters exceed the height of the levee and flow over its crest. As the water passes over the top, it may erode the levee, worsening the flooding and potentially causing a breach. In order to determine the annual probability that an earthen levee breaches due to overtopping, multiple flood characteristics such as peak flood water level, or related peak flow discharge, and flood duration need to be characterized statistically by using multivariate statistics. In this study, critical conditions for levee failure are described by using a Clayton copula relating peak flow discharge to flood duration. The obtained model is tested over a real river site located along the Panaro River, in northern Italy, where a 52-year time series of hourly flow discharge and a normal flow rating curve are available. The developed model makes it possible to delimitate the levee failure region within the population of flood events and to statistically describe earthen levee breach due to overtopping. Breach probability is found to be underestimated when the statistical association between peak flow discharge and flood duration is neglected. The proposed copula-based model is therefore important to support the design and construction of earthen levees, and to identify the actions needed to save lives and property when a flood exceeding the levee design limit occurs.

Abstract # 4281 Subjective social standing is related to inflammation during pregnancy

In this chapter, we review the problem of modeling correlated count data. Among the several methods that can be used for this scope, we focus on the copula approach, illustrating its advantages, but also possible limitations and issues arising in the discrete context if compared to the continuous case. After introducing the basic notions about copulas, the construction of a multivariate joint distribution is discussed and pseudorandom simulation and point estimation of copula-based models for count data are then outlined. Results related to minimum and maximum correlation between two assigned discrete marginal distributions are also described and put in connection with the choice of the copula to be used for modeling correlated counts. A numerical example and an application to a real dataset are provided.

A Deep Learning-Based Feature Extraction Framework for System Security Assessment

The ongoing decarbonisation of modern electricity systems has led to a substantial increase of operational uncertainty, particularly due to the large-scale integration of renewable energy generation. However, the expanding space of possible operating points renders necessary the development of novel security assessment approaches. In this paper we focus on the use of security rules where classifiers are trained offline to characterize previously unseen points as safe or unsafe. This paper proposes a novel deep learning-based feature extraction framework for building security rules. We show how deep autoencoders can be used to transform the space of conventional state variables (e.g., power flows) to a small number of dimensions where we can optimally distinguish between safe and unsafe operation. The proposed framework is data-driven and can be useful in multiple applications within the context of security assessment. To achieve high accuracy, a novel objective-based loss function is proposed to address the issue of imbalanced safe/unsafe classes that characterize electricity system operation. Furthermore, an R-vine copula-based model is proposed to sample historical data and generate large populations of anticipated system states for training. The superior performance of the proposed framework is demonstrated through a series of case studies and comparisons using the load and wind generation data from the French transmission system, which have been mapped to the IEEE 118-bus system.

Modeling the concurrent impact of extreme rainfall and reservoir storage on Kerala floods 2018: a Copula approach

This study investigates the concurrent impact of reservoir storage and extreme rainfall on the devastating Kerala floods 2018 using Copulas. First, the effect of storage of four major reservoirs Idukki, Idamalayar, Kakki and Kallada is investigated by developing storage curves. It is noticed that in the four reservoirs, the storage level was above the decadal mean storage in most of the days of July–August period of 2018. An analysis on the storage levels of the four reservoirs showed that the water level crossed the spillway crest level of the respective reservoirs by July 26, 2018 and the controlled release might have helped in alleviating the flood disasters. In the Copula-based modeling, first, the marginal distribution of monthly storage and rainfall of the four reservoirs are developed and it was found that the generalized Pareto (GP) distribution was fitting best for all the storage data. The exponential fit was found to be the best for rainfall of all reservoirs except that of Kakki. Subsequently, the joint probability of reservoir storage and rainfall is investigated using three popular Copulas of Archimedean family namely Gumbel–Hougaard (GH), Clayton and Frank and the best Copula is identified for each case. The return period curves are developed for both joint and conditional cases using the best fitted Copula. The sensitivity analysis of joint return period showed that the rainfall of Kakki and Idukki at 2018 was extreme and the effect of rainfall is dominant in these reservoirs.

Estimation under copula-based Markov normal mixture models for serially correlated data

We propose an estimation method under a copula-based Markov model for serially correlated data. Motivated by the fat-tailed distribution of financial assets, we select a normal mixture distribution for the marginal distribution. Based on the normal mixture distribution for the marginal distribution and the Clayton copula for serial dependence, we obtain the corresponding likelihood function. In order to obtain the maximum likelihood estimators, we apply the Newton-Raphson algorithm with appropriate transformations and initial values. We conduct simulation studies to evaluate the performance of the proposed method. In the empirical analysis, the stock price of Dow Jones Industrial Average is analyzed for illustration.

Geostatistical Analysis with Copula-based Models of Madograms, Correlograms and Variograms

This paper investigates models of stochastic dependence with geostatistical tools. Specifically, we use copulas to propose new models of stochastic spatial tools such as variograms, correlograms and the madograms. Copula versions of covariograms are provided both in second stationnary and intrinsic frameworks. Moreover, some usual families of models of variograms are claried with the corresponding parameters.

Bivariate probabilistic quantification of drought impacts on terrestrial vegetation dynamics in mainland China

Frequent droughts in a warming climate may exert more negative influences on ecosystems. Unlike previous studies that have investigated the vegetation response to drought generally in a deterministic way, a copula-based model is developed for quantifying drought impacts on terrestrial vegetation and identifying drought-vulnerable regions for mainland China from a probabilistic perspective in this study. The Normalized Difference Vegetation Index (NDVI) is firstly correlated with the Standardized Precipitation Evapotranspiration Index (SPEI) at varying timescales from 1 month to 24 months to determine the response time of vegetation to water variability. Then, the dependence structure of vegetation vigor and water availability is modeled through the bivariate copula analysis. Furthermore, conditional probabilities of vegetation decline under moderate, severe and extreme drought scenarios are systematically estimated using copula-based conditional probability distributions. Results indicate that spatial patterns of vegetation response time present distinct seasonality, with faster response to water variation in the southern part than in the northern proportion of mainland China during the non-growing season while the inverse pattern is observed for the growing season. The higher conditional probabilities of the below-average vegetation status in the dry condition evidence that water deficits overwhelming water surplus play a more profound role in diminishing vegetation vigor across more than 80% of mainland China. Specifically, when moderate droughts develop into extreme ones, the average probability of vegetation status below the 50th percentile escalates by 6.9%. Moreover, extreme droughts are noted to exaggerate probabilities of vegetation activity falling below the increasingly lower (40th, 30th, 20th and 10th) percentiles by 8.8%, 10.8%, 12.7% and 13.7% in comparison with moderate counterparts, thereby suggesting higher likelihood of the deteriorating drought conditions inducing vegetation losses, especially major vegetation decline. As for the drought-vulnerable region, North China, particularly the central Inner Mongolia, is recognized with vegetation decline probabilities being 28.1% and 68.8% greater than the mainland average given drought conditions (quantified as SPEI ≤ −1), respectively. Results of the study may improve our understanding of climatic extreme influence on vegetation status and benefit the effective drought preparedness and mitigation.

Periodic Copula Autoregressive Model Designed to Multivariate Streamflow Time Series Modelling

It is a challenge to develop models that can represent the stochastic behaviour of rivers and basins. Currently used streamflow models were constructed under rigid hypotheses. Hence, these models are limited in their ability to represent nonlinear dependencies and/or unusual distributions. Copulas help overcome these limitations and are being employed widely for modelling hydrological data. For instance, pure copula-based models have been proposed to simulate univariate hydrological series. However, there have been few studies on the use of copulas to model multivariate inflow series. Thus, the aim of this study is to develop a pure copula-based model for simulating periodic multivariate streamflow scenarios, wherein temporal and spatial dependencies are considered. The model was employed in a set of 11 affluent natural energy series from Brazil. We used the model to simulate many scenarios and analyze them through statistical tests such as Levene’s test, the Kolmogorov-Smirnov test, Kupiec test, and t-test. In addition, we investigated the spatial and temporal dependence of the scenarios. Finally, the critical periods of the simulated scenarios were investigated. The results indicated that the proposed model is capable of simulating scenarios that preserve historical features observed in the original data.