Density, Derived from Measured Distances, for Studying the Spatial Patterns

The purpose of the paper is to propose the use of bivariate index$(\hat{\theta},\hat{\theta}_{2})$as a summary description of the spatial pattern. For this we exploit the following fact. In a plane, for patterns formed by points, two types of distances (i) after selecting a random location, the distance at which the nearest point is situated (ii) after reaching the said point in (i) which may be called random point, the distance between random point and the nearest neighbour point to it, can be used to calculate two different estimates$\hat{\lambda}_{1}$and$\hat{\lambda}_{2}$. Let$\hat{\lambda}_{1}$be the estimate, which gives an idea about the density of clusters, arrived at by using the distances as measured in (i), and$\hat{\lambda}_{2}$be the second estimate, which gives an idea about the density within the clusters, arrived at by using mainly the distances as measured in (ii). For a uniform Poisson pattern we have$\lambda _{1}=\lambda _{2}$, for an aggregate pattern$\lambda _{1}\lambda _{2}$. Thus if$\hat{\theta}_{1}=(1/\hat{\lambda}_{1})$and$\hat{\theta}_{2}=(1/\hat{\lambda}_{2})$the bivariate index$(\hat{\theta}_{1},\hat{\theta}_{2})$by itself will throw more light than a single condensed density estimate. Theory for estimating$\theta _{1}$and$\theta _{2}$along with large sample tests for testing$\theta _{1}$against$\theta _{2}$is developed. Some published patterns are analysed.