Modeling Random Directions in 2D Simplex Data

We propose models and algorithms for learning about random directions in two-dimensional simplex data, and apply our methods to the study of income level proportions and their changes over time in a geostatistical area. There are several notable challenges in the analysis of simplex-valued data: the measurements must respect the simplex constraint and the changes exhibit spatiotemporal smoothness while allowing for possible heterogeneous behaviors. To that end, we propose Bayesian models that rely on and expand upon building blocks in circular and spatial statistics by exploiting suitable transformation based on the polar coordinates for circular data. Our models also account for spatial correlation across locations in the simplex and the heterogeneous patterns via mixture modeling. We describe some properties of the models and model fitting via MCMC techniques. Our models and methods are illustrated via a thorough simulation study, and applied to an analysis of movements and trends of income categories using the Home Mortgage Disclosure Act data.