Copula-based Method Research Articles

Exploring copulas for the imputation of complex dependent data

In this work we introduce a copula-based method for imputing missing data by using conditional density functions of the missing variables given the observed ones. In theory, such functions can be derived from the multivariate distribution of the variables of interest. In practice, it is very difficult to model joint distributions and derive conditional distributions, especially when the margins are different. We propose a natural solution to the problem by exploiting copulas so that we derive conditional density functions through the corresponding conditional copulas. The approach is appealing since copula functions enable us (1) to fit any combination of marginal distribution functions, (2) to take into account complex multivariate dependence relationships and (3) to model the marginal distributions and the dependence structure separately. We describe the method and perform a Monte Carlo study in order to compare it with two well-known imputation techniques: the nearest neighbour donor imputation and the regression imputation by EM algorithm. Our results indicate that the proposal compares favourably with classical methods in terms of preservation of microdata, margins and dependence structure.

Assessing the correlated performance functions of an engineering system via probabilistic analysis

An important issue regarding the use of probabilistic predictions for complex engineering systems is characterising the dependence structure among its correlated performance functions, which are driven by dependent or independent basic random variables. The interrelationship of these performance functions can be attributed to the same random variables and the cross correlation among the input parameters. An assessment of joint failure probability for an engineering system is proposed, which is associated with the correlated performance functions using a copula-based method by conveying the dependence structure of the performance functions. The method is demonstrated with four simple engineering problems, i.e., (a) bivariate distribution in which two predetermined performance functions are associated with each other; (b) pile bearing capacity in which the performance functions are related with the soil internal friction and the compressive strength of a concrete pile; (c) pipe flow in which the performance function of three pipes in a sewer system is assessed with six independent random variables; and (d) retaining wall in which the failure criteria for defining the performance functions include overturning failure about the toe point, sliding failure along the base, and bearing capacity instability considering uncertain soil properties. The computational efficiency is evaluated using the results based on the conventional bounding methods. The joint failure probability expressed by copulas provides a means to obtain the joint probabilities of multiple failure modes, which pave the way for an objective description of the overall failure probability of a practical engineering problem.

Infilling missing precipitation records – A comparison of a new copula-based method with other techniques

Summary Infilling missing data might be an unpleasant and tedious task, but is necessary for analysis and water resources management, so it should not be done in a lackadaisical manner. The important thing about the infilled values is that they need to be as good as possible, because poor infilling is likely to lead to poor decisions. Traditionally, a range of methods is routinely employed, e.g. Nearest Neighbor substitution through to Kriging, but few methods attach a quality estimate to the infilled values. In this paper a new copula based method is developed for infilling missing daily and monthly rain gauge data and is compared against six other commonly used methods, in a semi-arid environment with a range of rain-rates and interstation distances, in the Southern Cape region of South Africa. For daily data it is clear that the copula-based methods are superior to the others in terms of point estimation and have the added benefit of providing an estimate of the precision of the interpolation, not provided by the others. In our case, the addition of atmospheric circulation patterns designed to add information for infilling has a relatively small positive effect on the quality of the estimation. The main reason for this is that a small number of wet days does not allow a good estimation of the conditional distribution of precipitation amounts; we note that the average probability of a dry day in this region is 86%. An improvement of the estimate of the probability of a dry day was however observed. In other regions, with a higher number of wet days, an atmospheric Circulation Pattern (CP) based method is likely to lead to further improvements. Using copula-based methods, the estimated probabilities of a dry day correspond well to the observed frequencies of dry days. The monthly data yield the same conclusion, with the qualification that the Expectation Maximisation [EM] algorithm performs as well as the copula method (because of the low count of dry months in this region) but its relative disadvantage is that it does not offer as valuable a precision estimate.

A Copula-Based Method for Estimating Shear Strength Parameters of Rock Mass

The shear strength parameters (i.e., the internal friction coefficientfand cohesionc) are very important in rock engineering, especially for the stability analysis and reinforcement design of slopes and underground caverns. In this paper, a probabilistic method, Copula-based method, is proposed for estimating the shear strength parameters of rock mass. The optimal Copula functions between rock mass qualityQandf,Qandcfor the marbles are established based on the correlation analyses of the results of 12 sets of in situ tests in the exploration adits of Jinping I-Stage Hydropower Station. Although the Copula functions are derived from the in situ tests for the marbles, they can be extended to be applied to other types of rock mass with similar geological and mechanical properties. For another 9 sets of in situ tests as an extensional application, by comparison with the results from Hoek-Brown criterion, the estimated values offandcfrom the Copula-based method achieve better accuracy. Therefore, the proposed Copula-based method is an effective tool in estimating rock strength parameters.

Using copulas to characterise the dependency of GCL shear strengths

ABSTRACT: The sensitivity of landfill design to shear strength variability is controlled by the measured internal shear strength of the geosynthetic clay liner and the interface shear strength between the geosynthetic clay liner and the geomembrane. Based on the Mohr–Coulomb criterion, the measured shear strength values (including the internal and interface shear strength values) depend on the cohesion and friction angle. Here, a novel and efficient mathematical tool for analysing the properties of the bivariate and the correlated geosynthetic shear strength was constructed. The arbitrary marginal probability density distribution of the componential shear strength was organised as a bivariate joint distribution with a copula function to characterise the dependency between the different shear strengths. Generally, a copula-based method provides a simple and powerful framework for modelling interdependence among variables. Thus, the construction of copulas as an alternative to correlation coefficients is addressed. In addition, the shear strength marginal distributions were described by a non-normal distribution, which was identified by the Akaike information criterion. The dependency of these margins is implemented from several empirical copulas, which were chosen from the most frequently used Elliptical and Archimedean families. The simulated correlated material properties that were generated by the copula-based sampling method were used as input values to calculate the performance of a landfill lining problem. Moreover, the computed reliability indexes were compared with the reliability indexes that were obtained by the first-order second moment method. In addition, a sensitivity study with various correlation coefficients was conducted to determine the influence of the correlation measurements on the reliability index.

Impact of copulas for modeling bivariate distributions on system reliability

A copula-based method is presented to investigate the impact of copulas for modeling bivariate distributions on system reliability under incomplete probability information. First, the copula theory for modeling bivariate distributions as well as the tail dependence of copulas are briefly introduced. Then, a general parallel system reliability problem is formulated. Thereafter, the system reliability bounds of the parallel systems are generalized in the copula framework. Finally, an illustrative example is presented to demonstrate the proposed method. The results indicate that the system probability of failure of a parallel system under incomplete probability information cannot be determined uniquely. The system probabilities of failure produced by different copulas differ considerably. Such a relative difference in the system probabilities of failure associated with different copulas increases greatly with decreasing component probability of failure. The maximum ratio of the system probabilities of failure for the other copulas to those for the Gaussian copula can happen at an intermediate correlation. The tail dependence of copulas has a significant influence on parallel system reliability. The copula approach provides new insight into the system reliability bounds in a general way. The Gaussian copula, commonly used to describe the dependence structure among variables in practice, produces only one of the many possible solutions of the system reliability and the calculated probability of failure may be severely biased.

Bivariate distribution models using copulas for reliability analysis

The modeling of joint probability distributions of correlated variables and the evaluation of reliability under incomplete probability information remain a challenge that has not been studied extensively. This article aims to investigate the effect of copulas for modeling dependence structures between variables on reliability under incomplete probability information. First, a copula-based method is proposed to model the joint probability distributions of multiple correlated variables with given marginal distributions and correlation coefficients. Second, a reliability problem is formulated and a direct integration method for calculating probability of failure is presented. Finally, the reliability is investigated to demonstrate the effect of copulas on reliability. The joint probability distribution of multiple variables, with given marginal distributions and correlation coefficients, can be constructed using copulas in a general and flexible way. The probabilities of failure produced by different copulas can differ considerably. Such a difference increases with decreasing probability of failure. The reliability index defined by the mean and standard deviation of a performance function cannot capture the difference in the probabilities of failure produced by different copulas. In addition, the Gaussian copula, often adopted out of expedience without proper validation, produces only one of the various possible solutions of the probability of failure and such a probability of failure may be biased towards the non-conservative side. The tail dependence of copulas has a significant influence on reliability.

Reliability Research of k-out-of-n: G Supply Chain System Based on Copula

Supply chain management has received considerable attention in the literature and it is meaningful and important to be able to measure the reliability of supply chains. In the article, the suppliers in the supply chain systems are not independent of each other and the dependency relation may be either linear or nonlinear correlation. From the view of the distribution service process, a copula-based method is proposed for analyzing the reliability of supply chains. In this article, by introducing the model of k-out-of-n: G system into the studies of supply chains, an evaluation method is suggested and the reliability indexes are obtained. Finally, a numerical example is presented to illustrate the results obtained in this article.

Uncertainty analysis of correlated non-normal geotechnical parameters using Gaussian copula

Determining the joint probability distribution of correlated non-normal geotechnical parameters based on incomplete statistical data is a challenging problem. This paper proposes a Gaussian copula-based method for modelling the joint probability distribution of bivariate uncertain data. First, the concepts of Pearson and Kendall correlation coefficients are presented, and the copula theory is briefly introduced. Thereafter, a Pearson method and a Kendall method are developed to determine the copula parameter underlying Gaussian copula. Second, these two methods are compared in computational efficiency, applicability, and capability of fitting data. Finally, four load-test datasets of load-displacement curves of piles are used to illustrate the proposed method. The results indicate that the proposed Gaussian copula-based method can not only characterize the correlation between geotechnical parameters, but also construct the joint probability distribution function of correlated non-normal geotechnical parameters in a more general way. It can serve as a general tool to construct the joint probability distribution of correlated geotechnical parameters based on incomplete data. The Gaussian copula using the Kendall method is superior to that using the Pearson method, which should be recommended for modelling and simulating the joint probability distribution of correlated geotechnical parameters. There exists a strong negative correlation between the two parameters underlying load-displacement curves. Neglecting such correlation will not capture the scatter in the measured load-displacement curves. These results substantially extend the application of the copula theory to multivariate simulation in geotechnical engineering.

The spatial structure of European wind storms as characterized by bivariate extreme-value Copulas

Abstract. The winds associated with extra-tropical cyclones are amongst the costliest natural perils in Europe. Re/insurance companies typically have insured exposure at multiple locations and hence the losses they incur from any individual storm crucially depend on that storm's spatial structure. Motivated by this, this study investigates the spatial structure of the most extreme windstorms in Europe. The data consists of a carefully constructed set of 135 of the most damaging storms in the period 1972–2010. Extreme value copulas are applied to this data to investigate the spatial dependencies of gusts. The copula method is used to investigate three aspects of windstorms. First, spatial maps of expected hazard damage between large cities and their surrounding areas are presented. Second, we demonstrate a practical application of the copula method to benchmark catalogues of artificial storms for use in the re/insurance sector. Third, the copula-based method is used to investigate the sensitivity of spatially aggregated damage to climate variability. The copula method allows changes to be expressed in terms of storm frequency, local intensity, and storm spatial structure and gives a more detailed view of how climate variability may affect multi-location risk in Europe.

Dependence-Preserving Approach to Synthesizing Household Characteristics

One effective approach to study day-to-day traveler behavior is through the activity-based traffic demand model, in which all travelers are treated as individual agents and interact under a computation-intensive framework. Nevertheless, because of high survey costs, low response rate, and privacy concerns, detailed household and personal characteristics are usually unavailable. Various population synthesizers were therefore proposed to reconstruct a methodologically rigorous estimate of household characteristics from different surveys. For instance, the iterative proportional fitting (IPF) algorithm is used to synthesize the full population from the public use microdata sample (PUMS) and Census Summary File 3 (SF3) in the popular activity-based traffic demand model, TRANSIMS. However, some fundamental limitations of IPF (e.g., zero cells in the contingency table as a result of small sample size) have drawn sufficient attention and resulted in the development of enhanced IPF algorithms and other strategies. This paper proposes a copula-based method to synthesize household characteristics that preserves marginal distributions and dependence structure between variables. The proposed method is tested for the state of Iowa, and the results are compared with the IPF approach of TRANSIMS. The synthesized households resulted in the same local SF3 statistics at each block group. But having similar inter-variable correlations as described in the PUMS suggest the applicability of the copula-based approach. Because marginal distributions and dependence structure can be faithfully preserved, the proposed method could be a suitable alternative to synthesize realistic agent characteristics for further activity-based traffic demand modeling.

Copula-based statistical refinement of precipitation in RCM simulations over complex terrain

Abstract. This paper presents a new Copula-based method for further downscaling regional climate simulations. It is developed, applied and evaluated for selected stations in the alpine region of Germany. Apart from the common way to use Copulas to model the extreme values, a strategy is proposed which allows to model continuous time series. As the concept of Copulas requires independent and identically distributed (iid) random variables, meteorological fields are transformed using an ARMA-GARCH time series model. In this paper, we focus on the positive pairs of observed and modelled (RCM) precipitation. According to the empirical copulas, significant upper and lower tail dependence between observed and modelled precipitation can be observed. These dependence structures are further conditioned on the prevailing large-scale weather situation. Based on the derived theoretical Copula models, stochastic rainfall simulations are performed, finally allowing for bias corrected and locally refined RCM simulations.

Characterizing impulsive wave-in-deck loads on coastal bridges by probabilistic models of impact maxima and rise times

When waves impact a seawall, a vertical breakwater, an exposed jetty, a pier or a coastal bridge, they abruptly transfer their momentum into the structure. This energy transfer can be very violent and its duration exceptionally short. In the case of coastal bridges, whose spans are designed to have very short vibration period, wave impacts might have duration comparable to the natural period of oscillation of the structure, which therefore becomes prone to damage and failure. Previous forensic studies have documented the relative importance of impulsive loads on deck suspended structure, demonstrating the need to assess the effect of wave impacts on both the stability and the integrity of structural members since the early stages of the design. This requires the estimation of the dynamic characteristics of the loading pattern, and in particular the wave impulse and corresponding impact maxima and rise times. Based on the conservation of momentum, functional relationships between these parameters have been identified since pioneering work dating back to the late '30s of the 20th century. The complexity of the loading process, however, results in a significantly large variability of wave impact maxima and rise times even under similar conditions, suggesting the need for a probabilistic approach to the definition of the relationship between these two variables, to be applied when estimating the dynamic properties of wave for use in structural analysis of coastal structures. In the recent past, some effort has been made to identify functional relationships between such quantities; these require the assessment of the conditional quantiles (or similarly the conditional distribution) of wave impact maxima given the rise times. In this paper, we compare three different statistical methods proposed in the literature to accomplish this task, in order to assess the reliability of the approach and suggest guidelines for practical applications. A copula-based method, Generalized Additive Models for Location, Scale and Shape (GAMLSS), and quantile regression are applied to measurements from large-scale 3-dimensional physical model tests. The investigation suggests that quantile regression gives the simplest results to be used in practice; copula approach and GAMLSS are possible alternative when semi-parametric or fully parametric modeling is needed.

Copula-Based Method for Addressing Endogeneity in Models of Severity of Traffic Crash Injuries: Application to Two-Vehicle Crashes

An emerging copula-based methodology is used to address endogeneity in models of crash injury severity. Specifically, two important sources of endogeneity are addressed: (a) endogeneity due to correlations between the injury severity of the two drivers involved in two-vehicle crashes and (b) endogeneity of collision type and injury severity outcomes. Two sets of copula-based joint model systems are formulated and estimated by using data on two-vehicle crashes from the 2007 Generalized Estimates System: a copula-based joint ordered logit–ordered logit model of injury severities of the two drivers involved in two-vehicle crashes and a copula-based joint multinomial logit–ordered logit model of collision type and injury severity outcomes of two-vehicle crashes. The model estimation results and elasticity estimates underscore the importance of accommodating endogeneity in models of crash injury severity. The results shed new light on the determinants of injury severity in two-vehicle accidents.

A General Approach to Integrated Risk Management with Skewed, Fat-Tailed Risk

The goal of integrated risk management in a financial institution is to measure and manage risk and capital across a range of diverse business activities. This requires an approach for aggregating risk types (market, credit, and operational) whose distributional shapes vary considerably. In this paper, we construct the joint risk distribution for a typical large, internationally active bank using the method of copulas. This technique allows us to incorporate realistic marginal distributions that capture some of the essential empirical features of these risks like skewness and fat-tails while allowing for a rich dependence structure. We explore the impact of business mix and inter-risk correlations on total risk, whether measured by value-at-risk or expected shortfall. We find that given a risk type, total risk is more sensitive to differences in business mix or risk weights than to differences in inter-risk correlations. There is a complex relationship between volatility and fat-tails in determining the total risk: depending on the setting, they either offset or reinforce each other. The choice of copula (normal versus Student-t), which determines the level of tail dependence, has a more modest effect on risk. We then compare the copula-based method with several conventional approaches to computing risk, each of which may be thought of as an approximation. One easily implemented approximation, which uses empirical correlations and quantile estimates, tracks the copula approach surprisingly well. In contrast, the additive approximation, which assumes no diversification benefit, typically overestimates risk by about 30-40%.

A general approach to integrated risk management with skewed, fat-tailed risks

Integrated risk management for financial institutions requires an approach for aggregating risk types (market, credit, and operational) whose distributional shapes vary considerably. We construct the joint risk distribution for a typical large, internationally active bank using the method of copulas. This technique allows us to incorporate realistic marginal distributions that capture essential empirical features of these risks such as skewness and fat-tails while allowing for a rich dependence structure. We explore the impact of business mix and inter-risk correlations on total risk. We then compare the copula-based method with several conventional approaches to computing risk.