# 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)
```

##### 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"`

) gi - 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

##### 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"`

) 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 r the vector of values of the argument $r$ at which the function $K$ has been estimated theo the theoretical value $K(r) = \pi r^2$ or $L(r)=r$ for a stationary Poisson process - together with columns containing the values of the
local pair correlation function for each point in the pattern.
Column
`i`

corresponds to the`i`

th point. The last two columns contain the`r`

and`theo`

values.

##### See Also

##### Examples

```
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 1.25-5, License: GPL (>= 2)*