Uses composite likelihood or generalized least squares cross-validation to select a smoothing bandwidth for the kernel estimation of the inhomogeneous pair correlation function.
bw.pcfinhom(X, lambda=NULL, ..., rmax=NULL, nr=10000,
cv.method=c("compLik", "leastSQ", "oracle"),
leaveoneout=TRUE, simple=TRUE,
fast=TRUE, srange=NULL, ns=32, use.count=TRUE,
gtrue=NULL,
verbose=FALSE, warn=TRUE)A numerical value giving the selected bandwidth.
The result also belongs to the class "bw.optim"
which can be plotted.
This function selects an appropriate bandwidth bw
for the kernel estimator of the pair correlation function
of a point process intensity computed by pcfinhom.
With cv.method="leastSQ", the bandwidth
\(h\) is chosen to minimise an unbiased
estimate of the integrated mean-square error criterion
\(M(h)\) defined in equation (4) in Guan (2007a).
The code implements the fast algorithm of Jalilian and Waagepetersen
(2018).
With cv.method="compLik", the bandwidth
\(h\) is chosen to maximise a likelihood
cross-validation criterion \(CV(h)\) defined in
equation (6) of Guan (2007b).
$$ M(b) = \frac{\mbox{MSE}(\sigma)}{\lambda^2} - g(0) $$
With cv.method="oracle", the true pair correlation function
must be provided as the argument gtrue. The bandwidth
\(h\) is chosen to minimise the integrated squared difference
between the pcf estimate and the true pcf,
$$
M(h) = \int_0^{\mbox{rmax}} (\hat{g}(r) - g(r))^2 dr
$$
The result is a numerical value giving the selected bandwidth.
Baddeley, A., Davies, T.M. and Hazelton, M.L. (2025) An improved estimator of the pair correlation function of a spatial point process. Biometrika, to appear.
Guan, Y. (2007a). A composite likelihood cross-validation approach in selecting bandwidth for the estimation of the pair correlation function. Scandinavian Journal of Statistics, 34(2), 336--346.
Guan, Y. (2007b). A least-squares cross-validation bandwidth selection approach in pair correlation function estimations. Statistics & Probability Letters, 77(18), 1722--1729.
Jalilian, A. and Waagepetersen, R. (2018) Fast bandwidth selection for estimation of the pair correlation function. Journal of Statistical Computation and Simulation, 88(10), 2001--2011. https://www.tandfonline.com/doi/full/10.1080/00949655.2018.1428606
pcfinhom
b <- bw.pcfinhom(japanesepines)
plot(pcfinhom(japanesepines, bw=b))
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