A Lattice-Based Smoother for Regions with Irregular Boundaries and Holes

We consider the problem of estimating a smooth function over a spatial region that is delineated by an irregular boundary and potentially contains holes within the boundary. Methods commonly used for spatial function estimation are well-known to suffer from bias along such boundaries. The estimator we propose is a kernel regression estimator where the kernel is an approximation to a two-dimensional diffusion process contained within the region of interest. The diffusion process is approximated by the distribution of length-<i>k</i> random walks originating from each observation location and constrained to stay within the domain boundaries. We propose using a cross-validation criterion to find the optimal walk length <i>k</i>, which controls the smoothness of the resulting estimate. Simulations show that the method outperforms the soap film smoother of Wood et al. (2008) in many realistic scenarios, when data are noisy and borders are highly irregular. We illustrate the practical use of the estimator using measurements of soil manganese concentration around Port Moller, Alaska.