In the analysis of spatial point processes, it is generally assumed that the underlying spatial point process is "isotropic," i.e. that all characteristics are homogeneous with respect to direction. However, this is known in many applications not to be the case. For example, the distribution of plant seedling locations often exhibits directional asymmetry, or "anisotropy," due to factors such as prevailing wind direction and systematic migratory behavior of seed carriers. Failure to account for such directional inhomogeneity can result in erroneous inferences.
A special type of spatial point process is considered, namely the 2-dimensional Poisson cluster process with bivariate normal offspring dispersal (BVNPCP). Estimation of the parameters of a BVNPCP (the focus being the "cluster shape/scale parameter," the covariance matrix of the offspring dispersal distribution) is particularly challenging due to the substantial amount of latent data. The offspring relationships, number of parents and locations of parents are all unknown. Two approaches for testing for and estimating anisotropy are developed and applied to a collection of actual and simulated spatial point patterns.
The first approach considers the BVNPCP as a finite mixture model and combines EM algorithm parameter estimates, computed separately for different numbers of clusters, in a Bayesian model averaging type scheme. A "composite EM" estimator of the cluster shape/scale parameter is thus constructed, along with an estimated asymptotic variance computed from a combination of observed information matrices.
In the second approach, a reversible jump Markov chain Monte Carlo (RJMCMC) technique for 2-dimensional normal mixtures is developed. RJMCMC extends the traditional MCMC capabilities by providing for transitions between different parameter spaces, which are needed in our situation due to the unknown number of cluster. A new convergence assessment method, applicable to any RJMCMC situation in which distinct models can be identified, is designed and theoretically justified. Our analysis methods are also developed, including anisotropy testing/estimation, model checking and inference for number of clusters. The RJMCMC technique is flexible and has potential to apply to more complicated spatial point processes, and also other mixture-related problems.