# The SPP Procedure

### Border Edge Correction for Distance Functions

To compute the edge correction factors that appear in the formulas of the distance functions, the SPP procedure implements border edge correction  (Illian et al. 2008; Ripley 1988; Baddeley 2007). Border edge correction is necessary because the data are given for a bounded observation window W, but the pattern itself is assumed to extend beyond the observation window. However, because you can observe only what is within the window, a disc of radius r around a point x that lies close to the boundary of W might extend outside W. Because the original process X is not observed outside W, the number of points of X in is not observable (Baddeley 2007). Ignoring the fact that the observable quantity is less than or equal to leads to a bias that is caused by edge effects. The border edge corrector is a simple strategy to eliminate the bias that is caused by edge effects. Under the border method, the window W is replaced by a reduced window,

where denotes the minimum distance from X to a point on the boundary. The reduced window contains all the points in W that are at least r units away from the boundary .

Based on the preceding definition, the border edge corrected F, K, and G functions are

where ; is the observed nearest-neighbor distance, , for the ith point ; and is the distance from to the boundary . For more information about these border-edge-corrected functions, see  Baddeley (2007).