# pcfcross

##### Multitype pair correlation function (cross-type)

Calculates an estimate of the cross-type pair correlation function for a multitype point pattern.

- Keywords
- spatial, nonparametric

##### Usage

`pcfcross(X, i, j, ...)`

##### Arguments

- X
- The observed point pattern, from which an estimate of the cross-type pair correlation function $g_{ij}(r)$ will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor).
- i
- Number or character string identifying the type (mark value)
of the points in
`X`

from which distances are measured. - j
- Number or character string identifying the type (mark value)
of the points in
`X`

to which distances are measured. - ...
- Arguments passed to
`pcf.ppp`

.

##### Details

The cross-type pair correlation function
is a generalisation of the pair correlation function `pcf`

to multitype point patterns.

For two locations $x$ and $y$ separated by a distance $r$,
the probability $p(r)$ of finding a point of type $i$ at location
$x$ and a point of type $j$ at location $y$ is
$$p(r) = \lambda_i \lambda_j g_{i,j}(r) \,{\rm d}x \, {\rm d}y$$
where $\lambda_i$ is the intensity of the points
of type $i$.
For a completely random Poisson marked point process,
$p(r) = \lambda_i \lambda_j$
so $g_{i,j}(r) = 1$.
Indeed for any marked point pattern in which the points of type `i`

are independent of the points of type `j`

,
the theoretical value of the cross-type pair correlation is
$g_{i,j}(r) = 1$.
For a stationary multitype point process, the cross-type pair correlation
function between marks $i$ and $j$ is formally defined as
$$g_{i,j}(r) = \frac{K_{i,j}^\prime(r)}{2\pi r}$$
where $K_{i,j}^\prime$ is the derivative of
the cross-type $K$ function $K_{i,j}(r)$.
of the point process. See `Kest`

for information
about $K(r)$.

The command `pcfcross`

computes a kernel estimate of
the cross-type pair correlation function between marks $i$ and
$j$. It uses `pcf.ppp`

to compute kernel estimates
of the pair correlation functions for several unmarked point patterns,
and uses the bilinear properties of second moments to obtain the
cross-type pair correlation.

See `pcf.ppp`

for a list of arguments that control
the kernel estimation.

The companion function `pcfdot`

computes the
corresponding analogue of `Kdot`

.

##### 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 $g_{i,j}$ has been estimated theo the theoretical value $g_{i,j}(r) = 1$ for independent marks. - together with columns named
`"border"`

,`"bord.modif"`

,`"iso"`

and/or`"trans"`

, according to the selected edge corrections. These columns contain estimates of the function $g_{i,j}$ obtained by the edge corrections named.

##### See Also

Mark connection function `markconnect`

.

Multitype pair correlation `pcfdot`

.
Pair correlation `pcf`

,`pcf.ppp`

.
`Kcross`

##### Examples

```
data(amacrine)
p <- pcfcross(amacrine, "off", "on")
p <- pcfcross(amacrine, "off", "on", stoyan=0.1)
plot(p)
```

*Documentation reproduced from package spatstat, version 1.23-3, License: GPL (>= 2)*