Uses composite likelihood or generalized least squares cross-validation to select a smoothing bandwidth for the kernel estimation of pair correlation function.

```
bw.pcf(X, rmax=NULL, lambda=NULL, divisor="r",
kernel="epanechnikov", nr=10000, bias.correct=TRUE,
cv.method=c("compLik", "leastSQ"), simple=TRUE, srange=NULL,
..., verbose=FALSE, 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"`

).- rmax
Numeric. Maximum value of the spatial lag distance \(r\) for which \(g(r)\) should be evaluated.

- lambda
Optional. Values of the estimated intensity function. A vector giving the intensity values at the points of the pattern

`X`

.- divisor
Choice of divisor in the estimation formula: either

`"r"`

(the default) or`"d"`

. See`pcf.ppp`

.- kernel
Choice of smoothing kernel, passed to

`density`

; see`pcf`

and`pcfinhom`

.- nr
Integer. Number of subintervals for discretization of [0, rmax] to use in computing numerical integrals.

- bias.correct
Logical. Whether to use bias corrected version of the kernel estimate. See Details.

- cv.method
Choice of cross validation method: either

`"compLik"`

or`"leastSQ"`

(partially matched).- simple
Logical. Whether to use simple removal of spatial lag distances. See Details.

- srange
Optional. Numeric vector of length 2 giving the range of bandwidth values that should be searched to find the optimum bandwidth.

- ...
Other arguments, passed to

`pcf`

or`pcfinhom`

.- verbose
Logical value indicating whether to print progress reports during the optimization procedure.

- warn
Logical. If

`TRUE`

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

The bandwidth `bw`

returned by `bw.pcf`

is the standard deviation of the smoothing kernel,
following the standard convention in R.
As mentioned in the documentation for
`density.default`

and `pcf.ppp`

,
this differs from other definitions of bandwidth that can be
found in the literature. The scale parameter
`h`

, which is called the bandwidth in some literature,
is defined differently.
For example for the Epanechnikov kernel, `h`

is the half-width
of the kernel, and `bw=h/sqrt(5)`

.

Rasmus Waagepetersen and Abdollah Jalilian. Adapted for spatstat by 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 `bw`

for the kernel estimator of the pair correlation function
of a point process intensity computed by `pcf.ppp`

(homogeneous case) or `pcfinhom`

(inhomogeneous case).

With `cv.method="leastSQ"`

, the bandwidth
\(h\) is chosen to minimise an unbiased
estimate of the integrated mean-square error criterion
\(M(h)\) defined in equation (4) in Guan (2007a).
The code implements the fast algorithm of Jalilian and Waagepetersen
(2018).

With `cv.method="compLik"`

, the bandwidth
\(h\) is chosen to maximise a likelihood
cross-validation criterion \(CV(h)\) defined in
equation (6) of Guan (2007b).

$$ M(b) = \frac{\mbox{MSE}(\sigma)}{\lambda^2} - g(0) $$

The result is a numerical value giving the selected bandwidth.

Guan, Y. (2007a).
A composite likelihood cross-validation approach in selecting
bandwidth for the estimation of the pair correlation function.
*Scandinavian Journal of Statistics*,
**34**(2), 336--346.

Guan, Y. (2007b).
A least-squares cross-validation bandwidth selection approach
in pair correlation function estimations.
*Statistics & Probability Letters*,
**77**(18), 1722--1729.

Jalilian, A. and Waagepetersen, R. (2018)
Fast bandwidth selection for estimation of the pair correlation
function.
*Journal of Statistical Computation and Simulation*,
**88**(10), 2001--2011.
https://www.tandfonline.com/doi/full/10.1080/00949655.2018.1428606

`pcf.ppp`

,
`pcfinhom`

```
b <- bw.pcf(redwood)
plot(pcf(redwood, bw=b))
```

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