Copula Technique Research Articles

Copulas and Other Multivariate Models of Personal Exposures to VOC Mixtures

ABSTRACT While exposures to most pollutants occur in mixtures with other pollutants, joint distributions of pollutants have not been characterized, even for pollutants that frequently occur together. This study investigates exposures to volatile organic compounds (VOCs), and presents and evaluates innovative ways of modeling mixtures. Personal exposures to ten VOCs for 655 individuals were evaluated using a semi-parametric framework that estimates probabilities of mixtures containing components greater than specified levels. Common mixtures were identified using positive matrix factorization. Copulas were tested for their ability to model inter-pollutant dependencies, focusing on the most exposed individuals. For highly correlated VOCs, Gumbel and Gaussian copulas best described the joint probability. For VOCs with low correlations, distributions could be described by the product copula, among other approaches. High concentration mixtures usually resulted from strong sources (e.g., residence with attached garage), certain behaviors (e.g., cigarette smoking), and sometimes from multiple but unrelated sources (e.g., mothballs, solvents). This is the first article to characterize exposures to VOC mixtures measured using personal monitoring and a population-based sample, and it appears to be the first to apply the copula technique, which allows accurate modeling of joint distributions with only one additional parameter beyond those needed to represent the marginal distributions.

Interperiod Dependence of Ground-Motion Prediction Equations: A Copula Perspective

Abstract The conventional approach for developing empirical prediction equations of multivariate ground-motion parameters (e.g., response spectra at a set of vibration periods) is based on regression analysis of the individual parameters; their dependence is subsequently characterized by evaluating the linear correlation coefficient of the logarithm of the ground-motion parameters that are corrected by using a ground-motion prediction equation. A copula technique, which offers a flexible way of describing nonlinear dependence among multivariate data in isolation from their marginal distribution functions, can be used to validate this two-step approach. This new perspective on the multivariate aspects of the development of ground-motion prediction equations is explored through analysis of the strong ground-motion data associated with the Boore and Atkinson (2008) Pacific Earthquake Engineering Research–Next Generation Attenuation of Ground Motions (PEER-NGA) relation. The analysis results demonstrate that multivariate ground-motion parameters can be marginally modeled as a lognormal variate, and their interperiod dependence can be captured by using the normal copula. This finding validates the conventional two-step approach.

A Multivariate Jump-Driven Financial Asset Model

In this paper, we propose a multivariate financial model for nancial assets which incorporates jumps, skewness, kurtosis and stochastic volatility, and discuss its applications in the context of equity and credit risk. In the former case we describe the stochastic behavior of a series of stocks or indexes, in the latter we apply the model in a multi-firm, value-based default model. Starting from a independent Brownian world, we will introduce jumps and other deviations from normality, as well as non-Gaussian dependence, by the simple but very strong technique of stochastic time-changing. We work out the details in the case of a Gamma time-change, thus obtaining a multivariate Variance Gamma (VG) setting. We are able to characterize the model from an analytical point of view, by writing down the joint distribution function of the assets at any point in time and by studying their association via the copula technique. The model is also computationally friendly, since numerical results require a modest amount of time and the number of parameters grows linearly with the number of assets. The main feature of the model however is the fact that - opposite to other, non jointly Gaussian settings - its risk neutral dependence can be calibrated from univariate derivative prices. Examples from the equity and credit market show the goodness of fit attained.