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,

upper W Subscript circled-dash r Baseline equals upper W circled-dash b left-parenthesis 0 comma r right-parenthesis equals StartSet x element-of upper W colon StartAbsoluteValue EndAbsoluteValue x minus partial-differential upper W StartAbsoluteValue EndAbsoluteValue greater-than-or-equal-to r EndSet

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

ModifyingAbove upper F With caret left-parenthesis r right-parenthesis equals StartFraction 1 Over lamda StartAbsoluteValue upper W Subscript circled-dash r Baseline EndAbsoluteValue EndFraction sigma-summation Underscript g Subscript j Baseline element-of upper W Subscript circled-dash r Baseline Endscripts bold 1 StartSet d left-parenthesis g Subscript j Baseline comma x right-parenthesis less-than-or-equal-to r EndSet

ModifyingAbove upper K With caret left-parenthesis r right-parenthesis equals StartFraction sigma-summation Underscript i equals 1 Overscript n Endscripts sigma-summation Underscript j not-equals i Endscripts bold 1 StartSet StartAbsoluteValue EndAbsoluteValue x Subscript i Baseline minus x Subscript j Baseline StartAbsoluteValue EndAbsoluteValue less-than-or-equal-to r EndSet Over ModifyingAbove beta With caret n left-parenthesis x intersection upper W Subscript circled-dash r Baseline right-parenthesis EndFraction

ModifyingAbove upper G With caret left-parenthesis r right-parenthesis equals StartFraction sigma-summation Underscript x Subscript i Baseline element-of upper W Subscript circled-dash r Baseline Endscripts bold 1 StartSet StartAbsoluteValue EndAbsoluteValue x Subscript i Baseline minus upper X slash x Subscript i Baseline StartAbsoluteValue EndAbsoluteValue less-than-or-equal-to r EndSet Over n left-parenthesis upper X intersection upper W Subscript circled-dash r Baseline right-parenthesis EndFraction

ModifyingAbove upper G With caret left-parenthesis r right-parenthesis equals StartFraction sigma-summation Underscript i Endscripts bold 1 StartSet d Subscript i Baseline less-than-or-equal-to r comma b Subscript i Baseline greater-than-or-equal-to r EndSet Over sigma-summation Underscript i Endscripts bold 1 StartSet b Subscript i Baseline greater-than-or-equal-to r EndSet EndFraction

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).

Last updated: December 09, 2022