# 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
- Number or character string identifying the type (mark value)
of the points in
`X`

from which distances are measured. Defaults to the first level of`marks(X)`

. - j
- Number or character string identifying the type (mark value)
of the points in
`X`

to which distances are measured. 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

r the values of the argument $r$ at which the function $L_{ij}(r)$ has been estimated theo the theoretical value of $L_{ij}(r)$ for a marked Poisson process, identically equal to `r`

- together with a column or columns named
`"border"`

,`"bord.modif"`

,`"iso"`

and/or`"trans"`

, according to the selected edge corrections. These columns contain estimates of the function $L_{ij}(r)$ obtained by the edge corrections named.

##### Warnings

The arguments `i`

and `j`

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

. Beware of the usual
trap with factors: numerical values are not
interpreted in the same way as character values.

##### References

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

##### See Also

##### Examples

```
# Lansing Woods data
data(lansing)
lansing <- lansing[seq(1,lansing$n, by=10)]
ma <- split(lansing)$maple
wh <- split(lansing)$whiteoak
# method (1): estimate intensities by nonparametric smoothing
lambdaM <- density.ppp(ma, sigma=0.15, at="points")
lambdaW <- density.ppp(wh, sigma=0.15, at="points")
L <- Lcross.inhom(lansing, "whiteoak", "maple", lambdaW, lambdaM)
# method (2): fit parametric intensity model
fit <- ppm(lansing, ~marks * polynom(x,y,2))
# evaluate fitted intensities at data points
# (these are the intensities of the sub-processes of each type)
inten <- fitted(fit, dataonly=TRUE)
# split according to types of points
lambda <- split(inten, lansing$marks)
L <- Lcross.inhom(lansing, "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, X$window), lambdaJ=lamB)
```

*Documentation reproduced from package spatstat, version 1.16-2, License: GPL (>= 2)*