Empirical Copula Research Articles

Environmental contours using nonparametric copulas

The use of nonparametric copulas for modeling joint probability distributions of metocean variables and building environmental contours is examined. Provided that no assumption is made about a particular functional form, nonparametric copulas may be an alternative to improve modeling dependence patterns, which is critical to develop environmental contours. The transformation method with local log-quadratic likelihood kernel estimator and nearest-neighbor bandwidth (TLL2nn), the mirror-reflection and beta kernel methods, and Bernstein polynomials are used for nonparametric estimation. For comparison purposes, well known parametric copulas are also estimated. Hindcast data from tropical cyclones and storms at three sites in the southern Gulf of Mexico is used. The adequacy of the copula models is assessed first using graphical diagnostics tools. Data fitting is then evaluated using Cramér-von Mises statistics, and the Akaike and Bayesian information criteria are applied for model selection. The estimated copulas are also compared in terms of their adequacy to model upper and lower tail dependence. The results showed that whereas different parametric copulas were needed to model the different data sets, the nonparametric TLL2nn model was able to give adequate results for the three sites. Environmental contours using the nonparametric, parametric, and empirical copulas are compared.

Spatially coherent statistical simulation of widespread flooding events under climate change

Abstract Simulating rare widespread hydrological events can be difficult even with the use of modelled data such as the UKCP18 12 km regional climate projections. To generate larger event sets for application in catastrophe modelling, two statistical approaches are highlighted and applied to widespread GB-generated flooding events using a grid-based hydrological model and UKCP18 regional projections. An Empirical Copula method was applied on a national scale, generating over 600,000 events across two time-slices (1980–2010 and 2050–2080). This was compared to model-generated events and showed good matching across time-slices and ensemble members, although lacked some ability to describe the least-rare events. The Empirical Copula was also compared to an implementation of a conditional exceedance model. This model was much more computationally intensive so was restricted to Northwest England but offered the ability to be tuned more finely through choices of marginal distributions. Analysing over 11,000 events, it also matched well with the Empirical Copula and model-generated events but under-represented the smallest events. Both approaches require a broad dataset to draw from but showed reasonable efficacy. For simple statistics, the Empirical Copula shows the potential to be a powerful tool in exploring spatial structure over large regions or at a fine spatial resolution.

The Devil Is in the Tail Dependence: An Assessment of Multivariate Copula‐Based Frameworks and Dependence Concepts for Coastal Compound Flood Dynamics

AbstractA hurricane event can often produce both intense rainfall and a storm tide that can cause a major compound flooding threat to coastlines. This paper examined applications of multivariate copula‐based time series models using data observed during Hurricane Irma (2017) along the coastlines of Florida, Georgia, and South Carolina, United States. Multivariate time series models were developed using bivariate copulas wherein storm tide and rainfall data were modeled using LOWESS‐based autoregressive moving average (ARMA). n samples of observed data were then synthesized using a Monte Carlo approach in which the empirical copula and the parametric estimate of the copula were obtained to approximate two‐sided p‐values using the Rosenblatt probability integral transform method. Analysis suggested that proper selection of the underlying LOWESS‐based ARMA model was the crucial aspect for modeling compound flooding wherein Archimedean, Elliptical, and Extreme Value copulas all offered consistent flexibility in terms of dependence modeling. As a backdrop to compound flood probabilities, this research also outlined both theoretical and applied frameworks for the calculation of non‐exceedance probabilities in a multidimensional environment using classical isofrequency probability assumptions for the “AND” (a bivariate joint probability) and Survival Kendall definitions. Random realizations from storm copulas combined with multivariate non‐exceedance probability definitions ultimately showed there were periods of temporal yet cyclical high intensities that lasted 1–2 hr. Lastly, a discussion is presented on the broader application of the proposed methodology within the field of engineering design and risk management which may serve as a catalyst for the continued research in compound flooding.

Online prediction based on the Copula function for the intrinsic viscosity of polyester fibre

AbstractThe polymerization process, as the starting point of polyester fibre production, directly affects the fibre properties. However, the polymerization process has complex reaction mechanisms and involves many variables, making online modelling more challenging. In this paper, we introduce the Copula function into the modelling of the polymerization process. First, a feature selection method based on kernel principal component analysis (KPCA)‐Copula is proposed, and then, the Copula function to obtain the nonlinear characteristics among the variables is used. On this basis, an online prediction model of intrinsic viscosity based on just‐in‐time learning (JITL) was established, that is, empirical Copula‐JITL. In order to improve the computational efficiency of the empirical Copula‐JITL, the t‐Copula function is introduced. Furthermore, by combining the t‐Copula function with the empirical Copula function, we get a JITL model based on the hybrid Copula function, that is, hybrid Copula‐JITL. Finally, actual polymerization process data collected from fibre production are provided to demonstrate the effectiveness of the proposed algorithms.

CHARACTERISTICS OF JOINT DISTRIBUTION MODELS AND RETURN PERIOD OF EXTREME RAINFALL VOLUME AND INTENSITY

This article highlights a new approach to finding a copula model that most closely matches extreme rainfall characteristics in South Sulawesi, Indonesia. In climatology research, data that does not meet normal assumptions are often found due to extreme observations; therefore, a method is needed to overcome the dependency of variables that do not meet the normality assumption. The copula approach is a method that can determine the relationship between such variables. Copula can describe the dependency structure between variables with different margins and model its tail dependencies. This study discusses the application of Archimedean copula in modeling the dependency structure of both variables, namely the intensity of the 75th percentile (I75) and volume of the 75th percentile (P75). For explaining the dependencies of both variables, the best copula model was selected using the empirical copula method. The results showed that 16 stations followed the Clayton copula model, 23 stations followed the Gumbel copula model, and 14 stations followed the Frank copula model. After knowing the distribution model characteristics and the intensity and volume of extreme rainfall, the return period value on the intensity and volume of extreme rainfall was calculated. The return period value can be useful as information and evaluation valuable material for preparing natural hazard mitigation programs.

Postprocessing Ensemble Weather Forecasts for Introducing Multisite and Multivariable Correlations Using Rank Shuffle and Copula Theory

Abstract Statistical methods have been widely used to postprocess ensemble weather forecasts for hydrological predictions. However, most of the statistical postprocessing methods apply to a single weather variable at a single location, thus neglecting the intersite and intervariable dependence structures of forecast variables. This study synthesized a multisite and multivariate (MSMV) postprocessing framework that extends the univariate method to the MSMV version by directly rearranging the postprocessed ensemble members (post-reordering strategy) or by rearranging the latent variables used in the univariate method (pre-reordering strategy). Based on the univariate generator-based postprocessing (GPP) method, the two reordering strategies and three dependence reconstruction methods [rank shuffle (RS), Gaussian copula (GC), and empirical copula (EC)] totaling six MSMV methods (RS-Pre, GC-Pre, EC-Pre, RS-Post, GC-Post, and EC-Post) were evaluated in postprocessing ensemble precipitation and temperature forecasts for the Xiangjiang basin in China using the 11-member ensemble forecasts from the Global Ensemble Forecasting System (GEFS). The results showed that raw GEFS forecasts tend to be biased for both the forecast ensembles and the intersite and intervariable dependencies. The univariate method can improve the univariate performance of ensemble mean and spread but misrepresent the intersite and intervariable dependence among the forecast variables. The MSMV framework can well utilize the advantages of the univariate method and also reconstruct the intersite and intervariable dependencies. Among the six methods, RS-Pre, RS-Post, GC-Post, and EC-Post perform better than the others with respect to reproducing the univariate statistics and multivariable dependences. The post-reordering strategy is recommended to combine the univariate method (i.e., GPP) and reconstruction methods.

Testing bivariate independence based on α-divergence by improved probit transformation method for copula density estimation

Independence test based on empirical copula does not perform well in the presence of weak dependency or when dependency occurs only in the tails. The copula density, which is estimated by a local likelihood probit transformation method, is used to detect the independence. In this article, three nonparametric tests of independence based on α-divergence and copula density are introduced. These tests are capable of considering weak dependency. The asymptotic consistency of the copula-based α-divergence estimator is also derived. In addition, the empirical powers of the proposed tests are computed through extensive simulations. The results show that the new tests outperform in small sample sizes or weak dependencies. Finally, an application in uranium exploration is presented to illustrate the applicability of the proposed tests.

Asymmetry quantification in cross modal retrieval using copulas

Copulas are used to describe and explain the asymmetry between image-text and text-image retrieval observed in different values of the mean average precision (MAP). We use empirical copulas to quantify the asymmetry in a general framework of cross-modal retrieval via suitable asymmetry measures. Several experiments are done on real world dataset feautures to prove the relevance of our analysis.

Conditional empirical copula processes and generalized measures of association

We study the weak convergence of conditional empirical copula processes indexed by general families of conditioning events that have non zero probabilities. Moreover, we also study the case where the conditioning events are chosen in a data-driven way. The validity of several bootstrap schemes is stated, including the exchangeable bootstrap. We define general multivariate measures of association, possibly given some fixed or random conditioning events. By applying our theoretical results, we prove the asymptotic normality of the estimators of such measures. We illustrate our results with financial data.

Electricity‐heat‐gas integrated demand response dependency assessment based on BOXCOX‐Pair Copula model

With the continuous development of Regional Integrated Energy System (RIES), demand response (DR) is composed of diversified loads including electric load, heat load and gas load. Their cross-dependencies reflecting the nonlinear coupling complementary relationship between each load type are one of the key factors to improve multi-energy flow optimisation modelling and utilisation efficiency on the demand side. Accordingly, this paper proposes a DR dependency assessment method considering electricity-heat-gas loads based on the BOXCOX-Pair Copula model. BOXCOX transformation is introduced to convert various probability distribution statistics of electricity-heat-gas loads into Gaussian distributed variables. C-vine Pair Copula is used to characterise an ensemble-of-trees of high-dimensional dependency structure among multi-energy demand modalities. Then the combined model of BOXCOX transformation and C-vine Pair Copula can be employed to determine the complex coupling dependency among electricity-heat-gas loads according to different DR statistics. Some metrics between the original empirical distribution and BOXCOX-Pair Copula distribution are introduced to assess the dependency evaluation precision of the proposed model. Finally, the novel dependency assessment model is numerically tested utilising the electricity load, heat load and gas load data sequences in a real RIES. The results illustrate that the cross-dependency of electricity-heat-gas integrated DR based on the BOXCOX-C-vine copula model is closer to that of actual sample data, which verify the effectiveness and superiority of the proposed approach.

Dependencia entre garantías en el ramo autos de una empresa aseguradora. Un análisis a través de la Teoría de cópulas

En el presente trabajo se trata de analizar las relaciones existentes entre distintas garantías del seguro de autos a través de la aplicación de la teoría de cópulas. El análisis de los datos permite arrojar luz sobre las relaciones de dependencia existentes entre los riesgos cubiertos por las pólizas a través de diversas funciones y, cómo de ellas se derivan importantes efectos sobre la segmentación de los clientes y las formas de tarificación.

On weak conditional convergence of bivariate Archimedean and Extreme Value copulas, and consequences to nonparametric estimation

Looking at bivariate copulas from the perspective of conditional distributions and considering weak convergence of almost all conditional distributions yields the notion of weak conditional convergence. At first glance, this notion of convergence for copulas might seem far too restrictive to be of any practical importance – in fact, given samples of a copula C the corresponding empirical copulas do not converge weakly conditional to C with probability one in general. Within the class of Archimedean copulas and the class of Extreme Value copulas, however, standard pointwise convergence and weak conditional convergence can even be proved to be equivalent. Moreover, it can be shown that every copula C is the weak conditional limit of a sequence of checkerboard copulas. After proving these three main results and pointing out some consequences, we sketch some implications for two recently introduced dependence measures and for the nonparametric estimation of Archimedean and Extreme Value copulas.

Tests of multivariate copula exchangeability based on Lévy measures

AbstractThis paper introduces tests for the symmetry of the copula of random vector. The proposed statistics are based on the copula characteristic function and the weight function that appears naturally in their definition are assumed to belong to the general family of Lévy measures. The proposed test statistics are rank‐based and expresses as weighted ‐norms computed from a vector of empirical copula characteristic functions. Their nondegenerate asymptotic distributions under the null hypothesis and general alternatives, as well as the validity of a multiplier bootstrap for the computation of p‐values, are derived using nonstandard arguments. Extended Monte–Carlo experiments show that the new tests hold their size well and are powerful against a wide range of alternatives, and appear to be more powerful than a Cramér–von Mises test based on empirical copulas.

Functional, randomized and smoothed multivariate quantile regions

The mass transportation approach to multivariate quantiles in Chernozhukov et al. (2017) was modified in Faugeras and Rüschendorf (2017) by a two steps procedure. In the first step, a mass transportation problem from a spherical reference measure to the copula is solved and combined in the second step with a marginal quantile transformation in the sample space. Also, generalized quantiles given by suitable Markov morphisms are introduced there.In the present paper, this approach is further extended by a functional approach in terms of membership functions, and by the introduction of randomized quantile regions. In addition, in the case of continuous marginals, a smoothed version of the empirical quantile regions is obtained by smoothing the empirical copula. All three extended approaches give empirical quantile ares of exact level and improved stability. The resulting depth areas give a valid representation of the central quantile areas of a multivariate distribution and provide a valuable tool for their analysis.

A Sample Generation of Scenario Earthquake Shaking Maps via a Combination of Modal Decomposition and Empirical Copula toward Seismic Hazard Assessment

ABSTRACTTo quantify variations in seismic hazard assessment, a large number of scenario earthquake shaking maps (hereafter referred to as s-EQ maps for short) are required. In this article, we propose a method to easily generate a large number of s-EQ maps by combining modal decomposition and empirical copula. First, by applying modal decomposition to a set of existing s-EQ maps, we identify a scenario set with a parameter space to reduce the dimensionality of models. Next, we model a probability distribution of the modal coordinates, which we regard as random variables to allow us to induce a probability measure on the parameter space. By applying an empirical copula for modeling the probability distributions, the dependence structure between modal coordinates and the probability distribution of an individual modal coordinate can be discussed independently. Some simulations show that the dependence structure between modal coordinates is important to not distort the seismic hazard assessment. The proposed method can be applied to actual existing s-EQ maps to easily generate a large number of new s-EQ maps that follow the original probability distribution. We also show that variations can be added to the newly generated s-EQ maps by adjusting the empirical copula. Furthermore, we give a generalization of the proposed method to cases with nonequal probabilities of occurrence of existing scenarios.

Evaluating Aortic Stenosis Using the Archimedean Copula Methodology

In modeling and analyzing multivariate data, the conventionally used measure of dependence structure is the Pearson’s correlation coeffi cient. However use of the correlation as a dependence measure has several pitfalls. Copulas recently have emerged as an alternative measure of the de pendence, overcoming most of the drawbacks of the correlation. We discuss Archimedean copulas and their relationships with tail dependence. An algo rithm to construct empirical and Archimedean copulas is described. Monte Carlo simulations are carried out to replicate and analyze data sets by iden tifying the appropriate copula. We apply the Archimedean copula based methodology to assess the accuracy of Doppler echocardiography in deter mining aortic valve area from the Aortic Stenosis: Simultaneous Doppler – Catheter Correlative study carried out at the King Faisal Specialist Hospital and Research Centre, Riyadh, KSA

A Multivariate Approach to Generate Synthetic Short‐To‐Medium Range Hydro‐Meteorological Forecasts Across Locations, Variables, and Lead Times

AbstractThe use of hydro‐meteorological forecasts in water resources management holds great promise as a soft pathway to improve system performance. Methods for generating synthetic forecasts of hydro‐meteorological variables are crucial for robust validation of forecast use, as numerical weather prediction hindcasts are only available for a relatively short period (10–40 years) that is insufficient for assessing risk related to forecast‐informed decision‐making during extreme events. We develop a generalized error model for synthetic forecast generation that is applicable to a range of forecasted variables used in water resources management. The approach samples from the distribution of forecast errors over the available hindcast period and adds them to long records of observed data to generate synthetic forecasts. The approach utilizes the Skew Generalized Error Distribution (SGED) to model marginal distributions of forecast errors that can exhibit heteroskedastic, auto‐correlated, and non‐Gaussian behavior. An empirical copula is used to capture covariance between variables, forecast lead times, and across space. We demonstrate the method for medium‐range forecasts across Northern California in two case studies for (1) streamflow and (2) temperature and precipitation, which are based on hindcasts from the NOAA/NWS Hydrologic Ensemble Forecast System (HEFS) and the NCEP GEFS/R V2 climate model, respectively. The case studies highlight the flexibility of the model and its ability to emulate space‐time structures in forecasts at scales critical for water resources management. The proposed method is generalizable to other locations and computationally efficient, enabling fast generation of long synthetic forecast ensembles that are appropriate for risk analysis.

Research on Revenue Insurance Premium Ratemaking of Jujube Based on Copula-Stochastic Optimization Model

During the process of jujube planting, there are not only natural risks caused by natural disasters but also market risks caused by price factors. In the study, firstly, wavelet analysis method was used to stabilize the jujube yield per unit area and the jujube price from 1997 to 2018 in Aksu region, Xinjiang, China. Secondly, EasyFit software was used to fit the distribution functions of yield per unit area and price, respectively. Thirdly, the optimal Copula function which connects the marginal distribution functions and its joint distribution function was selected with the principle of “the minimum square distance from the empirical Copula function.” Finally, taking the premium rate and the insurance amount as two decision variables, the farmer’s risk minimization as the objective function, around the four constraints of functions and role of insurance, the nonspeculative nature of insurance, the sustainability of insurance, and the moral hazard factors and the farmers’ willing to participate in insurance, the Copula-stochastic optimization model was set up to determine the premium rate of jujube revenue insurance in Aksu region.

Novel multivariate quantile mapping methods for ensemble post-processing of medium-range forecasts

Statistical post-processing is an indispensable tool for providing accurate weather forecasts and early warnings for weather extremes. Most statistical post-processing is univariate, with dependencies introduced via use of an empirical copula. Standard empirical methods take a dependence template from either the raw ensemble output (ensemble copula coupling, ECC) or the observations (Schaake Shuffle, SSh). There are drawbacks to both methods. In ECC it is assumed that the raw ensemble simulates the dependence well, which is not always the case (e.g. 2-meter temperature in The Netherlands). The Schaake Shuffle is not able to capture flow dependent changes to the dependence and the choice of observations is key. Here we compare a reshuffled standard ensemble model output statistics (EMOS) approach with two multivariate bias adjustment approaches that have not been used before in a post-processing context: 1) the multivariate bias correction with N-dimensional probability density function transform (MBCn) and 2) multivariate ranks that are defined with optimal assignment (OA). These methods have the advantage that they are able to explicitly capture the dependence structure that is present in the observations. We apply ECC, the Schaake Shuffle, MBCn and OA to 2-m and dew point temperature forecasts at seven stations in The Netherlands. Forecasts are verified with both univariate and multivariate methods, and using a heat index derived from both variables, the wet-bulb globe temperature (WBGT). Our results demonstrate that the spatial and inter-variable dependence structure is more realistic in MBCn and OA compared to ECC or the Schaake Shuffle. The variogram score shows that while ECC is most skilful for the first two days, at moderate lead times MBCn is most skilful and at the longest lead times OA is more skilful than both ECC and MBCn. Overall, we highlight the importance of considering the dependence between variables and locations in the statistical post-processing of weather forecasts.

Evaluating Aortic Stenosis Using the Archimedean Copula Methodology

In modeling and analyzing multivariate data, the conventionally used measure of dependence structure is the Pearson's correlation coeffi- cient. However use of the correlation as a dependence measure has several pitfalls. Copulas recently have emerged as an alternative measure of the de- pendence, overcoming most of the drawbacks of the correlation. We discuss Archimedean copulas and their relationships with tail dependence. An algo- rithm to construct empirical and Archimedean copulas is described. Monte Carlo simulations are carried out to replicate and analyze data sets by iden- tifying the appropriate copula. We apply the Archimedean copula based methodology to assess the accuracy of Doppler echocardiography in deter- mining aortic valve area from the Aortic Stenosis: Simultaneous Doppler - Catheter Correlative study carried out at the King Faisal Specialist Hospital and Research Centre, Riyadh, KSA.