Bayesian smoothing and step functions in the nonparametric estimation of curves and surfaces - Heikkinen, Juha (1997)
Various problems are considered mainly in the field of spatial statistics: image restoration, modelling of interactions in a spatial point pattern, and estimation of Poisson intensities both in time and space with and without covariates. These tasks are tackled following a new nonparametric Bayesian approach to the estimation of curves and surfaces. It is based on a model approximation where the approximating functions are piecewise constant. The partition of the domain to the subregions of constant function values is either fixed (static version) or random (dynamic version). In the latter case random partitions are generated as Voronoi tessellations of random point patterns. Estimates produced using the dynamic version are not necessarily step functions; for example the pointwise posterior means typically form a smooth continuous curve or surface. Smoothing between nearby function values is applied by means of a locally dependent Markov random field prior in the spirit of Bayesian image analysis. Markov chain Monte Carlo methods are proposed for the numerical estimation. This includes modification and combination of earlier Monte Carlo maximum likelihood algorithms for posterior mode estimation, and application of recently developed methods for sampling in a variable dimensional space. The approach is demonstrated in a number of examples with both real and synthetic data sets. The most notable real applications are the estimation of biogeographical ranges from atlas data, and the modelling of spatial variation in plant abundance using concomitant variables.