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)`

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

which can be plotted.

- 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.

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`

.

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

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.

`density.ppp`

,
`bw.ppl`

,
`bw.scott`

,
`bw.CvL`

,
`bw.frac`

.

```
attach(split(lansing))
b <- bw.diggle(hickory)
plot(b, ylim=c(-2, 0), main="Cross validation for hickories")
if(interactive()) {
plot(density(hickory, b))
}
```

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