# Lcross.inhom

##### Inhomogeneous Cross Type L Function

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

##### 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

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.

##### References

`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. }

`Lcross`

,
`Linhom`

,
`Kcross.inhom`

# 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 1.37-0, License: GPL (>= 2)*