Choice Of Copula Research Articles

Influence of data source and copula statistics on estimates of compound flood extremes in a river mouth environment

Abstract. Coastal and riverine floods are major concerns worldwide as they can impact highly populated areas and result in significant economic losses. In a river mouth environment, interacting hydrological and oceanographical processes can enhance the severity of floods. The compound flood hazards from high sea levels and high river discharge are often estimated using copulas, among other methods. Here, we systematically investigate the influence of different data sources coming from observations and models as well as the choice of copula on extreme water level estimates. While we focus on the river mouth at the city of Halmstad (Sweden), the approach presented is easily transferable to other sites. Our results show that the choice of data sources can considerably impact the results up to 10 % and 15 % for the river time series and 3 % to 4.6 % for the sea level time series under the 5- and 30-year return periods, respectively. The choice of copula can also strongly influence the outcome of such analyses up to 13 % and 9.5 % for the 5-year and 30-year return periods. Each percentage refers to the normalized difference in return level results we can expect when choosing a certain copula or input dataset. The copulas found to statistically best fit our datasets are the Clayton, BB1, and Gaussian (once) ones. We also show that the compound occurrence of high sea levels and river runoff may lead to heightened flood risks as opposed to considering them independent processes and that, in the current study, this is dominated by the hydrological driver. Our findings contribute to framing existing studies, which typically only consider selected copulas and datasets, by demonstrating the importance of considering uncertainties.

Classification of hyperspectral images with copulas

In the last decade, supervised learning methods for the classification of remotely sensed images (RSI) have grown significantly, especially for hyper-spectral (HS) images. Recently, deep learning-based approaches have produced encouraging results for the land cover classification of HS images. In particular, the Convolutional Neural Networks (CNN) and Recurrent Neural Networks (RNN) have shown good performance. However, these methods suffer for the problem of the hyperparameter optimization or tuning that requires a high computational cost; moreover, they are sensitive to the number of observations in the learning phase. In this work we propose a novel supervised learning algorithm based on the use of copula functions for the classification of hyperspectral images called CopSCHI (Copula Supervised Classification of Hyperspectral Images). In particular, we start with a dimensionality reduction technique based on Singular Value Decomposition (SVD) in order to extract a small number of relevant features that best preserve the characteristics of the original image. Afterward, we learn the classifier through a dynamic choice of copulas that allows us to identify the distribution of the different classes within the dataset. The use of copulas proves to be a good choice due to their ability to recognize the probability distribution of classes and hence an accurate final classification with low computational cost can be conducted. The proposed approach was tested on two benchmark datasets widely used in literature. The experimental results confirm that CopSCHI outperforms the state-of-the-art methods considered in this paper as competitors.

Characterizing multivariate coastal flooding events in a semi-arid region: the implications of copula choice, sampling, and infrastructure

Abstract. Multivariate coastal flooding is characterized by multiple flooding pathways (i.e., high offshore water levels, streamflow, energetic waves, precipitation) acting concurrently. This study explores the joint risks caused by the co-occurrence of high marine water levels and precipitation in a highly urbanized semi-arid, tidally dominated region. A novel structural function developed from the multivariate analysis is proposed to consider the implications of flood control infrastructure in multivariate coastal flood risk assessments. Univariate statistics are analyzed for individual sites and events. Conditional and joint probabilities are developed using a range of copulas, sampling methods, and hazard scenarios. The Nelsen, BB1, BB5, and Roch–Alegre were selected based on a Cramér–von Mises test and generally produced robust results across a range of sampling methods. The impacts of sampling are considered using annual maximum, annual coinciding, wet-season monthly maximum, and wet-season monthly coinciding sampling. Although annual maximum sampling is commonly used for characterizing multivariate events, this work suggests annual maximum sampling may substantially underestimate marine water levels for extreme events. Water level and precipitation combinations from wet-season monthly coinciding sampling benefit from a dramatic increase in data pairs and provide a range of physically realistic pairs. Wet-season monthly coinciding sampling may provide a more accurate multivariate flooding risk characterization for long return periods in semi-arid regions. Univariate, conditional, and bivariate results emphasize the importance of proper event definition as this significantly influences the associated event risks.

Copula based Bayesian data analysis of loss reserving

Prediction of loss reserves corresponding to dependent lines of business is one of the most important problems in the actuarial sciences. In this paper, we propose a class of copula based multivariate distributions to model the losses with the heavy tailed distribution in the run-off triangles to predict unpaid losses. We set up ANOVA, ANCOVA, and state space models with four choices of copulas, Clayton, Frank, Gumbel, and Gaussian to provide a new procedure for analyzing run-off triangle tables. We use the Hamiltonian Monte Carlo sampler to perform a Bayesian analysis to estimate the parameters. We apply the proposed models to the data set consists of two lines of business of paid losses data from the Schedule P of the National Association of Insurance Commissioners (NAIC) database. Using some well known criteria, we compare the prediction accuracy of the mean models. As a result, the ANCOVA model with the Clayton copula dominates the other models.

Implicit Copulas: An Overview

Implicit copulas are the most common copula choice for modeling dependence in high dimensions. This broad class of copulas is introduced and surveyed, including elliptical copulas, skew t copulas, factor copulas, time series copulas and regression copulas. The common auxiliary representation of implicit copulas is outlined, and how this makes them both scalable and tractable for statistical modeling. Issues such as parameter identification, extended likelihoods for discrete or mixed data, parsimony in high dimensions, and simulation from the copula model are considered. Bayesian approaches to estimate the copula parameters, and predict from an implicit copula model, are outlined. Particular attention is given to implicit copula processes constructed from time series and regression models, which is at the forefront of current research. Two econometric applications—one from macroeconomic time series and the other from financial asset pricing—illustrate the advantages of implicit copula models.

Frequency effects and the study of Limon’s Spanish copulas: ser and estar

This paper presents a brief but comprehensive review of what has been done regarding the Spanish copula. It explores how frequency effects can help account for the unexplained variance found in the different proposed models, based on the Exemplar Model. This is done to propose a theoretical framework for the study of the Spanish copula in contact situations that include frequency as a variable to understand whether change is ongoing or stable, and whether the change of estar is extending to the contexts of ser, ser is extending to the contexts of estar, or both phenomena is taking place. I explore how each of the schema components interacts with each other, how these interactions fit into the traditional description of copula choice, and move away from the generative view of language as a set of rules to begin looking at it as a system highly affected by the frequency of use of its components. I then propose a procedure on how to explain copula behavior in Spanish that includes a multi-level analysis of this phenomenon.

Copula Choice is Associated with Discourse Integration Skills in Spanish-Speaking Children

AbstractThe present study investigates the relationship between discourse integration skills and Spanish copula choice in monolingual Chilean Spanish-speaking children. Previous research has focused on determining the age at which children associateestarto a transitory property andserto an inherent property (Holtheuer, Carolina. 2012. Spanish-speaking children do not always overuse estar.Revista Signos45(78). 3–19, Holtheuer, Carolina & Johanna Rendle-Short. 2013. Ser and estar: Corrective input to children’s errors of the Spanish copula verbs.First Language33(2). 155–167, Schmitt, Cristina & Karen Miller. 2007. Making discourse dependent decisions: The case of the copulas ser and estar in Spanish.Lingua117(11). 1907–1929, Sera, Maria. 1992. To be or to be: Use and acquisition of the Spanish copulas.Journal of Memory and Language31. 408–427, Requena, Pablo, Astrid Román-Hernández & Karen Miller. 2015. Children’s knowledge of the Spanish copulas ser and estar with novel adjectives.Language Acquisition: A Journal of Developmental Linguistics22. 193–207), showing that by 2 years of age children know the categorial distribution of the two copulas, but that at 5 years of age only some children, but not all children, have knowledge of the transitory/inherent distinction that often arises when the same adjective occurs with one or the other copula. The present study seeks to extend the latter research by further investigating the factors that might influence why only some children associateestarto transitory properties at 5 years of age, a question that will not only shed light onwhenthe subtle meaning differences of the copulas are acquired, but alsohowthey are acquired. Maienborn, Claudia (2005. A discourse-based account of Spanish ser/estar.Linguistics43(1). 155–180) notes thatestaris discourse-linked, whileseris not. Specifically, the use ofestarwith an adjectival predicate to compare changes to a person across different stages of their life relies on one’s ability to integrate the larger discourse (i.e., these various stages) into their statement. In the present study, to determine whether Spanish-speaking children’s production of the copulaestarwith adjectival predicates was associated with their discourse integration abilities, children were presented with both a Copula Elicitation Task and a Discourse Production Task. Analyses revealed an association between children’s discourse integration skills and their use ofestarwith adjectives to express transitory properties, a finding that indicates that children’s acquisition ofestaris mastered late – at least in part because of their late development of discourse integration skills more generally. Most studies on the acquisition ofestarwith adjectival complements have focused primarily on comprehension. Experimental studies on children’sproductionofestarare rare. As such, this experimental study is one of the few that examines children’s use ofestarin production and the first, as far as we know, that provides empirical support for the link between discourse integration skills in children and their use ofestarwith adjectival predicates.

A Mixture of Regular Vines for Multiple Dependencies

To uncover complex hidden dependency structures among variables, researchers have used a mixture of vine copula constructions. To date, these have been limited to a subclass of regular vine models, the so-called drawable vine, fitting only one type of bivariate copula for all variable pairs. However, the variation of complex hidden correlations from one pair of variables to another is more likely to be present in many real datasets. Single-type bivariate copulas are unable to deal with such a problem. In addition, the regular vine copula model is much more capable and flexible than its subclasses. Hence, to fully uncover and describe complex hidden dependency structures among variables and provide even further flexibility to the mixture of regular vine models, a mixture of regular vine models, with a mixed choice of bivariate copulas, is proposed in this paper. The model was applied to simulated and real data to illustrate its performance. The proposed model shows significant performance over the mixture of R-vine densities with a single copula family fitted to all pairs.

Modelling mortality dependence: An application of dynamic vine copula

Vine copula, constructed from bivariate copulas, provides great flexibility in modelling complex high-dimensional dependence. When applied to multi-population mortality modelling, vine copula yields significant improvement over traditional multivariate copulas. In this paper, we propose to capture time-varying features in mortality dependence with dynamic regular vine (R-vine) copula which is built from bivariate copulas with time-varying dependence parameters. We develop two dependence dynamics for R-vine copulas and illustrate the selection and estimation of dynamic R-vine copulas using mortality data from eight populations. The estimated R-vine copulas using the proposed dependence dynamics are shown to yield better goodness of fit than both static and regime-switching vine copulas. We further demonstrate the simulation of mortality paths using dynamic R-vine copulas and examine the impact of vine copula choice on the assessed effectiveness of longevity hedge.

Trivariate copula to design coastal structures

Abstract. Some coastal structures must be redesigned in the future due to rising sea levels caused by climate change. The design of structures subjected to the actions of waves requires an accurate estimate of the long return period of such parameters as wave height, wave period, storm surge and more specifically their joint exceedance probabilities. The simplified Defra method that is currently used in particular for European coastal structures makes it possible to directly connect the joint exceedance probabilities to the product of the univariate probabilities by means of a single factor. These schematic correlations do not, however, represent all the complexity of the reality because of the use of this single factor. That may lead to damaging errors in coastal structure design. The aim of this paper is therefore to remedy the lack of robustness of these current approaches. To this end, we use copula theory with a copula function that aggregates joint distribution functions to their univariate margins. We select a bivariate copula that is adapted to our application by the likelihood method. In order to integrate extreme events, we also resort to the notion of tail dependence. The optimal copula parameter is estimated through the analysis of the tail dependence coefficient, the likelihood method and the mean error. The most robust copulas for our practical case with applications in Saint-Malo and Le Havre (in northern France) are the Clayton copula and the survival Gumbel copula. The originality of this paper is the creation of a new and robust trivariate copula with an analysis of the sensitivity to the method of construction and to the choice of the copula. Firstly, we select the best fitting of the bivariate copula with its parameter for the two most correlated univariate margins. Secondly, we build a trivariate function. For this purpose, we aggregate the bivariate function with the remaining univariate margin with its parameter. We show that this trivariate function satisfies the mathematical properties of the copula. We finally represent joint trivariate exceedance probabilities for a return period of 10, 100 and 1000 years. We finally conclude that the choice of the bivariate copula is more important for the accuracy of the trivariate copula than its own construction.

Consistent Spread Dynamics for CVA Risk Charge and Historical Value-at-Risk by Means of Cross Sectional / Consolidated Bucket Link Copula Simulation

This paper describes the modelling of spread risk, in case of missing or illiquid market data, by using a subset of good quality liquid bond/credit default swap (CDS) spread time series. The proposed method links copula simulation to the actual historical spread dynamics. This is important when calculating credit valuation adjustment (CVA) risk charge and Value-at-Risk (VaR) with historical simulation. The methodology center around buckets of similar spreads. Buckets with good data, are straight forward, whereas buckets without data rely on a cross sectional model spread based on other buckets with good data. Residuals from regression of the bucket spread returns against market index returns are used to derive a link for each bucket. The link is subsequently used for simulating the spread dynamics in case of missing or illiquid spreads, using a modified one factor copula. A link between the actual and simulated residuals maintaining the risk dynamics is thus ensured. The result of the copula simulation is transformed into quantiles that are plugged into residual distributions from actual quality data, thereby maintaining the properties of actual market data such that the choice of copula only affects the risk dynamics, not the distributions of the risk factors.

A copula-based method of classifying individuals into binary disease categories using dependent biomarkers

Classification of a disease often depends on more than one test, and the tests can be interrelated. Under the incorrect assumption of independence, the test result using dependent biomarkers can lead to a conflicting disease classification. We develop a copula-based method for this purpose that takes dependency into account and leads to a unique decision. We first construct the joint probability distribution of the biomarkers considering Frank’s, Clayton’s and Gumbel’s copulas. We then develop the classification method and perform a comprehensive simulation. Using simulated data sets, we study the statistical properties of joint probability distributions and determine the joint threshold with maximum classification accuracy. Our simulation study results show that parameter estimates for the copula-based bivariate distributions are not biased. We observe that the thresholds for disease classification converge to a stationary distribution across different choices of copulas. We also observe that the classification accuracy decreases with the increasing value of the dependence parameter of the copulas. Finally, we illustrate our method with a real data example, where we identify the joint threshold of Apolipoprotein B to Apolipoprotein A1 ratio and total cholesterol to high-density lipoprotein ratio for the classification of myocardial infarction. We conclude, the copula-based method works well in identifying the joint threshold of two dependent biomarkers for an outcome classification. Our method is flexible and allows modeling broad classes of bivariate distributions that take dependency into account. The threshold may allow clinicians to classify uniquely individuals at risk of developing the disease and plan for early intervention.

A bivariate power generalized Weibull distribution: A flexible parametric model for survival analysis.

We are concerned with the flexible parametric analysis of bivariate survival data. Elsewhere, we argued in favour of an adapted form of the 'power generalized Weibull' distribution as an attractive vehicle for univariate parametric survival analysis. Here, we additionally observe a frailty relationship between a power generalized Weibull distribution with one value of the parameter which controls distributional choice within the family and a power generalized Weibull distribution with a smaller value of that parameter. We exploit this relationship to propose a bivariate shared frailty model with power generalized Weibull marginal distributions linked by the BB9 or 'power variance function' copula, then change it to have adapted power generalized Weibull marginals in the obvious way. The particular choice of copula is, therefore, natural in the current context, and the corresponding bivariate adapted power generalized Weibull model a novel combination of pre-existing components. We provide a number of theoretical properties of the models. We also show the potential of the bivariate adapted power generalized Weibull model for practical work via an illustrative example involving a well-known retinopathy dataset, for which the analysis proves to be straightforward to implement and informative in its outcomes.

Nonparametric Estimation of Copula Regression Models With Discrete Outcomes

Multivariate discrete outcomes are common in a wide range of areas including insurance, finance, and biology. When the interplay between outcomes is significant, quantifying dependencies among interrelated variables is of great importance. Due to their ability to accommodate dependence flexibly, copulas are being applied increasingly. Yet, the application of copulas on discrete data is still in its infancy; one of the biggest barriers is the nonuniqueness of copulas, calling into question model interpretations and predictions. In this article, we study copula estimation with discrete outcomes in a regression context. As the marginal distributions vary with covariates, inclusion of continuous regressors expands the region of support for consistent estimation of copulas. Because some properties of continuous outcomes do not carry over to discrete outcomes, specification of a copula model has been a problem. We propose a nonparametric estimator of copulas to identify the “hidden” dependence structure for discrete outcomes and develop its asymptotic properties. The proposed nonparametric estimator can also serve as a diagnostic tool for selecting a parametric form for copulas. In the simulation study, we explore the performance of the proposed estimator under different scenarios and provide guidance on when the choice of copulas is important. The performance of the estimator improves as discreteness diminishes. A practical bandwidth selector is also proposed. An empirical analysis examines a dataset from the Local Government Property Insurance Fund (LGPIF) in the state of Wisconsin. We apply the nonparametric estimator to model the dependence among claim frequencies from different types of insurance coverage. Supplementary materials for this article are available online.

Do bilinguals generalizeestarmore than monolinguals and what is the role of conceptual transfer?

Aims and objectives/purpose/research questions:Among the questions that remain open is whether bilingualism leads to simplification of alternatives in language in order to reduce cognitive load. This hypothesis has been supported by evidence showing that bilinguals generalize the Spanish copula estar ‘to be’ faster than monolinguals. Yet, other studies found no such clear trend. While conceptual transfer could account for the conflicting evidence in the literature, its role has not been demonstrated. Our study aims to fill this gap by testing simplification in Spanish copula choice among bilinguals and, in particular, the role of transfer.Design/methodology/approach:We used a contextualized copula choice task, comprising 28 sentences.Data and analysis:Sixty Romani–Spanish bilinguals from Mexico responded to the questionnaire in both Spanish and Romani. A control group of 62 Mexican Spanish monolinguals responded in Spanish. We constructed generalized linear mixed-effects models to analyse the results.Findings/conclusions:Analysis of the results reveals greater extension of estar among bilinguals for individual-level predicates as well as for traits not susceptible to change. Comparison of the responses of bilinguals (in Romani and in Spanish) and of Spanish monolinguals indicates that Romani could be reinforcing the generalization of estar in the Spanish responses of bilinguals.Originality:To our knowledge, this is the first study to examine copula choice in bilingual mode. In addition, it brings evidence from an under-researched community with little normative pressure.Significance/implications:Our study shows that conceptual transfer may be driving the extension of estar among bilinguals.

Quantifying the association between progression-free survival and overall survival in oncology trials using Kendall's τ.

This paper considers methods for estimating the association between progression‐free and overall survival in oncology trials. Copula‐based, nonparametric, and illness‐death model–based methods are reviewed. In addition, the approach based on an underlying illness‐death model is generalized to allow general parametric models. The performance of these methods, in terms of bias and efficiency, is investigated through simulation and also illustrated using data from a clinical trial of treatments for colon cancer. The simulations suggest that the illness‐death model–based method provides good estimates of Kendall's τ across several scenarios. In some situations, copula‐based methods perform well but their performance is sensitive to the choice of copula. The Clayton copula is most appropriate in scenarios, which might realistically reflect an oncology trial, but the use of copula models in practice is questionable.

Copula + Adjective

Researchers sometimes, unknowingly, exclude the existence of the phenomenon in the population under study to develop investigations that try to explain discrepancies observed in the literature. Explanations justifying each study are based on theories; or conjecture related to specific independent variables and how researchers disagree with the construct proposed originally. These disagreements create variants of the original study, but not proper replications of it. This practice produces what Ottenbacher (1996: 275) called 'a contradictory research literature, apparently dynamic, but failing to resolve uncertainties; therefore, failing to establish statistical consensus'. To resolve these uncertainties, this paper addresses the concerns that arise by not reporting two statistics: power and effect size. Power calculations were conducted for the studies on copula choice in bilingual Spanish. Results show that studies have a low power due to decisions that were made during their design, or due to the aggregation of naturally independent data. due to the aggregation of naturally independent data.

Assessing the copula selection for bivariate frequency analysis based on the tail dependence test

The flood characteristics, namely, peak, duration and volume provide important information for the design of hydraulic structures, water resources planning, reservoir management and flood hazard mapping. Flood is a complex phenomenon defined by strongly correlated characteristics such as peak, duration and volume. Therefore, it is necessary to study the simultaneous, multivariate, probabilistic behaviour of flood characteristics. Traditional multivariate parametric distributions have widely been applied for hydrological applications. However, this approach has some drawbacks such as the dependence structure between the variables, which depends on the marginal distributions or the flood variables that have the same type of marginal distributions. Copulas are applied to overcome the restriction of traditional bivariate frequency analysis by choosing the marginals from different families of the probability distribution for flood variables. The most important step in the modelling process using copula is the selection of copula function which is the best fit for the data sample. The choice of copula may significantly impact the bivariate quantiles. Indeed, this study indicates that there is a huge difference in the joint return period estimation using the families of extreme value copulas and no upper tail copulas (Frank, Clayton and Gaussian) if there exists asymptotic dependence in the flood characteristics. This study suggests that the copula function should be selected based on the dependence structure of the variables. From the results, it is observed that the result from tail dependence test is very useful in selecting the appropriate copula for modelling the joint dependence structure of flood variables. The extreme value copulas with upper tail dependence have proved that they are appropriate models for the dependence structure of the flood characteristics and Frank, Clayton and Gaussian copulas are the appropriate copula models in case of variables which are diagnosed as asymptotic independence.

Graphical Methods For Detecting Dependence


 
 
 Copulas have become a useful tool for modeling data when the dependence among random variables exists and the multivariate normality assumption is not fulfilled. The copulas have been applied in several fields. In finance, copulas are used in asset modeling and risk management. In biomedical studies, copulas are used to model correlated lifetimes and competitive risks [1]. In engineering, copulas are used in multivariate process control and hydrological modeling [2]. The interest in modeling multivariate problems involving dependent variables is generalized in several areas, making this methodology in a convenient way to model the dependence structure of random variables. However, in practice there is not a standard method for selecting a copula among several possible models, so that the choice of an appropriate copula is one of the greatest challenges facing the researcher. In this paper some graphical methods for detecting dependencies among random variables are discussed. 
 
 

Model Risk in FRTB: The Case of the Default Risk Charge

The recent Fundamental Review of the Trading Book (FRTB) resulted in revised standards for capital requirements for market risks in a bank’s trading book. As part of the ruleset, default risk needs to be measured and capitalized through a dedicated Default Risk Charge (DRC). With the DRC as an extreme tail risk measure at 99.9% confidence level for portfolio default losses at a one-year horizon, there is inherent model risk associated with the reflection of joint defaults. Wilkens and Predescu (2017) proposed an overall framework for modeling the DRC that is based on a Gaussian factor copula model to capture the coincidence of defaults. This paper assesses the resulting model risk by analyzing alternative copulas (Gaussian, Student t, and Clayton) and the influence on the DRC figures with the help of a set of example portfolios. The copula choice can affect the DRC considerably, especially for directional and less diversified portfolios; the influence on typical larger-scale, diversified portfolios is much less pronounced. The uncertainty arising from the calibration of any copula from only a few data points – as implied by the regulation – is at least of equal importance as the selection of the dependence model itself.