R: Cross Validated Bandwidth Selection for Pair Correlation...

bw.pcf {spatstat.core} | R Documentation

## 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, warn=TRUE)

### 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.
`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 `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) = \int_{0}^{rmax} \hat{g}^2(r;b) r dr - ∑_{u,v}

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.pcf` 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 Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.

### 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

      b <- bw.pcf(redwood)
      plot(pcf(redwood, bw=b))

[Package spatstat.core version 2.4-4 Index]