Delaunay tesselation as a scale-free description of spatial aggregation: How to minimize edge-effect problems

Background/Question/Methods

One key challenge in spatial statistics is the inherent scale-dependency of many metrics of density or aggregation.  For example, calculating spatial density (organisms/unit area) requires defining an areal scale (e.g., cm², m²) for the denominator.  Measuring density at that scale may obscure patterns at smaller scales, along with potentially important ecological interactions.  One solution is to use a scale-free measure of spatial aggregation, such as nearest-neighbor distances. A rigorous method for calculating nearest-neighbor distances is Delaunay tesselation, which defines a polygonal neighborhood around each individual in a landscape.  The tesselation also creates ‘spokes’ radiating from each individual to its nearest neighbors.  Mean spoke length is then a useful scale-free density metric.  However, the Delaunay method suffers from serious edge effects.  Neighborhoods on the edge of a spatial domain are unconstrained and spoke lengths for those outlying individuals are spurious.  The standard recommendation for this problem is to define a buffer zone of outer individuals and only examine neighbor distances among individuals in the domain interior.  This requires discarding large amounts of valuable data.  We examined several different alternative ‘fixes’ to the edge effect problem in order to find a method that does not introduce bias in spoke-length calculations.

Results/Conclusions

The best edge-effect correction mechanism involves defining a ring of simulated proxy individuals around the perimeter of the spatial domain.  Those proxies are included in the tesselation, and properly constrain the neighborhoods of all of the actual individuals in the domain.  The spokes extending to the proxies are then excluded from all calculations of nearest-neighbor distance.  The accuracy of this method increases asymptotically with the number of proxies, so it is simple to determine the minimum number of proxies to obtain maximum accuracy for any dataset.  This method provides an unbiased estimate of mean and variance in spoke length (and thus nearest-neighbor distance) and does not require excluding any data points.