# pcfcross.inhom

##### Inhomogeneous Multitype Pair Correlation Function (Cross-Type)

Estimates the inhomogeneous cross-type pair correlation function for a multitype point pattern.

- Keywords
- spatial, nonparametric

##### Usage

```
pcfcross.inhom(X, i, j, lambdaI = NULL, lambdaJ = NULL, ...,
r = NULL, breaks = NULL,
kernel="epanechnikov", bw=NULL, stoyan=0.15,
correction = c("isotropic", "Ripley", "translate"),
sigma = NULL, varcov = NULL)
```

##### Arguments

- X
- The observed point pattern, from which an estimate of the inhomogeneous 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. - lambdaI
- Optional.
Values of the estimated intensity function of the points of type
`i`

. Either a vector giving the intensity values at the points of type`i`

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

) giving the i - lambdaJ
- Optional.
Values of the estimated intensity function of the points of type
`j`

. A numeric vector, pixel image or`function(x,y)`

. - r
- Vector of values for the argument $r$ at which $g_{ij}(r)$ should be evaluated. There is a sensible default.
- breaks
- Optional. An alternative to the argument
`r`

. Not normally invoked by the user. - kernel
- Choice of smoothing kernel, passed to
`density.default`

. - bw
- Bandwidth for smoothing kernel, passed to
`density.default`

. - ...
- Other arguments passed to the kernel density estimation
function
`density.default`

. - stoyan
- Bandwidth coefficient; see Details.
- correction
- Choice of edge correction.
- sigma,varcov
- Optional arguments passed to
`density.ppp`

to control the smoothing bandwidth, when`lambdaI`

or`lambdaJ`

is estimated by kernel smoothing.

##### Details

The inhomogeneous cross-type pair correlation function $g_{ij}(r)$ is a summary of the dependence between two types of points in a multitype spatial point process that does not have a uniform density of points.

The best intuitive interpretation is the following: the probability $p(r)$ of finding two points, of types $i$ and $j$ respectively, at locations $x$ and $y$ separated by a distance $r$ is equal to $$p(r) = \lambda_i(x) lambda_j(y) g(r) \,{\rm d}x \, {\rm d}y$$ where $\lambda_i$ is the intensity function of the process of points of type $i$. For a multitype Poisson point process, this probability is $p(r) = \lambda_i(x) \lambda_j(y)$ so $g_{ij}(r) = 1$.

The command `pcfcross.inhom`

estimates the inhomogeneous
pair correlation using a modified version of
the algorithm in `pcf.ppp`

.

If the arguments `lambdaI`

and `lambdaJ`

are missing or
null, they are estimated from `X`

by kernel smoothing using a
leave-one-out estimator.

##### Value

- A function value table (object of class
`"fv"`

). Essentially a data frame containing the variables r the vector of values of the argument $r$ at which the inhomogeneous cross-type pair correlation function $g_{ij}(r)$ has been estimated theo vector of values equal to 1, the theoretical value of $g_{ij}(r)$ for the Poisson process trans vector of values of $g_{ij}(r)$ estimated by translation correction iso vector of values of $g_{ij}(r)$ estimated by Ripley isotropic correction - as required.

##### See Also

##### Examples

```
data(amacrine)
plot(pcfcross.inhom(amacrine, "on", "off", stoyan=0.1),
legendpos="bottom")
```

*Documentation reproduced from package spatstat, version 1.23-5, License: GPL (>= 2)*