A Riemannian Tool for Clustering of Geo-Spatial Multivariate Data

Abstract

Geological modeling is essential for the characterization of natural phenomena and can be done in two steps: (1) clustering the data into consistent groups and (2) modeling the extent of these groups in space to define domains, honoring the labels defined in the previous step. The clustering step can be based on the information of continuous multivariate data in space instead of relying on the geological logging provided. However, extracting coherent spatial multivariate information is challenging when the variables show complex relationships, such as nonlinear correlation, heteroscedastic behavior, or spatial trends. In this work, we propose a method for clustering data, valid for domaining when multiple continuous variables are available and robust enough to deal with cases where complex relationships are found. The method looks at the local correlation matrix between variables at sample locations inferred in a local neighborhood. Changes in the local correlation between these attributes in space can be used to characterize the domains. By endowing the space of correlation matrices with a manifold structure, matrices are then clustered by adapting the K-means algorithm to this manifold context, using Riemannian geometry tools. A real case study illustrates the methodology. This example demonstrates how the clustering methodology proposed honors the spatial configuration of data delivering spatially connected clusters even when complex nonlinear relationships in the attribute space are shown.