A Truly Multivariate Normal Score Transform Based on Lagrangian Flow

Abstract

In many geostatistical applications, a transformation to standard normality is a first step in order to apply standard algorithms in two-point geostatistics. However, in the case of a set of collocated variables, marginal normality of each variable does not imply multivariate normality of the set, and a joint transformation is required. In addition, current methods are not affine equivariant, as should be required for multivariate regionalized data sets without a unique, canonical representation (e.g., vector-valued random fields, compositional random fields, layer cake models). This contribution presents an affine equivariant method of Gaussian anamorphosis based on a flow deformation of the joint sample space of the variables. The method numerically solves the differential equation of a continuous flow deformation that would transform a kernel density estimate of the actual multivariate density of the data into a standard multivariate normal distribution. Properties of the flow anamorphosis are discussed for a synthetic application, and the implementation is illustrated via two data sets derived from Western Australian mining contexts.