John M. Castelloe
Ph. D. Thesis, December 1998
Department of Statistics and Actuarial Science, University of Iowa
Full manuscript: PDF, zipped postscript
Abstract: In the analysis of spatial point patterns, 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). In this process, ``parent'' events are assumed to be located uniformly in some region. Each parent event gives rise to a collection of ``offspring'' events, displaced according to a common bivariate normal distribution. The resulting point pattern is taken to be the collection all offspring events, with no information about parents recorded. If the covariance matrix (called the ``cluster shape/scale parameter'') of the bivariate normal distribution is a multiple of the identity matrix, then isotropy holds, with clusters having a circular shape. Otherwise, the process is anisotropic with elliptical clusters.
Estimation of the parameters of a BVNPCP is particularly challenging due to the substantial amount of latent data. The offspring relationships, number of parents and locations of parents are all unknown. In this thesis, two approaches for testing for and estimating anisotropy are developed and applied to a collection of actual and simulated spatial point patterns. The cluster shape/scale parameter is re-parameterized in terms of anisotropy strength, anisotropy direction, and cluster size to allow for more transparent interpretation of results.
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 clusters. A new convergence assessment method, applicable to
any RJMCMC situation in which distinct models can be identified, is designed and theoretically justified. A ``model'' in our case is a given number of clusters, in other words, the number of components in a mixture. Output 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.