  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.
