bw.pcfinhom: Cross Validated Bandwidth Selection for Inhomogeneous Pair...

In spatstat.explore: Exploratory Data Analysis for the 'spatstat' Family

Cross Validated Bandwidth Selection for Inhomogeneous Pair Correlation Function

Description

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

Usage

  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)

Arguments

X	A point pattern (object of class "ppp").
lambda	Optional. Values of the estimated intensity function. Either a vector giving the intensity values at the points of the pattern X, a pixel image (object of class "im") giving the intensity values at all locations, a fitted point process model (object of class "ppm", "kppm" or "dppm") or a function(x,y) which can be evaluated to give the intensity value at any location.
...	Additional arguments passed to pcfinhom.
rmax	Optional. Numeric value. Maximum value of the spatial lag distance r for which g(r) should be evaluated.
nr	Integer. Number of subintervals for discretization of [0, rmax] to use in computing numerical integrals.
cv.method	Choice of cross validation method: either "compLik", "leastSQ" or "oracle" (partially matched).
leaveoneout	Logical value specifying whether to use leave-one-out estimators. See Details.
simple	Logical. Whether to use simple removal of spatial lag distances. See Details.
fast	Logical value indicating whether to find the optimal value by an optimization algorithm (fast=TRUE, the default) or by evaluating the objective function on an equally-spaced grid of bandwidth values (fast=FALSE).
srange	Optional. Numeric vector of length 2 giving the range of bandwidth values that should be searched to find the optimum bandwidth.
ns	Integer. Number of values of bandwidths at which to evaluate the objective function, when fast=FALSE.
use.count	Logical value specifying the benchmark for the calculation when cv.method="martin". In this calculation, the sum of values 1/g(d_{i,j}) over all pairwise distances is compared to a benchmark. If use.count=TRUE (the default), the benchmark is simply the total number of pairs of points contributing to the sum. If use.count=FALSE, the theoretical expected value is used as the benchmark.
gtrue	Function in the R language giving the true pair correlation function, when cv.method="oracle".
verbose	Logical value indicating whether to print progress reports during the optimization procedure.
warn	Logical. If TRUE, issue a warning if the optimum value of the cross-validation criterion occurs at one of the ends of the search interval.

Details

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.

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.pcfinhom is the standard deviation of the smoothing kernel, following the standard convention in R. As mentioned in the documentation for density.default and pcf.ppp, this differs from other definitions of bandwidth that can be found in the literature. The scale parameter h, which is called the bandwidth in some literature, is defined differently. For example for the Epanechnikov kernel, h is the half-width of the kernel, and bw=h/sqrt(5).

Author(s)

Rasmus Waagepetersen and Abdollah Jalilian. Adapted for spatstat by \spatstatAuthors. Hacked by \adrian, \martinH and \tilman.

References

\smoothpcfpaper

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

pcfinhom

Examples

  b <- bw.pcfinhom(japanesepines)
  plot(pcfinhom(japanesepines, bw=b))

spatstat.explore documentation built on April 4, 2025, 2:49 a.m.