Chooses the standard deviation of an isotropic Gaussian kernel intensity
estimate by minimising \(M(r) = (1/\lambda - 2K(r))/(\pi r^2) +
J(r)/(\pi r^2)^2\), \(J(r) = \int_0^{2r} \phi(t, r) dK(t)\),
\(\phi(t, r) = 2r^2(acos(y) - y\sqrt{1 - y^2})\), \(y = t/(2r)\),
with \(\sigma = r/2\) (Diggle 1985; Berman and Diggle 1989), as
spatstat.explore::bw.diggle for a rectangular window: isotropic K
with \(\lambda^2 = n(n-1)/|W|^2\) on nr radii up to
min(shortside/4, sqrt(1000/(pi lambda))) (or 4 hmax), J as the
left-point Stieltjes sum, M for r up to rmax/2, and the grid minimiser.
Usage
BandwidthDiggle(points, window, nr = 512L, hmax = NULL)
Arguments
- points
Two-column matrix of coordinates.
- window
c(xmin, xmax, ymin, ymax).
- nr
Number of radii.
- hmax
Largest bandwidth considered.
Value
List with sigma, h, criterion, J,
lambda_hat, at_boundary.
References
Berman, M. and Diggle, P. (1989). Estimating weighted
integrals of the second-order intensity of a spatial point process.
Journal of the Royal Statistical Society B 51, 81-92.
Diggle, P. J. (1985). A kernel method for smoothing point process data.
Applied Statistics 34, 138-147.
Examples
P <- cbind(c(.1, .4, .35, .8, .7, .55, .2, .9, .15, .6), c(.2, .8, .3, .6, .15, .5, .65, .9,
.95, .85))
BandwidthDiggle(P, c(0, 1, 0, 1))[c("sigma", "at_boundary")]
#> $sigma
#> [1] 0.06237769
#>
#> $at_boundary
#> [1] TRUE
#>
