spatstat (version 1.60-1)

bw.pcf: Cross Validated Bandwidth Selection for Pair Correlation Function

Description

Uses composite likelihood or generalized least squares cross-validation to select a smoothing bandwidth for the kernel estimation of pair correlation function.

Usage

bw.pcf(X, rmax=NULL, lambda=NULL, divisor="r", 
         kernel="epanechnikov", nr=10000, bias.correct=TRUE, 
         cv.method=c("compLik", "leastSQ"), simple=TRUE, srange=NULL,
	 …, verbose=FALSE)

Arguments

X

A point pattern (object of class "ppp").

rmax

Numeric. Maximum value of the spatial lag distance \(r\) for which \(g(r)\) should be evaluated.

lambda

Optional. Values of the estimated intensity function. A vector giving the intensity values at the points of the pattern X.

divisor

Choice of divisor in the estimation formula: either "r" (the default) or "d". See pcf.ppp.

kernel

Choice of smoothing kernel, passed to density; see pcf and pcfinhom.

nr

Integer. Number of subintervals for discretization of [0, rmax] to use in computing numerical integrals.

bias.correct

Logical. Whether to use bias corrected version of the kernel estimate. See Details.

cv.method

Choice of cross validation method: either "compLik" or "leastSQ" (partially matched).

simple

Logical. Whether to use simple removal of spatial lag distances. See Details.

srange

Optional. Numeric vector of length 2 giving the range of bandwidth values that should be searched to find the optimum bandwidth.

…

Other arguments, passed to pcf or pcfinhom.

verbose

Logical value indicating whether to print progress reports during the optimization procedure.

Value

A numerical value giving the selected bandwidth. The result also belongs to the class "bw.optim" which can be plotted.

Definition of bandwidth

The bandwidth bw returned by bw.pcf corresponds to the standard deviation of the smoothoing kernel. As mentioned in the documentation of density.default and pcf.ppp, this differs from the scale parameter h of the smoothing kernel which is often considered in the literature as the bandwidth of the kernel function. For example for the Epanechnikov kernel, bw=h/sqrt(h).

Details

This function selects an appropriate bandwidth bw for the kernel estimator of the pair correlation function of a point process intensity computed by pcf.ppp (homogeneous case) or pcfinhom (inhomogeneous case).

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).

$$ M(b) = \frac{\mbox{MSE}(\sigma)}{\lambda^2} - g(0) $$

The result is a numerical value giving the selected bandwidth.

References

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

See Also

pcf.ppp, pcfinhom

Examples

# NOT RUN {
  b <- bw.pcf(redwood)
  plot(pcf(redwood, bw=b))
# }