  An income distribution describes how an entity's total wealth is distributed
amongst its population. A problem of interest to regional economics researchers
is to understand the spatial homogeneity of income distributions among
different regions. In economics, the Lorenz curve is a well-known functional
representation of income distribution. In this article, we propose a mixture of
finite mixtures (MFM) model as well as a Markov random field constrained
mixture of finite mixtures (MRFC-MFM) model in the context of spatial
functional data analysis to capture spatial homogeneity of Lorenz curves. We
design efficient Markov chain Monte Carlo (MCMC) algorithms to simultaneously
infer the posterior distributions of the number of clusters and the clustering
configuration of spatial functional data. Extensive simulation studies are
carried out to show the effectiveness of the proposed methods compared with
existing methods. We apply the proposed spatial functional clustering method to
state level income Lorenz curves from the American Community Survey Public Use
Microdata Sample (PUMS) data. The results reveal a number of important
clustering patterns of state-level income distributions across US.
