Edge-correction for spatial kernel smoothing methods? When is it necessary?

A limitation to the practical implementation of some kernel smoothing methods for spatial data is the need for edge correction. This applies particularly to kernel density estimation. Here we demonstrate by simulation the extent of the bias introduced when edge correction is not applied for realisations of both homogeneous and inhomogeneous Poisson processes. This shows the overwhelming importance of edge-correction. We also argue that edge correction is usually not necessary for kernel regression. Introduction Kernel density estimation (KDE) is widely used in spatial epidemiology. Its primary use is for producing smoothed density maps of point patterns. For example, kernel estimation of farm density provides a visual impression of the spatial distribution of farms in a region of interest, whilst acknowledging the uncertainty associated with the geo-referencing of individual farms. Another example is in estimating spatial variation in disease prevalence, in which context it is often reasonable to assume that long-term prevalence varies continuously over the geographical region of interest. A problem with KDE arises when points of interest are close to the boundary of the study area. Current algorithms for edge-correction are either difficult to apply or computationally expensive, especially for complex borders. Accordingly edge-corrected kernel estimation is not currently implemented in standard GIS software. Here we describe a more efficient algorithm for the implementation of a particular approach to edge-corrected spatial KDE. We also explain in the discussion section why edgecorrection is usually not necessary for the related problem of spatial kernel regression estimation (KRE). Material and Methods A non-edge-corrected kernel density estimator, based on data x1,x2,...,xn where the points xi lie within a spatial region A, takes the form ƒ ∧ (x) = nh Σ w{(x – xi)/h}, where h is the bandwidth and w(u) is a spatial probability density function. We have conducted a simulation study to investigate the effects of edge-correction on spatial kernel density estimation, using both Gaussian and quartic kernels with five different bandwidths chosen as 1/4, 1/2, 1, 2, and 4 times h0, where is chosen by Scott’s rule in R. We simulated point patterns as realisations of two different spatial point processes, each of which is conditioned to generate 1000 points in the unit square with the left-bottom vertex at the origin. The first point process is a homogeneous Poisson process, for which the density is spatially constant; the second is an inhomogeneous Poisson process with density proportional to 1+0.7cos[2π(u – 0.5)] where u is the distance of the point to the origin. We generated 100 replications of each process. VLA Gisvet Canada AmQ Justifie 9/12/04 2:19 PM Page 40