Clayton Copula Research Articles

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.

Multivariate analysis of traffic flow using copula-based model at an isolated road intersection

Investigating the correlation relationship among traffic flow parameters (speed, time headway, volume) and accurately estimating their distribution function at urban road intersections are critical to provide theoretical support for traffic simulation in traffic management and control. Previous studies usually investigated distributions of traffic flow parameters separately, and they did not take into account the correlation between these parameters. In this study, a copula-based approach is proposed to analyze the dependence structure and model the multivariate distribution of traffic flow parameters during peak and off-peak hours in three directions at the intersections. First, the correlation between the traffic flow parameters/variables is examined by Kendall’s tau and Spearman’s rho to measure the correlation relationship. Then, fitting marginal distributions for traffic variables is conducted. The distribution function with the best fitting performance of each traffic variable is selected based on AIC and K–S tests. Moreover, copula-based models that employ Gaussian, FGM, Gumbel, Clayton, Frank and AMH copula functions are used to construct the bivariate joint distribution. The optimal copula model is selected by using the log-likelihood and AIC to evaluate fitting results. Finally, by comparing the observed data with the same size simulated data generated from the best fitting copula model, we find that the copula-based model could accurately construct the bivariate joint distribution and reproduce the dependence structure between the variables. Overall, the findings of this study provide an in-depth perspective on the dependence structure of traffic variables at urban road intersections.

Spatio-temporal wind speed prediction based on Clayton Copula function with deep learning fusion

Accurate prediction of wind speed plays an important role in increasing the power generation of wind turbines and realizing efficient use of wind energy. However, due to the large number of wind turbines in the wind farm and the complex wake effects between the units, the coupling degree and spatial correlation of the wind speed of the wind turbines are increased. Accordingly, this paper proposes a wind speed prediction model based on spatio-temporal dependency analysis. The proposed model first uses long short-term memory(LSTM) neural network to predict the wind speed of each wind turbine to obtain its residuals, which can extract the time correlation of the wind-speed series; Then by applying the Clayton Copula function to analyze the correlation between the residual series and wind-speed series to get the joint-distribution function. The joint-distribution function can be used to calculate the prediction error of the wind speed and complete the wind speed prediction. The validity of the method in this work is verified using the measured wind-speed data of a wind farm. Experimental results show that the method effectively solves the problem of finding spatio-temporal dependency of wind speed and significantly improves the prediction accuracy of wind speed.

Computational methods for a copula-based Markov chain model with a binomial time series

A copula-based Markov chain model can flexibly capture serial dependence in a time series. However, the computational developments for copula-based Markov models remain insufficient for discrete marginal models compared with continuous ones. In this article, we develop computational methods for a binomial time series under the Clayton and Joe copulas. The methods include the data-generation, parameter estimation, model selection, and goodness-of-fit tests. We implement the methods in our R package Copula.Markov (https://CRAN.R-project.org/package=Copula.Markov). We conduct simulations to see the performance of the developed methods. Finally, the proposed method is illustrated by a real dataset.

A novel four-parameter log-logistic model: mathematical properties and applications to breaking stress, survival times and leukemia data

In this paper, we introduce a new continuous log-logistic extension. Several of its properties are established. A numerical analysis for skewness and kurtosis is presented. The new failure rate can be "bathtub or U shaped", "increasing", "decreasing-constant", "J shaped", "constant" and "decreasing". Many bivariate and Multivariate type distributions are derived using the Clayton Copula and the Morgenstern family. To assess of the finite sample behavior of the estimators, we performed a graphical simulation. Some useful applications are considered for supporting the new model.

A copula incorporated cellular automata module for modeling the spatial distribution of oasis recovered by ecological water diversion: An application to the Qingtu Oasis in Shiyang River basin, China

Assessing the impact of ecological water diversion on oasis restoration is critical for water resources and environmental management in arid inland basins. This study proposed a copula incorporated cellular automata (CA) module for modeling the oasis’s spatial distribution driven by ecological water diversion. Given the effects of groundwater depth and community expansion on oasis evolution, a water availability index and a neighbor vegetation index were proposed. These two indices were joint using the Clayton copula function, producing a vegetation suitability index that quantified vegetation growth’s habitat quality and provided the CA module’s basis to identify cellular categories. The CA module was integrated into a previously developed conceptual lumped ecohydrological model (CLEM), resulting in an improved CA-CLEM that outputted groundwater depth, oasis area, oasis NDVI, and oasis distribution data. The CLEM provided the oasis area changes for the CA module, which specified the spatial distribution of these changes. The CA-CLEM had a simple structure, low computation cost, and good performance, indicated by the application in the Shiyang River basin in Northwest China. Results show that an ecological water diversion of 30 million m3, with an average groundwater depth of 2.95 m, an oasis area of 25 km2, and an oasis NDVI of 0.36, would successfully recover the terminal Qingtu Oasis in 2020. The oasis distribution and its dynamic evolution processes were irregular because of the spatial heterogeneity of vegetation habitat suitability. The present Qingtu Oasis was fragile, and its sustainability relied on ecological water diversion. The reduction in water diversion would cause water table decline, oasis area shrinkage, vegetation degradation, and oasis fragmentation. The CA-CLEM combined with scenario analysis provides practical and reliable implications for managers and policymakers to improve ecological water diversion strategies.

A copula-based approach to accommodate intra-household interaction in workers’ daily maintenance activity stop generation modeling

A new stop frequency model was developed to predict the daily number of maintenance activity stops made by individual household heads during a typical weekday. This new model was based on the modification of a conventional multivariate ordered probit (MOP) model by maintaining the probit assumption for the marginal distributions while introducing non-normal dependence among the error terms using copula functions. Therefore, the copula-based MOP model would relieve the restriction of imposing joint normality on the error terms in the conventional MOP model. The new MOP model would not only account for the intra-household interactions in stop-making decisions, but also allow the best functional form to be determined for representing dependencies among household heads. Using the New York Metropolitan Transportation Council's 2010/2011 regional household travel survey data, the copula-based MOP model was employed to examine stop-making behavior for individual household heads residing in New York City and its adjacent counties in Mid-Hudson Valley and New Jersey. Empirical results provided useful insights into the observed effects of socio-demographics, land use density, transportation service, and work schedule together with potential unobserved common effects on the inter-relatedness of spousal stop-making decisions at the household level. The results show that the MOP model with a Clayton copula structure provides the best data fits and there is a very strong positive dependence among error terms of stop-making equations. Furthermore, the dependence among the maintenance activity propensities of household heads is asymmetric, with a stronger tendency of household heads to simultaneously have low maintenance activity levels than to simultaneously have high maintenance activity levels.

Multivariate Hydrologic Risk Analysis for River Thames

This study analyzed the multivariate flood risk for the river Thames at Kingston based on historical flood data from the National River Flow Archive (NRFA) website. The bivariate risk analysis framework was prepared from the joint return periods of the peak flow (m3/s) and 3-day annual maximum flow (m3/s) flood pair. A total of 137 samples of flood pairs from 1883 to 2019 were adopted for risk analysis. The multivariate return periods were characterized depending on the quantification of the bivariate flood frequency analysis of the pair through copulas methods. The unknown parameter of each copula was estimated using the method-of-moment (MOM) estimator based on Kendall’s tau inversion, in which the Clayton copula performed best to model the dependence of the two flood variables. Then, the bivariate hydrologic risk was characterized based on the joint return period in AND, established from the Clayton copula method. The results reveal that the flood pair would keep a constant hydrologic risk value for some time then moderately decrease as the 3-day AMAX flow increases from 700 m3/s. This hydrologic risk indicator was analyzed under four service time scenarios and three peak flows whose return periods were positioned at 50, 100, and 150 years. The outcomes from the bivariate risk analysis of the flood pairs can be used as decision support during the design of flood defenses and hydraulic facilities.

The Bivariate Defective Gompertz Distribution Based on Clayton Copula with Applications to Medical Data

In medical studies, it is common the presence of a fraction of patients who do not experience the event of interest. These patients are people who are not at risk of the event or are patients who were cured during the research. The proportion of immune or cured patients is known in the literature as cure rate. In general, the traditional existing lifetime statistical models are not appropriate to model data sets with cure rate, including bivariate lifetimes. In this paper, it is proposed a bivariate model based on a defective Gompertz distribution and also using a Clayton copula function to capture the possible dependence structure between the lifetimes. An extensive simulation study was carried out in order to evaluate the biases and the mean squared errors for the maximum likelihood estimators of the parameters associated to the proposed distribution. Some applications using medical data are presented to show the usefulness of the proposed model. Maximum likelihood and Bayesian methods were used to estimate the parameters of the model.

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.

Copula-Based Estimation Methods for a Common Mean Vector for Bivariate Meta-Analyses

Traditional bivariate meta-analyses adopt the bivariate normal model. As the bivariate normal distribution produces symmetric dependence, it is not flexible enough to describe the true dependence structure of real meta-analyses. As an alternative to the bivariate normal model, recent papers have adopted “copula” models for bivariate meta-analyses. Copulas consist of both symmetric copulas (e.g., the normal copula) and asymmetric copulas (e.g., the Clayton copula). While copula models are promising, there are only a few studies on copula-based bivariate meta-analysis. Therefore, the goal of this article is to fully develop the methodologies and theories of the copula-based bivariate meta-analysis, specifically for estimating the common mean vector. This work is regarded as a generalization of our previous methodological/theoretical studies under the FGM copula to a broad class of copulas. In addition, we develop a new R package, “CommonMean.Copula”, to implement the proposed methods. Simulations are performed to check the proposed methods. Two real dataset are analyzed for illustration, demonstrating the insufficiency of the bivariate normal model.

Dynamic signal selection strategies

This paper concerns the selection of a small number of predictors from a much larger set of potential stock signals. Selection is based on choosing predictors with high levels of dependency in stock returns. Eight different models of pairwise dependency are employed, including the Gaussian, t , Clayton, Frank and Gumbel copulas. In addition, predictive factors and returns are mapped to gamma, beta and standard bilateral gamma marginals with dependency constructed by using the magnitude of fractional common components from the same distributions. Each dependency model is used to select predictors. The predictions are used both directly and as required returns to build a measure for the value of an invested dollar. Stocks are ranked by these two metrics daily. For the mean-reversion strategy, we take short positions in a quarter of the top-ranked stocks and long positions in a quarter of the bottom-ranked stocks. The positioning is reversed under momentum. Trading performance results are presented for a variety of dependencies and sectors over the period from June 2006 to January 2021. The selection procedures are observed to deliver a reasonable set of trading strategies.

The implications of dependence, tail dependence, and bounds’ measures for counterparty credit risk pricing

This paper investigates the counterparty credit risk of interest rate swaps positions using the credit valuation adjustment (CVA) measure, and examines the potential dependence relationships between the probability of default (PD) and exposure at default (EAD). We empirically tested, using interest rate swaption implied market volatilities, six tail dependence models: a Basel III Committee independent model, a Gaussian, Student’s t, Gumbel, Clayton, and a wrong way risk (WWR) copula with dependence approach. The results show that the CVA underestimation when using a Clayton copula for modelling the dependence of PD and EAD is about 10%–856% compared to using WWR, and the underestimation between using the standardised Basel independent model and using the Gaussian copula is about 552%–2621%, including the period of the 2007/2008 crisis, signalling that dependence is more important than tail dependence for CVA measurement. This has important implications for regulators, financial institutions, and credit risk managers when calculating counterparty risk.

Probabilistic Seismic Demand Models and Life-Cycle Fragility Estimates for High-Rise Buildings

This paper presents a probabilistic framework for estimating the time-variant seismic fragility of high-rise buildings under carbonation-induced corrosion with a multiple damage model. A 42-story steel frame RC core tube (SFRCT) building was selected as a representative high-rise building to illustrate the proposed method. The main novelties of this study are that (1) a Bayesian updating method is used to determine the posterior probability distributions of the unknown parameters in the time-varying demand models of the maximum interstory drift and the residual drift at the top floor, (2) an analytical derivation method is used to compute the probability distribution of the corrosion initiation time in reinforcements, (3) the uncertainties associated with the corrosion initiation time and unknown parameters in demand models are incorporated into fragility estimates based on total probability theory, and (4) the elliptical copula and Archimedean copula are used to develop bivariate fragility estimates that consider the correlations among the limit states associated with maximum drifts and residual drifts, respectively. The results indicate that the seismic demand and fragility estimates of this high-rise building increase as the corrosion initiation time, which highlights the necessity of accounting for the uncertainty in the corrosion initiation time. In addition, the Gaussian copula and the Clayton copula (special cases of the elliptical copula and the Archimedean copula, respectively) produce similar joint probability distributions for the two-dimensional limit states; however, the Clayton copula can be used in a more general condition. The proposed methodology is applicable for evaluating the seismic life-cycle fragility of both existing and new high-rise buildings under multicriteria performance indicators.

Modelling to Engineering Data Using a New Class of Continuous Models

In this paper, a new flexible generator of continuous lifespan models referred to as the Topp-Leone Weibull G (TLWG) family is developed and studied. Several mathematical characteristics have been investigated. The new hazard rate of the new model can be “monotonically increasing,” “monotonically decreasing,” “bathtub,” and “J shape.” The Farlie Gumbel Morgenstern (FGM) and the modified FGM (MFGM) families and Clayton Copula (CCO) are used to describe and display simple type Copula. We discuss the estimation of the model parameters by the maximum likelihood (MLL) estimations. Simulations are carried out to show the consistency and efficiency of parameter estimates, and finally, real data sets are used to demonstrate the flexibility and potential usefulness of the proposed family of algorithms by using the TLW exponential model as example of the new suggested family.

Shared parameter and copula models for analysis of semicontinuous longitudinal data with nonrandom dropout and informative censoring.

Analysis of longitudinal semicontinuous data characterized by subjects' attrition triggered by nonrandom dropout is complex and requires accounting for the within-subject correlation, and modeling of the dropout process. While methods that address the within-subject correlation and missing data are available, approaches that incorporate the nonrandom dropout, also referred to informative right censoring, in the modeling step are scarce due to the computational intensity and possible intractable integration needed for its implementation. Appreciating the complexity of this problem and the need for a new methodology that is feasible for implementation, we propose to extend a framework of likelihood-based marginalized two-part models to account for informative right censoring. The censoring process is modeled using two approaches: (1) Poisson censoring for the count of visits before dropout and (2) survival time to dropout. Novel consideration was given to the proposed joint modeling approaches for the semicontinuous and censoring components of the likelihood function which included (1) shared parameter, and (2) Clayton copula. The cross-part and within-part correlations were accounted for through a complex random effect structure that models correlated random intercepts and slopes. Feasibility of implementation, and accuracy of these approaches were investigated using extensive simulation studies and clinical application.

Reliability estimation of [formula omitted]-out-of-[formula omitted] system in a multicomponent stress–strength dependent model based on copula function

In reliability analysis of the traditional stress–strength models, the stress and strength are often assumed to be independent variables. However the case where this two variables are dependent is more realistic in engineering. To evaluate multicomponent system reliability in such case, the stress and strength are assumed to have dependent Kumaraswamy variable and unit Gompertz variable based on Clayton copula. Then the maximum likelihood (ML), least squares (LS), maximum product of spacings (MPS), Cramér–von-Mises and LS (CL) hybrid estimates of the unknown parameters and system reliability are derived using two-step estimation procedure under order sample. Also the asymptotic distribution of the ML estimation and the parametric bootstrap percentile method are used to construct approximate confidence intervals (CI). Moreover, Monte Carlo simulations are implemented to compare the performance of the proposed methods. Finally, one real data set is analyzed for illustrative purposes.

Vine Copula Based Modeling

With the availability of massive multivariate data comes a need to develop flexible multivariate distribution classes. The copula approach allows marginal models to be constructed for each variable separately and joined with a dependence structure characterized by a copula. The class of multivariate copulas was limited for a long time to elliptical (including the Gaussian and t-copula) and Archimedean families (such as Clayton and Gumbel copulas). Both classes are rather restrictive with regard to symmetry and tail dependence properties. The class of vine copulas overcomes these limitations by building a multivariate model using only bivariate building blocks. This gives rise to highly flexible models that still allow for computationally tractable estimation and model selection procedures. These features made vine copula models quite popular among applied researchers in numerous areas of science. This article reviews the basic ideas underlying these models, presents estimation and model selection approaches, and discusses current developments and future directions.

Design loads and reliability assessment of marine structures considering statistical models of metocean data

The objective of this study is to investigate the effect of statistical models for constructing the bivariate distribution of metocean data on design loads and reliability assessment of offshore structures. First, the conventional conditional joint distribution and single copula models are utilized for characterizing the statistical data. Additionally, a novel mixed copula model is developed using a linear combination of Gaussian, Clayton, Gumbel and Frank copula considering various tails characteristics, the parameter estimation is performed using expectation-maximization (EM) algorithm. Moreover, a mixed joint model based on linear combinations of simple statistical models is proposed. Then, the environmental contours are constructed based on various joint probability distribution of environmental variables. Finally, the reliability of marine structures is estimated using direct Monte Carlo simulation considering dependence structure and joint models of statistical data. The results indicate the reliability of offshore structure differs significantly because the bivariate distribution models of environmental variables could not be defined uniquely, the dependence structure has non-negligible effect on reliability assessment. A mixed copula can be suitable for describing variables with symmetrical or asymmetric, independent or correlative, upper or lower tails, or complex characteristics. The mixed joint model provides satisfactory results for marginal and joint fitting.

On the Use of Copula for Quality Control Based on an AR(1) Model

Manufacturing for a multitude of continuous processing applications in the era of automation and ‘Industry 4.0’ is focused on rapid throughput while producing products of acceptable quality that meet customer specifications. Monitoring the stability or statistical control of key process parameters using data acquired from online sensors is fundamental to successful automation in manufacturing applications. This study addresses the significant problem of positive autocorrelation in data collected from online sensors, which may impair assessment of statistical control. Sensor data collected at short time intervals typically have significant autocorrelation, and traditional statistical process control (SPC) techniques cannot be deployed. There is a plethora of literature on techniques for SPC in the presence of positive autocorrelation. This paper contributes to this area of study by investigating the performance of ‘Copula’ based control charts by assessing the average run length (ARL) when the subsequent observations are correlated and follow the AR(1) model. The conditional distribution of yt given yt−1 is used in deriving the control chart limits for three different categories of Copulas: Gaussian, Clayton, and Farlie-Gumbel-Morgenstern Copulas. Preliminary results suggest that the overall performance of the Clayton Copula and Farlie-Gumbel-Morgenstern Copula is better compared to other Archimedean Copulas. The Clayton Copula is the more robust with respect to changes in the process standard deviation as the correlation coefficient increases.