Copula-based Model Research Articles

Copula-based modeling and stochastic simulation of seasonal intermittent streamflows for arid regions

Streamflow is often intermittent in arid and semi-arid regions. Stochastically simulated data play a key role in managing water resources with intermittent streamflows. The stochastic modeling of intermittent streamflow that incorporates the seasonality of key statistics is a difficult task. In the current study, the product model was tested to simulate the intermittent monthly streamflow by employing the periodic Markov chain (PMC) model for occurrence and the periodic gamma autoregressive (PGAR) and copula models for amount. The copula models were tested in a previous study for the simulation of yearly streamflow, resulting in successful replication of the key and operational statistics of historical data; however, the copula models have never been tested on a monthly time scale. The intermittent models were applied to the Colorado River system in the present study. A few drawbacks of the PGAR model were identified, such as significant underestimation of minimum values on an aggregated yearly time scale and restrictions of the parameter boundaries. Conversely, the copula models do not present such drawbacks but show feasible reproduction of key and operational statistics. We concluded that the copula models combined with the PMC model is a feasible method for the simulation of intermittent monthly streamflow time series.

Insurance ratemaking using a copula-based multivariate Tweedie model

The Tweedie distribution, featured with a mass probability at zero, is a convenient tool for insurance claims modeling and pure premium determination in general insurance. Motivated by the fact that an insurance policy typically provides multiple types of coverage, we propose a copula-based multivariate Tweedie regression for modeling the semi-continuous claims while accommodating the association among different types. The proposed approach also allows for dispersion modeling, resulting in a multivariate version of the double generalized linear model. We demonstrate the application in insurance ratemaking using a portfolio of policyholders of automobile insurance from the state of Massachusetts in the United States.

Composite likelihood for joint analysis of multiple multistate processes via copulas

A copula-based model is described which enables joint analysis of multiple progressive multistate processes. Unlike intensity-based or frailty-based approaches to joint modeling, the copula formulation proposed herein ensures that a wide range of marginal multistate processes can be specified and the joint model will retain these marginal features. The copula formulation also facilitates a variety of approaches to estimation and inference including composite likelihood and two-stage estimation procedures. We consider processes with Markov margins in detail, which are often suitable when chronic diseases are progressive in nature. We give special attention to the setting in which individuals are examined intermittently and transition times are consequently interval-censored. Simulation studies give empirical insight into the different methods of analysis and an application involving progression in joint damage in psoriatic arthritis provides further illustration.

Estimation procedures for exchangeable Marshall copulas with hydrological application

Complex phenomena in environmental sciences can be conveniently represented by several inter-dependent random variables. In order to describe such situations, copula-based models have been studied during the last year. In this paper, we consider a novel family of bivariate copulas, called exchangeable Marshall copulas. Such copulas describe both positive and (upper) tail association between random variables. Specifically, inference procedures for the family of exchangeable Marshall copulas are introduced, based on the estimation of their (univariate) generator. Moreover, the performance of the proposed methodologies is shown in a simulation study. Finally, an illustration describes how the proposed procedures can be useful in a hydrological application.

Copula-based modeling of degree-correlated networks

Dynamical processes on complex networks such as information exchange, innovation diffusion, cascades in financial networks or epidemic spreading are highly affected by their underlying topologies as characterized by, for instance, degree–degree correlations. Here, we introduce the concept of copulas in order to generate random networks with an arbitrary degree distribution and a rich a priori degree–degree correlation (or ‘association’) structure. The accuracy of the proposed formalism and corresponding algorithm is numerically confirmed, while the method is tested on a real-world network of yeast protein–protein interactions. The derived network ensembles can be systematically deployed as proper null models, in order to unfold the complex interplay between the topology of real-world networks and the dynamics on top of them.

A copula-based precipitation forecasting model: Investigating the interdecadal modulation of ENSO's impacts on monthly precipitation

[1] The influence of two large-scale circulation patterns (the El Nino Southern Oscillation (ENSO) and the Pacific Decadal Oscillation (PDO)), and the effect of the interdecadal modulation of ENSO on precipitation in the state of Texas, U.S., was explored. Texas, by virtue of its size, topography, and geographical location, spans a wide range of climatic regions. The state is divided into 10 climate divisions. The precipitation pattern in each division follows different probability distributions. The climate regimes which trigger this difference are discussed. The seasonal correlation between ENSO and PDO with precipitation anomaly in each climate division was established. Copula-based models were developed to examine the dependence structure between the large-scale climate indices and average monthly seasonal precipitation. The choice of copula is discussed in light of the dependence structure. The selected copulas were then used to simulate precipitation anomalies in three climate divisions: one which has a semiarid climate, one located in the wettest region, and one straddling the subtropical humid and subtropical subhumid regions of the state. The statistical performance of bivariate models for ENSO and precipitation, and trivariate models for ENSO, PDO, and precipitation, in simulating precipitation anomalies were compared. In general, inclusion of PDO was found to improve simulation results. The most notable improvement was in simulating negative precipitation anomalies during La Nina and negative PDO. The copula models were also tested for their abilities to predict precipitation anomalies in these three regions. Again, the trivariate models performed better, especially in predicting droughts due to La Nina and negative PDO.

Copulaedas: AnRPackage for Estimation of Distribution Algorithms Based on Copulas

The use of copula-based models in EDAs (estimation of distribution algorithms) is currently an active area of research. In this context, the copulaedas package for R provides a platform where EDAs based on copulas can be implemented and studied. The package offers complete implementations of various EDAs based on copulas and vines, a group of well-known optimization problems, and utility functions to study the performance of the algorithms. Newly developed EDAs can be easily integrated into the package by extending an S 4 class with generic functions for their main components. This paper presents copulaedas by providing an overview of EDAs based on copulas, a description of the implementation of the package, and an illustration of its use through examples. The examples include running the EDAs defined in the package, implementing new algorithms, and performing an empirical study to compare the behavior of different algorithms on benchmark functions and a real-world problem.

A Unifying Framework for Marginalised Random‐Intercept Models of Correlated Binary Outcomes

We demonstrate that many current approaches for marginal modeling of correlated binary outcomes produce likelihoods that are equivalent to the copula-based models herein. These general copula models of underlying latent threshold random variables yield likelihood-based models for marginal fixed effects estimation and interpretation in the analysis of correlated binary data with exchangeable correlation structures. Moreover, we propose a nomenclature and set of model relationships that substantially elucidates the complex area of marginalized random intercept models for binary data. A diverse collection of didactic mathematical and numerical examples are given to illustrate concepts.

Nonlinear Modeling of Temporal Wind Power Variations

Modeling wind speed time series (WSTS) is an essential part of network planning studies in order to generate synthetic wind power time series (WPTS). Hence, this paper proposes a methodology to equip planners with accurate simulation of wind speed and power variations as well as complete temporal dependence structure based on the copula theory. Unlike traditional autoregressive and Markov chain methods, the suggested technique is well-prepared to deal with “nonlinear long-memory temporal dependence” and “non-Gaussian empirical probability distributions” of the WSTS. Meanwhile, the proposed statistical modeling framework is compatible with the scenario-based analysis of active networks as well. Furthermore, a case study for optimal sizing of an autonomous wind/photovoltaic/battery system is presented. The purpose of the presented study is to fully examine the accuracy and effectiveness of the copula-based model of wind generation for planning nonmemoryless power systems. Among a list of commercially available system devices, the optimal number and type of units are calculated ensuring both a minimum 20-year round total system cost and a perfect reliability. The genetic algorithm is used in four wind generation scenarios consisting of real and simulated WPTS. Then, considering the corresponding optimal solutions, the autoregressive moving average (ARMA), nonparametric Markov and proposed copula-based simulations are compared against real data.

Total loss estimation using copula-based regression models

We present a joint copula-based model for insurance claims and sizes. It uses bivariate copulae to accommodate for the dependence between these quantities. We derive the general distribution of the policy loss without the restrictive assumption of independence. We illustrate that this distribution tends to be skewed and multi-modal, and that an independence assumption can lead to substantial bias in the estimation of the policy loss. Further, we extend our framework to regression models by combining marginal generalized linear models with a copula. We show that this approach leads to a flexible class of models, and that the parameters can be estimated efficiently using maximum-likelihood. We propose a test procedure for the selection of the optimal copula family. The usefulness of our approach is illustrated in a simulation study and in an analysis of car insurance policies.

Analytic Nakagami fading parameter estimation in dependent noise channel using copula

Abstract In this paper, the probability density function (PDF) estimation is introduced in the framework of estimating the Nakagami fading parameter. This approach provides an analytic procedure for finding the fading parameter. Using the copula theory, an accurate PDF estimate is obtained even when the desired signal is corrupted in a noisy environment. In the real world, the noise samples could be highly dependent on the main signal. Copula-based models are a general set of statistical models defined for any multivariate random variable. Thus, they depict the statistical behavior of a received signal including two dependent terms, representative of the desired signal and noise. Previous works in the Nakagami parameter determination have mainly examined estimation based on either a noiseless sample model or an independent trivial noisy one. In this paper, we consider a more comprehensive situation about the noise destruction and our investigation is done in low signal-to-noise ratios. The parametric bootstrap method approves the accuracy of the analytically estimated PDF, and simulation results show that the new estimator has superior performance over conventional estimators.

On the estimation of normal copula discrete regression models using the continuous extension and simulated likelihood

The continuous extension of a discrete random variable is amongst the computational methods used for estimation of multivariate normal copula-based models with discrete margins. Its advantage is that the likelihood can be derived conveniently under the theory for copula models with continuous margins, but there has not been a clear analysis of the adequacy of this method. We investigate the asymptotic and small-sample efficiency of two variants of the method for estimating the multivariate normal copula with univariate binary, Poisson, and negative binomial regressions, and show that they lead to biased estimates for the latent correlations, and the univariate marginal parameters that are not regression coefficients. We implement a maximum simulated likelihood method, which is based on evaluating the multidimensional integrals of the likelihood with randomized quasi-Monte Carlo methods. Asymptotic and small-sample efficiency calculations show that our method is nearly as efficient as maximum likelihood for fully specified multivariate normal copula-based models. An illustrative example is given to show the use of our simulated likelihood method.

Time-Varying Systemic Risk: Evidence from a Dynamic Copula Model of CDS Spreads

This paper proposes a new class of copula-based dynamic models for high dimension conditional distributions, facilitating the estimation of a wide variety of measures of systemic risk. Our proposed models draw on successful ideas from the literature on modeling high dimension covariance matrices and on recent work on models for general time-varying distributions. Our use of copula-based models enable the estimation of the joint model in stages, greatly reducing the computational burden. We use the proposed new models to study a collection of daily credit default swap (CDS) spreads on 100 U.S. firms over the period 2006 to 2012. We find that while the probability of distress for individual firms has greatly reduced since the financial crisis of 2008-09, the joint probability of distress (a measure of systemic risk) is substantially higher now than in the pre-crisis period.

Regional Air Quality Conformity in Transportation Networks with Stochastic Dependencies: A Theoretical Copula-Based Model

The air quality levels in various regions around the world remain a large public concern. Transportation is known to be a major contributor to reduced air quality levels. Until now, the modeling of the regional impact of transportation on air quality has been based on the assumption of determinism. On the other hand, it is well recognized that transportation systems are subject to both demand and supply uncertainties. In this paper, we relax the assumption of determinism and allow for capacity and link flow uncertainty. We introduce a probability measure – coined the conformity probability – to capture the full probabilistic behavior of vehicular emissions. Moreover, stochastic dependencies are modeled using copulas, generalizing other commonly used dependence modeling techniques in the transportation network modeling arena. In a case study we demonstrate that such a generalization is critical as the ranking of capacity expansion projects to improve air quality is shown to be dependent on the hypothesized dependence structure. Finally, we present some preliminary results that suggest that capacity uncertainty is more detrimental to the environment (i.e. leads to lower conformity probabilities) than demand uncertainty.

Analysis of directional dependence using asymmetric copula-based regression models

The directional dependence between variables using asymmetric copula regression has drawn much attention in recent years. There are, however, some critical issues which have not been properly addressed in regards to the statistical inference of the directional dependence. For example, the previous use of asymmetric copulas failed to fully capture the dependence patterns between variables, and the method used for the parameter estimation was not optimal. In addition, no method was considered for selecting a suitable asymmetric copula or for computing the general measurements of the directional dependence when there are no closed-form expressions. In this paper, we propose a generalized multiple-step procedure for the full inference of the directional dependence in joint behaviour based on the asymmetric copula regression. The proposed procedure utilizes several novel methodologies that have not been considered in the literature of the analysis of directional dependence. The performance and advantages of the proposed procedure are illustrated using two real data examples, one from biological research on histone genes, and the other from developmental research on attention deficit hyperactivity disorder.

Call-type dependence in multiskill call centers

The effect on multiskill call-center performance of pooling dependent call types is investigated. For this purpose, a copula-based modeling approach is used to provide multivariate models that take into account the call types’ asymmetric dependence structures found in empirical data. Then, the realistic input models of the call-type-dependent arrival processes are used in a simulation study to explore the sensitivity of the pooling decision to this dependence. We find that the widely used assumption of independence, as well as the misspecification of the dependence structure, can lead to substantial misestimation of call-center performance. This demonstrates the importance of modeling call-type dependence in stochastic simulation studies of call centers. We also show, through case studies, that pooling two asymmetric left-tail-dependent call types is more likely to lead to low agents occupancy; whereas the presence of right-tail dependence structure increases the risk of service-level shortfall. This work provides new managerial insights to improve decision making in determining which call types to merge in the same pool in multiskill call centers.

Analysis of ENSO-based climate variability in modulating drought risks over western Rajasthan in India

This paper investigates the role of El Nino-Southern Oscillation (ENSO)-based climate variability in modulating multivariate drought risks in the drought-prone region of Western Rajasthan in India. Droughts are multivariate phenomenon, often characterized by severity, duration and peak. By using multivariate ENSO index, annual drought events are partitioned into three climatic states – El Nino, La Nina and neutral phases. For multivariate probabilistic representation of drought characteristics, trivariate copulas are employed, which have the ability to preserve the dependence structure of drought variables under uncertain environment. The first copula model is developed without accounting the climate state information to obtain joint and conditional return periods of drought characteristics. Then, copula-based models are developed for each climate state to estimate the joint and conditional probabilities of drought characteristics under each ENSO state. Results of the study suggest that the inclusion of ENSO-based climate variability is helpful in knowing the associated drought risks, and useful for management of water resources in the region.

Mean-Variance Optimization Still Works! A Bayesian Methodology with Vine Copulas

Why do mean-variance (MV) models perform so poorly? In searching for an answer to this question, we estimate expected returns by sampling from a multivariate probability model that explicitly incorporates distributional asymmetries. Specifically, our empirical analysis shows that an application of copulas using marginal models that incorporate dynamic features such as autoregression, volatility clustering, and skewness to reduce estimation error in comparison to historical sampling windows. Using these copula-based models, we find that several MV-based rules exhibit statistically significant and superior performance improvements even after accounting for transaction costs. However, we find that outperforming the naive equally-weighted (1/N) strategy after accounting for transactions costs still remains an elusive task.

Joint Model of Freight Mode and Shipment Size Choice

Mode and shipment size choice are among the most critical logistics decisions but are typically studied separately in freight demand studies despite their strong correlations. The study described in this paper used an innovative copula-based framework to model freight mode and shipment size choice simultaneously as a joint decision-making problem. The study used a copula-based joint multinomial logit–multinomial logit model in which both mode choice and shipment size were modeled as discrete choices in a multinomial logit structure. The proposed copula-based model was intended to capture the potential effects of observed and unobserved factors that jointly affect both choices. The data used in this study were gathered through a large-scale establishment survey conducted in 2011 in the United States. The survey provided detailed information on more than 1,840 individual shipments that were used to develop the disaggregate models. Results of the estimated model underline the importance of the interrelationship between freight mode choice and shipment size and suggest that common unobserved factors influencing mode and shipment size choices exist. The model provides a better way to capture the effects of observed and unobserved factors that affect both choices simultaneously.

Weighted Cook-Johnson Copula and their Characterizations: Application to Probably Modeling of the Hot Spring Eruptions

Copulas have emerged as a practical method for multivariate modeling. A limited amount of work has been conducted regarding the application of copula-based modeling in context analysis. This study generalizes the Cook-Johnson copula under the appropriate weighted function and provides examples and the properties of the generalized Cook-Johnson copula. Results show that the generalized Cook-Johnson copula is suitable for probable modeling of hot spring eruption.