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

```
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)
```

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.

- 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
The type (mark value) of the points in

`X`

from which distances are measured. A character string (or something that will be converted to a character string). Defaults to the first level of`marks(X)`

.- j
The type (mark value) of the points in

`X`

to which distances are measured. A character string (or something that will be converted to a character string). Defaults to the second level of`marks(X)`

.- 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 intensity values at all locations, or a`function(x,y)`

which can be evaluated to give the intensity value at any location.- 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
This argument is for internal use only.

- 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.

Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner r.turner@auckland.ac.nz

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.

`pcf.ppp`

,
`pcfinhom`

,
`pcfcross`

,
`pcfdot.inhom`

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

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