# bw.CvL

##### Cronie and van Lieshout's Criterion for Bandwidth Selection for Kernel Density

Uses Cronie and van Lieshout's criterion based on Cambell's formula to select a smoothing bandwidth for the kernel estimation of point process intensity.

##### Usage

`bw.CvL(X, …, srange = NULL, ns = 16, sigma = NULL)`

##### Arguments

- X
A point pattern (object of class

`"ppp"`

).- …
Ignored.

- srange
Optional numeric vector of length 2 giving the range of values of bandwidth to be searched.

- ns
Optional integer giving the number of values of bandwidth to search.

- sigma
Optional. Vector of values of the bandwidth to be searched. Overrides the values of

`ns`

and`srange`

.

##### Details

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 discrepancy between the area of the observation window and the sum of reciprocal estimated intensity values at the points of the point process $$ \mbox{CvL}(\sigma) = (|W| - \sum_i 1/\hat\lambda(x_i))^2 $$ where the sum is taken over all the data points \(x_i\), and where \(\hat\lambda(x_i)\) is the kernel-smoothing estimate of the intensity at \(x_i\) with smoothing bandwidth \(\sigma\).

The value of \(\mbox{CvL}(\sigma)\) is computed
directly, using `density.ppp`

,
for `ns`

different values of \(\sigma\)
between `srange[1]`

and `srange[2]`

.

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 bandwidth selection criterion
as a function of `sigma`

.

##### Value

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

which can be plotted.

##### References

Cronie, O and Van Lieshout, M N M (2018) A non-model-based approach to
bandwidth selection for kernel estimators of spatial intensity functions,
*Biometrika*, **105**, 455-462.

##### See Also

##### Examples

```
# NOT RUN {
if(interactive()) {
b <- bw.CvL(redwood)
b
plot(b, main="Cronie and van Lieshout bandwidth criterion for redwoods")
plot(density(redwood, b))
plot(density(redwood, bw.CvL))
}
# }
```

*Documentation reproduced from package spatstat, version 1.57-1, License: GPL (>= 2)*