# Lcross.inhom

##### Inhomogeneous Cross Type L Function

For a multitype point pattern, estimate the inhomogeneous version of the cross-type \(L\) function.

- Keywords
- spatial, nonparametric

##### Usage

`Lcross.inhom(X, i, j, …)`

##### Arguments

- X
The observed point pattern, from which an estimate of the inhomogeneous cross type \(L\) function \(L_{ij}(r)\) will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor). See under Details.

- 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)`

.- …
Other arguments passed to

`Kcross.inhom`

.

##### Details

This is a generalisation of the function `Lcross`

to include an adjustment for spatially inhomogeneous intensity,
in a manner similar to the function `Linhom`

.

All the arguments are passed to `Kcross.inhom`

, which
estimates the inhomogeneous multitype K function
\(K_{ij}(r)\) for the point pattern.
The resulting values are then
transformed by taking \(L(r) = \sqrt{K(r)/\pi}\).

##### Value

An object of class `"fv"`

(see `fv.object`

).

Essentially a data frame containing numeric columns

the values of the argument \(r\) at which the function \(L_{ij}(r)\) has been estimated

the theoretical value of \(L_{ij}(r)\)
for a marked Poisson process, identically equal to `r`

##### Warnings

The arguments `i`

and `j`

are always interpreted as
levels of the factor `X$marks`

. They are converted to character
strings if they are not already character strings.
The value `i=1`

does **not**
refer to the first level of the factor.

##### References

Moller, J. and Waagepetersen, R. Statistical Inference and Simulation for Spatial Point Processes Chapman and Hall/CRC Boca Raton, 2003.

##### See Also

##### Examples

```
# NOT RUN {
# Lansing Woods data
woods <- lansing
# }
# NOT RUN {
ma <- split(woods)$maple
wh <- split(woods)$whiteoak
# method (1): estimate intensities by nonparametric smoothing
lambdaM <- density.ppp(ma, sigma=0.15, at="points")
lambdaW <- density.ppp(wh, sigma=0.15, at="points")
L <- Lcross.inhom(woods, "whiteoak", "maple", lambdaW, lambdaM)
# method (2): fit parametric intensity model
fit <- ppm(woods ~marks * polynom(x,y,2))
# evaluate fitted intensities at data points
# (these are the intensities of the sub-processes of each type)
inten <- fitted(fit, dataonly=TRUE)
# split according to types of points
lambda <- split(inten, marks(woods))
L <- Lcross.inhom(woods, "whiteoak", "maple",
lambda$whiteoak, lambda$maple)
# synthetic example: type A points have intensity 50,
# type B points have intensity 100 * x
lamB <- as.im(function(x,y){50 + 100 * x}, owin())
X <- superimpose(A=runifpoispp(50), B=rpoispp(lamB))
L <- Lcross.inhom(X, "A", "B",
lambdaI=as.im(50, Window(X)), lambdaJ=lamB)
# }
```

*Documentation reproduced from package spatstat, version 1.56-1, License: GPL (>= 2)*