# localpcf

##### Local pair correlation function

Computes individual contributions to the pair correlation function from each data point.

- Keywords
- spatial, nonparametric

##### Usage

```
localpcf(X, ..., delta=NULL, rmax=NULL, nr=512, stoyan=0.15)
localpcfinhom(X, ..., delta=NULL, rmax=NULL, nr=512, stoyan=0.15,
lambda=NULL, sigma=NULL, varcov=NULL,
update=TRUE, leaveoneout=TRUE)
```

##### Arguments

- X
A point pattern (object of class

`"ppp"`

).- delta
Smoothing bandwidth for pair correlation. The halfwidth of the Epanechnikov kernel.

- rmax
Optional. Maximum value of distance \(r\) for which pair correlation values \(g(r)\) should be computed.

- nr
Optional. Number of values of distance \(r\) for which pair correlation \(g(r)\) should be computed.

- stoyan
Optional. The value of the constant \(c\) in Stoyan's rule of thumb for selecting the smoothing bandwidth

`delta`

.- lambda
Optional. Values of the estimated intensity function, for the inhomogeneous pair correlation. Either a vector giving the intensity values at the points of the pattern

`X`

, a pixel image (object of class`"im"`

) giving the intensity values at all locations, a fitted point process model (object of class`"ppm"`

,`"kppm"`

or`"dppm"`

) or a`function(x,y)`

which can be evaluated to give the intensity value at any location.- sigma,varcov,…
These arguments are ignored by

`localpcf`

but are passed by`localpcfinhom`

(when`lambda=NULL`

) to the function`density.ppp`

to control the kernel smoothing estimation of`lambda`

.- leaveoneout
Logical value (passed to

`density.ppp`

or`fitted.ppm`

) specifying whether to use a leave-one-out rule when calculating the intensity.- update
Logical value indicating what to do when

`lambda`

is a fitted model (class`"ppm"`

,`"kppm"`

or`"dppm"`

). If`update=TRUE`

(the default), the model will first be refitted to the data`X`

(using`update.ppm`

or`update.kppm`

) before the fitted intensity is computed. If`update=FALSE`

, the fitted intensity of the model will be computed without re-fitting it to`X`

.

##### Details

`localpcf`

computes the contribution, from each individual
data point in a point pattern `X`

, to the
empirical pair correlation function of `X`

.
These contributions are sometimes known as LISA (local indicator
of spatial association) functions based on pair correlation.

`localpcfinhom`

computes the corresponding contribution
to the *inhomogeneous* empirical pair correlation function of `X`

.

Given a spatial point pattern `X`

, the local pcf
\(g_i(r)\) associated with the \(i\)th point
in `X`

is computed by
$$
g_i(r) = \frac a {2 \pi n} \sum_j k(d_{i,j} - r)
$$
where the sum is over all points \(j \neq i\),
\(a\) is the area of the observation window, \(n\) is the number
of points in `X`

, and \(d_{ij}\) is the distance
between points `i`

and `j`

. Here `k`

is the
Epanechnikov kernel,
$$
k(t) = \frac 3 { 4\delta} \max(0, 1 - \frac{t^2}{\delta^2}).
$$
Edge correction is performed using the border method
(for the sake of computational efficiency):
the estimate \(g_i(r)\) is set to `NA`

if
\(r > b_i\), where \(b_i\)
is the distance from point \(i\) to the boundary of the
observation window.

The smoothing bandwidth \(\delta\) may be specified.
If not, it is chosen by Stoyan's rule of thumb
\(\delta = c/\hat\lambda\)
where \(\hat\lambda = n/a\) is the estimated intensity
and \(c\) is a constant, usually taken to be 0.15.
The value of \(c\) is controlled by the argument `stoyan`

.

For `localpcfinhom`

, the optional argument `lambda`

specifies the values of the estimated intensity function.
If `lambda`

is given, it should be either a
numeric vector giving the intensity values
at the points of the pattern `X`

,
a pixel image (object of class `"im"`

) giving the
intensity values at all locations, a fitted point process model
(object of class `"ppm"`

, `"kppm"`

or `"dppm"`

)
or a `function(x,y)`

which
can be evaluated to give the intensity value at any location.
If `lambda`

is not given, then it will be estimated
using a leave-one-out kernel density smoother as described
in `pcfinhom`

.

##### Value

An object of class `"fv"`

, see `fv.object`

,
which can be plotted directly using `plot.fv`

.
Essentially a data frame containing columns

the vector of values of the argument \(r\) at which the function \(K\) has been estimated

the theoretical value \(K(r) = \pi r^2\) or \(L(r)=r\) for a stationary Poisson process

##### See Also

##### Examples

```
# NOT RUN {
data(ponderosa)
X <- ponderosa
g <- localpcf(X, stoyan=0.5)
colo <- c(rep("grey", npoints(X)), "blue")
a <- plot(g, main=c("local pair correlation functions", "Ponderosa pines"),
legend=FALSE, col=colo, lty=1)
# plot only the local pair correlation function for point number 7
plot(g, est007 ~ r)
gi <- localpcfinhom(X, stoyan=0.5)
a <- plot(gi, main=c("inhomogeneous local pair correlation functions",
"Ponderosa pines"),
legend=FALSE, col=colo, lty=1)
# }
```

*Documentation reproduced from package spatstat, version 1.64-1, License: GPL (>= 2)*