Avner Bar-Hen

Estimation of the density of a clustered point pattern using a distance method

Distance-based methods use point-to-point distances or random-location-to-point distances in a cloud of points to estimate characteristics of the point pattern. One such characteristics is the density of points. The difficulty with distance-based density estimators is that their distribution depends on the spatial pattern of points. In particular, the distribution of distances is untractable for usual clustered patterns, that are often observed in natural systems. Here, we propose a density estimator for clustered patterns, based on the random-location-to-pth-point distance Xp. An approximate expression for the distribution function, Fp, of Xp was obtained by identifying the first two moments of the count of individuals in disks for a given point process with the first two moments of a negative binomial distribution. The approximate expression of Fp was then used to derive a maximum-likelihood estimator of the intensity of the point process. The quality of the approximation of Fp was assessed for homogeneous Poisson processes (for which the expression of Fp is exact) and for Matérn processes. The intensity estimator based on Matérn processes was then used to estimate tree density in a tree savanna in Mali, and it compared favorably with six robust estimators found in the literature.

Influence and sensitivity measures in correspondence analysis

Since correspondence analysis appears to be sensitive to outliers, it is important to be able to evaluate the sensitivity of the data on the results. This article deals with measuring the influence of rows and columns on the results obtained with correspondence analysis. To establish the influence of individuals on the analysis, we use the notion of influence curve and we propose a general criterion based on the mean square error to measure the sensitivity of the correspondence analysis and its robustness. A numerical example is presented to illustrate the notions developed in this article.