Spatial mixed effects model for compositional data with applications to coal geology

• We derive lognormal kriging and block kriging equations.
• We extend the SME model to multivariate analysis.
• We show a log-normal approach to compositional data analysis.
• The implementation of our methods is illustrated on coal geochemical data.
• There is higher precision when estimating block averages than individual locations.

We analyze data on the geochemical make-up of coal samples throughout the state of Illinois. The goal is to estimate the geochemical properties at unobserved locations over a specified region. Multivariate spatial modeling requires characterization of both spatial and cross-spatial covariances. Reduced rank spatial models are popular in analyzing large spatial datasets. We develop a multivariate spatial mixed effects model for log-normal processes and show how to implement with compositional data to predict on point locations, as well as the average prediction over a finite area. We use log-normal kriging for the components of compositional data, and show how to obtain estimates and measures of precision in isometric log-ratio coordinates.