k-tree density estimation from sparse nearest-neighbor data

Background/Question/Methods

Ecologists frequently estimate population densities of plants and sessile animals from nearest-neighbor data using k-tree sampling methods: distances (and sometimes directions) from sample points to one or more nearest detected organisms. These methods typically assume a unique nearest neighbor to each sample point and a spatial distribution of individuals that is random and homogeneous. If these assumptions are satisfied, average nearest-neighbor distance can be used to reliably estimate population density. However, these assumptions often are not satisfied, and using procedures that assume unique nearest neighbors or homogeneity can introduce bias. Here, we derive the likelihood function for k-tree samples where k = 1, for both homogeneous and non-homogeneous Poisson point processes, and regardless of whether each sample point has a unique nearest neighbor. We modify an existing F function to provide a test for spatial homogeneity; find the maximum likelihood estimator (and its variance) of overall population density if the population is spatially homogeneous; and develop a kernel intensity estimation method that we use to estimate the local population density at a given sampled point (or small area) when the sample is from a non-homogeneous Poisson point process (NHPP). The tests for spatial pattern and associated local or population-level estimators are illustrated for populations of the purple pitcher plant, Sarracenia purpurea, surveyed in 64-m²plots with 100 regularly-spaced sample points at 77 bogs throughout New England.

Results/Conclusions

Approximately one-third of the sites had samples that were characterized by complete spatial randomness (CSR). For these 25 sites, our likelihood-based estimates of plant density (on average, 1.26 plants/m²) were substantially lower than those generated by commonly-used algorithms based on average nearest-neighbor distances. Among the remaining sites, 24 samples were clustered, eight were regular, and 20 were regular at small sample-grains but were clustered at large sample-grains. For sites with NHPP data, we estimated the local plant density by kernel intensity estimation using a bivariate Normal kernel with equal variances that were chosen by cross-validation. The overall population density was then calculated by dividing the sum of the values of the intensity function at each observed point (local density) by the sum of the values of the kernel intensity estimates over the area in which no plants were found.