Uses cross-validation to select a smoothing bandwidth for the kernel estimation of point process intensity.

`bw.diggle(X, ..., correction="good", hmax=NULL, nr=512, warn=TRUE)`

X

A point pattern (object of class `"ppp"`

).

…

Ignored.

correction

Character string passed to `Kest`

determining the edge correction to be used to
calculate the \(K\) function.

hmax

Numeric. Maximum value of bandwidth that should be considered.

nr

Integer. Number of steps in the distance value \(r\) to use in computing numerical integrals.

warn

Logical. If `TRUE`

, issue a warning if the minimum of
the cross-validation criterion occurs at one of the ends of the
search interval.

A numerical value giving the selected bandwidth.
The result also belongs to the class `"bw.optim"`

which can be plotted.

The smoothing parameter `sigma`

returned by `bw.diggle`

(and displayed on the horizontal axis of the plot)
corresponds to `h/2`

, where `h`

is the smoothing
parameter described in Diggle (2003, pages 116-118) and
Berman and Diggle (1989).
In those references, the smoothing kernel
is the uniform density on the disc of radius `h`

. In
`density.ppp`

, the smoothing kernel is the
isotropic Gaussian density with standard deviation `sigma`

.
When replacing one kernel by another, the usual
practice is to adjust the bandwidths so that the kernels have equal
variance (cf. Diggle 2003, page 118). This implies that `sigma = h/2`

.

This function selects an appropriate bandwidth `sigma`

for the kernel estimator of point process intensity
computed by `density.ppp`

.

The bandwidth \(\sigma\) is chosen to minimise the mean-square error criterion defined by Diggle (1985). The algorithm uses the method of Berman and Diggle (1989) to compute the quantity $$ M(\sigma) = \frac{\mbox{MSE}(\sigma)}{\lambda^2} - g(0) $$ as a function of bandwidth \(\sigma\), where \(\mbox{MSE}(\sigma)\) is the mean squared error at bandwidth \(\sigma\), while \(\lambda\) is the mean intensity, and \(g\) is the pair correlation function. See Diggle (2003, pages 115-118) for a summary of this method.

The result is a numerical value giving the selected bandwidth.
The result also belongs to the class `"bw.optim"`

which can be plotted to show the (rescaled) mean-square error
as a function of `sigma`

.

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, series B*
**51**, 81--92.

Diggle, P.J. (1985)
A kernel method for smoothing point process data.
*Applied Statistics* (Journal of the Royal Statistical Society,
Series C) **34** (1985) 138--147.

Diggle, P.J. (2003)
*Statistical analysis of spatial point patterns*,
Second edition. Arnold.

# NOT RUN { data(lansing) attach(split(lansing)) b <- bw.diggle(hickory) plot(b, ylim=c(-2, 0), main="Cross validation for hickories") if(interactive()) { plot(density(hickory, b)) } # }