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

Abstract

<p>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. (<a href="#cit0022" target="_blank">2008</a>) 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.</p