Uses the Cronie-Van Lieshout criterion to select the global smoothing bandwidth for adaptive kernel estimation of point process intensity.

```
bw.CvL.adaptive(X, ...,
hrange = NULL, nh = 16, h=NULL,
bwPilot = bw.scott.iso(X),
edge = FALSE, diggle = TRUE)
```

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

which can be plotted.

- X
A point pattern (object of class

`"ppp"`

).- ...
Additional arguments passed to

`densityAdaptiveKernel`

.- hrange
Optional numeric vector of length 2 giving the range of values of global bandwidth

`h`

to be searched.- nh
Optional integer giving the number of values of bandwidth

`h`

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

`nh`

and`hrange`

.- bwPilot
Pilot bandwidth. A scalar value in the same units as the coordinates of

`X`

. The smoothing bandwidth for computing an initial estimate of intensity using`density.ppp`

.- edge
Logical value indicating whether to apply edge correction.

- diggle
Logical. If

`TRUE`

, use the Jones-Diggle improved edge correction, which is more accurate but slower to compute than the default correction.

Marie-Colette Van Lieshout. Modified by Adrian Baddeley Adrian.Baddeley@curtin.edu.au.

This function selects an appropriate value of global bandwidth
`h0`

for adaptive kernel estimation of the intensity function
for the point pattern `X`

.

In adaptive estimation, each point in the point pattern is
subjected to a different amount of smoothing, controlled by
data-dependent or spatially-varying bandwidths.
The global bandwidth `h0`

is a scale factor
which is used to adjust all of the data-dependent bandwidths
according to the Abramson (1982) square-root rule.

This function considers each candidate value of bandwidth \(h\),
performs the smoothing steps described above, extracts the
adaptively-estimated intensity values
\(\hat\lambda(x_i)\) at each data point \(x_i\),
and calculates the Cronie-Van Lieshout criterion
$$
\mbox{CvL}(h) = \sum_{i=1}^n \frac 1 {\hat\lambda(x_i)}.
$$
The value of \(h\) which minimises the squared difference
$$
LP2(h) = (CvL(h) - |W|)^2
$$
(where `|W|`

is the area of the window of `X`

)
is selected as the optimal global bandwidth.

Bandwidths `h`

are physical distance values
expressed in the same units as the coordinates of `X`

.

Abramson, I. (1982)
On bandwidth variation in kernel estimates --- a square root law.
*Annals of Statistics*, **10**(4), 1217-1223.

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.

Van Lieshout, M.N.M. (2021)
Infill asymptotics for adaptive kernel estimators of spatial intensity.
*Australian and New Zealand Journal of Statistics*
**63** (1) 159--181.

`adaptive.density`

,
`densityAdaptiveKernel`

,
`bw.abram`

,
`density.ppp`

.

To select a *fixed* smoothing bandwidth
using the Cronie-Van Lieshout criterion, use `bw.CvL`

.

```
h0 <- bw.CvL.adaptive(redwood3)
plot(h0)
plot(as.fv(h0), CvL ~ h)
Z <- densityAdaptiveKernel(redwood3, h0)
plot(Z)
```

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