Simulation of non-Gaussian processes using fractile correlation

Abstract

The difficulties of simulating non-Gaussian stochastic processes to follow arbitrary product–moment covariance models and arbitrary non-Gaussian marginal distributions are well known. This paper proposes to circumvent these difficulties by prescribing a fractile correlation function, rather than the usual product–moment covariance function. This fractile correlation can be related to the product–moment correlation of a Gaussian process analytically. A Gaussian process with the requisite product–moment correlation can be simulated using the Karhunen–Loeve (K–L) expansion and transformed to satisfy any arbitrary marginal distribution using the usual CDF mapping. The fractile correlation of the non-Gaussian process will be identical to that of the underlying Gaussian process because it is invariant to monotone transforms. This permits the K–L expansion to be extended in a very general way to any second-order non-Gaussian processes. The simplicity of the proposed approach is illustrated numerically using a stationary squared exponential and a non-stationary Brown–Bridge fractile correlation function in conjunction with a shifted lognormal and a shifted exponential marginal distribution.