Copula-based Quadratic Point Estimate Method for probabilistic moments evaluation

Abstract

Numerous engineering problems often involve the evaluation of probabilistic moments of quantities of interest (QoI). Several computation techniques exist in the literature for such estimations, with sampling-based methods being the most generally applicable ones. However, typical sampling methods most often require numerous model-calls in high dimensional spaces to achieve a sufficient accuracy. A different approach to this problem is sought through the development of dedicated probabilistic techniques resembling quadrature methods, such as the Point Estimate Method (PEM) and the Unscented Transformation (UT). PEM and UT-based methods rely on deterministic, weighted sampling points, established by matching a few input moments, sometimes doing so by employing optimization techniques for each considered problem. Several methods have been thus accordingly suggested, mainly having either linear or exponential increase of the number of required samples with increasing dimensions, counterbalancing computational demands and estimation accuracy. Recently, the authors developed the Quadratic Point Estimate Method (QPEM) with 2n^2+1 sampling points, where n is the number of dimensions, providing analytical expressions for sample locations and weights in the Gaussian space without any optimization procedure requirements. QPEM can significantly improve the estimation accuracy of the output QoI moments, in relation to PEM and UT-based methods with linear samples increase, while at the same time having an affordable and competitive computational cost up to a considerable number of dimensions. The QPEM is further enhanced and generalized in this work by enabling copula integration into the framework, that enable effective modeling of the joint input probability density function (PDF) by estimating marginals and the random variables dependence structure. The copula-based QPEM involves three practical cases in this work, when: (a) the input PDF can be estimated from available data; (b) the input PDF is partially known, through marginals and covariance structures, and can be transformed to the Gaussian space; and (c) the input PDF is partially known but cannot be mapped to the Gaussian space. The validity and outstanding performance of copula-based QPEM are showcased against numerous other sampling methods, in various static and dynamic examples, also involving spatial stochastic fields and random excitations, with emphasis on several geotechnical applications and problems, where PEM related methods have a long-history of development and successful implementations.