The SPP procedure implements the following nearest-neighbor distance functions:
empty-space F function
nearest-neighbor G function
J function
A typical test that uses any nearest-neighbor function compares the empirical distribution function with the corresponding function for a homogeneous Poisson process that has first-order intensity . Usually, the first-order intensity is obtained as the number of observations per unit of area, .
The empty-space F function is defined as the empirical distribution function of the observed empty-space distances, , which is measured from a set of reference grid points g to the nearest point in the point pattern. The empty-space distance can be defined as
In practice, the computation of the empty-space F function also involves an edge correction. The edge-corrected empty-space F function is defined as
where is an edge correction. PROC SPP implements the border edge correction (Illian et al. 2008, p. 185–186) as described in the section Border Edge Correction for Distance Functions.
For a homogeneous Poisson process that has first-order intensity , the F function is
You compare the empirical and Poisson empty-space F function by using the EDF and the P-P plot in the F function summary panel plot. Values of suggest a regularly spaced pattern, and values of suggest a clustered pattern (Baddeley and Turner 2005).
The nearest-neighbor G function is the empirical distribution of the observed nearest-neighbor distance of the points within the point pattern. In practice, the G function also involves an edge correction and is defined as
where is the border edge correction (Illian et al. 2008, p. 185-186) as described in the section Border Edge Correction for Distance Functions and is the distance to the nearest neighbor for the ith point.
For a homogeneous Poisson process that has first-order intensity , the G function can be defined as
The interpretation of is opposite to the interpretation of . That is, values of imply a clustered pattern, and values of suggest a regular pattern (Baddeley and Turner 2005).
The third type of nearest-neighbor distance function is the J function, which is defined as a combination of both the F and G functions (Baddeley et al. 2000). The J function is defined for all distances r such that . The J function can be defined as
For a homogeneous Poisson process, . When takes values greater than 1, regularity is indicated; when takes values less than 1, the underlying process is more clustered than expected. As can be seen from the expression of , the estimate is an uncorrected estimate of the J-function and hence its computation does not require an edge correction (Baddeley et al. 2000).