Multivariate Modeling with Copulas and Engineering Applications

Abstract

This chapter reviews multivariate modeling using copulas with illustrative applications in engineering such as multivariate process control and degradation analysis. A copula separates the dependence structure of a multivariate distribution from its marginal distributions. Properties and statistical inferences of copula-based multivariate models are discussed in detail. Applications in engineering are illustrated via examples of bivariate process control and degradation analysis, using existing data in the literature. An R package copula facilitates developments and applications of copula-based methods. The major change from the last version (Yan 2006) is the update on the R package copula (Hofert et al. 2018). Section 46.1 provides the background and motivation of multivariate modeling with copulas. Most multivariate statistical methods are based on the multivariate normal distribution, which cannot meet the practical needs to fit non-normal multivariate data. Copula-based multivariate distributions offer much more flexibility in modeling various non-normal data. They have been widely used in insurance, finance, risk management, and medical research. This chapter focuses on their applications in engineering. Section 46.2 introduces the concept of copulas and its connection to multivariate distributions. The most important result about copulas is Sklar’s (1959) theorem which shows that any continuous multivariate distribution has a canonical representation by a unique copula and all its marginal distributions. Scale-invariant dependence measures for two variables, such as Kendall’s tau and Spearman’s rho, are completely determined by their copula. The extremes of these two concordance measures, − 1 and 1, are obtained under perfect dependence, corresponding to the Fréchet-Hoeffding lower and upper bounds of copulas, respectively. A general algorithm to simulate random vectors from a copula is also presented. Section 46.3 introduces two commonly used classes of copulas: elliptical copulas and Archimedean copulas. Elliptical copulas are copulas of elliptical distributions. Two most widely used elliptical copulas, the normal copula and the t copula, are discussed. Archimedean copulas are constructed without referring to distribution functions and random variables. Three popular Archimedean families, Clayton copula, Frank copula, and Gumbel copula, each having a mixture representation with a known frailty distribution, are discussed. Simulation algorithms are also presented. Section 46.4 presents the maximum likelihood inference of copula-based multivariate distributions given the data. Three likelihood approaches are introduced. The exact maximum likelihood approach estimates the marginal and copula parameters simultaneously by maximizing the exact parametric likelihood. The inference functions for margins approach is a two-step approach, which estimates the marginal parameters separately for each margin in a first step and then estimates the copula parameters given the the marginal parameters. The canonical maximum likelihood approach is for copula parameters only, using uniform pseudo-observations obtained from transforming all the margins by their empirical distribution functions. Section 46.5 presents two novel engineering applications. The first example is a bivariate process control problem, where the marginal normality seems appropriate, but joint normality is suspicious. A Clayton copula provides better fit to the data than a normal copula. Through simulation, the upper control limit of Hotelling’s T2 chart based on normality is shown to be misleading when the true copula is a Clayton copula. The second example is a degradation analysis, where all the margins are skewed and heavy-tailed. A multivariate gamma distribution with normal copula fits the data much better than a multivariate normal distribution. Section 46.6 concludes and points to references about other aspects of copula-based multivariate modeling that are not discussed in this chapter.