# bw.ppl

##### Likelihood Cross Validation Bandwidth Selection for Kernel Density

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

##### Usage

`bw.ppl(X, ..., srange=NULL, ns=16)`

##### 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.

##### 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 maximise the point process likelihood cross-validation criterion $$\mbox{LCV}(\sigma) = \sum_i \log\hat\lambda_{-i}(x_i) - \int_W \hat\lambda(u) \, {\rm d}u$$ where the sum is taken over all the data points $x_i$, where $\hat\lambda_{-i}(x_i)$ is the leave-one-out kernel-smoothing estimate of the intensity at $x_i$ with smoothing bandwidth $\sigma$, and $\hat\lambda(u)$ is the kernel-smoothing estimate of the intensity at a spatial location $u$ with smoothing bandwidth $\sigma$. See Loader(1999, Section 5.3).

The value of $\mbox{LCV}(\sigma)$ 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 (rescaled) mean-square error
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

Loader, C. (1999)
*Local Regression and Likelihood*.
Springer, New York.

##### See Also

##### Examples

```
b <- bw.ppl(redwood)
plot(b, main="Likelihood cross validation for redwoods")
plot(density(redwood, b))
<testonly>b <- bw.ppl(redwood, srange=c(0.03, 0.07), ns=2)</testonly>
```

*Documentation reproduced from package spatstat, version 1.41-1, License: GPL (>= 2)*