A Riemannian Approach to Multivariate Geostatistical Modeling

TL;DR

This paper introduces a Riemannian framework for geostatistical modeling of correlation matrices, enabling locally adaptive linear models and coherent interpolation of correlation data on a non-linear manifold.

Contribution

It proposes novel methods leveraging Riemannian geometry to improve modeling and clustering of multivariate geoscience data with non-linear correlation structures.

Findings

Effective interpolation of correlation matrices on the Riemannian manifold.

Locally varying coregionalization models improve data representation.

Enhanced clustering of multivariate geological data.

Abstract

In geosciences, the use of classical Euclidean methods is unsuitable for treating and analyzing some types of data, as this may not belong to a vector space. This is the case for correlation matrices, belonging to a subfamily of symmetric positive definite matrices, which in turn form a cone shape Riemannian manifold. We propose two novel applications for dealing with the problem of accounting with the non-linear behavior usually presented on multivariate geological data by exploiting the manifold features of correlations matrices. First, we employ an extension for the linear model of coregionalization (LMC) that alters the linear mixture, which is assumed fixed on the domain, and making it locally varying according to the local strength in the dependency of the coregionalized variables. The main challenge, once this relaxation on the LMC is assumed, is to solve appropriately the interpolation of the different known correlation matrices throughout the domain, in a reliable and coherent fashion. The present work adopts the non-euclidean framework to achieve our objective by locally averaging and interpolating the correlations between the variables, retaining the intrinsic geometry of correlation matrices. A second application deals with the problem of clustering of multivariate data.