The SPP Procedure

Border Edge Correction for Distance Functions

To compute the edge correction factors $e(x_ i,r)$ 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 $b(x,r)$ 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 $b(x,r)$ is not observable (Baddeley 2007). Ignoring the fact that the observable quantity $n(X\cap W \cap b(x,r))$ is less than or equal to $n(X\cap b(x,r))$ 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,

\[ W_{\circleddash r} = W \circleddash b(0,r) = \{ x \in W: ||x - \partial W||\geq r \} \]

where $||x- \partial W||$ 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 $\partial W$.

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

\[ \hat{F}(r) = \frac{1}{\lambda |W_{\circleddash r}|}\sum _{g_ j\in W_{\circleddash r}} \Strong{1}\{ d(g_ j,x) \leq r\} \]
\[ \hat{K}(r) = \frac{\sum _{i=1}^{n} \sum _{j\ne i} \Strong{1}\{ ||x_ i - x_ j|| \leq r\} }{\hat{\beta } n(x \cap W_{\circleddash r})} \]
\[ \hat{G}(r) = \frac{\sum _{x_ i \in W_{\circleddash r}} \Strong{1}\{ ||x_ i - X/\ x_ i|| \leq r \} }{n(X\cap W_{\circleddash r})} \]
\[ \hat{G}(r) = \frac{\sum _ i \Strong{1}\{ d_ i \leq r, b_ i \geq r \} }{\sum _ i\Strong{1}\{ b_ i \geq r\} } \]

where $\hat{\beta } = n(x) / \lambda |W|$; $||x_ i - X/\  x_ i||$ is the observed nearest-neighbor distance, $d_ i$, for the ith point $x_ i$; and $b_ i$ is the distance from $x_ i$ to the boundary $\partial W$. For more information about these border-edge-corrected functions, see  Baddeley (2007).