Formula: \(g(r) = K'(r) / (2 pi r)\). K is estimated with
the reduced-sample (border) correction and differentiated by a
central difference of half-width h,
g_hat(r) = (K_hat(r + h) - K_hat(r - h)) / (2 h) / (2 pi r),
which is deterministic – no kernel smoothing and no data-dependent
bandwidth rule. Under complete spatial randomness \(K(r) = pi
r^2\) so g(r) = 1 at every radius; above one means clustering
at that scale and below one regularity
Usage
Pcfunc(points, window, r, h = NULL)
Arguments
- points
Point coordinates, n by 2.
- window
c(xmin, ymin, xmax, ymax) or an (m, 2) vertex
matrix; NULL takes the bounding box of points.
- r
Radii, strictly positive.
- h
Half-width of the central difference; defaults to one
quarter of the smallest radius, which keeps r - h positive.
Value
List with g, r, K, h,
estimate, lambda_hat, n.
References
Stoyan, D. & Stoyan, H. (1994). Fractals, Random Shapes
and Point Fields, Wiley, chapter 14; Diggle, P. J. (2003).
Statistical Analysis of Spatial Point Patterns, 2nd edition,
Arnold, section 4.3.
Examples
set.seed(21)
pts <- matrix(runif(400), 200, 2)
round(Pcfunc(pts, window = c(0, 0, 1, 1), r = c(0.05, 0.1, 0.15))$g, 2) # near 1 under CSR
#> [1] 1.13 0.92 0.98
