Multivariate Modeling with Copulas and Engineering Applications

Abstract

This chapter reviews multivariate modeling with copulas and provides novel applications in engineering. 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. A software package has been developed to promote the development and application of copula-based methods. Section 51.1 introduces the concept of copulas and its connection to multivariate distributions. The most important result about copulas is Sklarʼs theorem, which shows that any continuous multivariate distribution has a canonical representation by a unique copula and all its marginal distributions. A general algorithm to simulate random vectors from a copula is also presented. Section 51.2 introduces two commonly used classes of copulas: elliptical copulas and Archimedean copulas. Simulation algorithms are also presented. Section 51.3 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 51.4 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 a better fit to the data than a normal copula. Through simulation, the upper control limit of Hotellingʼs 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 51.5 concludes and points to references about other aspects of copula-based multivariate modeling that are not discussed in this chapter. An open-source software package for the R project has been developed to promote copula-related methodology development and applications. An introduction to the package and illustrations are provided in the Appendix.