Do Rosenblatt and Nataf isoprobabilistic transformations really differ?

Abstract

This article is the third in a series dedicated to the mathematical study of isoprobabilistic transformations and their relationship with stochastic dependence modelling, see [R. Lebrun, A. Dutfoy, An innovating analysis of the Nataf transformation from the viewpoint of copula, Probabilistic Engineering Mechanics (2008). doi: 10.1016/j.probengmech.2008.08.001] for an interpretation of the Nataf transformation in term of normal copula and [R. Lebrun, A. Dutfoy, A generalization of the Nataf transformation to distributions with elliptical copula, Probabilistic Engineering Mechanics (24) (2009), 172–178. doi:10.1016/j.probengmech.2008.05.001] for a generalisation of the Nataf transformation to any elliptical copula. In this article, we explore the relationship between two isoprobabilistic transformations widely used in the community of reliability analysts, namely the Generalised Nataf transformation and Rosenblatt transformation. First, we recall the elementary results of the copula theory that are needed in the remaining of the article, as a preliminary section to the presentation of both the Generalized Nataf transformation and the Rosenblatt transformation in the light of the copula theory. Then, we show that the Rosenblatt transformation using the canonical order of conditioning is identical to the Generalised Nataf transformation in the normal copula case, which is the most usual case in reliability analysis since it corresponds to the classical Nataf transformation. At this step, we also show that it is not possible to extend the Rosenblatt transformation to distributions with general elliptical copula the way the Nataf transformation has been generalised. Furthermore, we explore the effect of the conditioning order of the Rosenblatt transformation on the usual reliability indicators obtained from a FORM or SORM method. We show that in the normal copula case, all these reliability indicators, excepted the importance factors, are unchanged whatever the conditioning order one chooses. In the last section, we conclude the article with two numerical applications that illustrate the previous results: the equivalence between both transformations in the normal copula case, and the effect of the conditioning order in the normal and non-normal copula case.