# Kcross.inhom

##### Inhomogeneous Cross K Function

For a multitype point pattern, estimate the inhomogeneous version of the cross $K$ function, which counts the expected number of points of type $j$ within a given distance of a point of type $i$, adjusted for spatially varying intensity.

- Keywords
- spatial, nonparametric

##### Usage

```
Kcross.inhom(X, i, j, lambdaI=NULL, lambdaJ=NULL, ..., r=NULL, breaks=NULL,
correction = c("border", "isotropic", "Ripley", "translate"),
sigma=NULL, varcov=NULL,
lambdaIJ=NULL)
```

##### Arguments

- X
- The observed point pattern, from which an estimate of the inhomogeneous cross type $K$ function $K_{ij}(r)$ 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)`

. - lambdaI
- Optional.
Values of the the estimated intensity of the sub-process of
points of type
`i`

. Either a pixel image (object of class`"im"`

), a numeric vector containing the intensity values at each of the type`i`

- lambdaJ
- Optional.
Values of the the estimated intensity of the sub-process of
points of type
`j`

. Either a pixel image (object of class`"im"`

), a numeric vector containing the intensity values at each of the type`j`

- r
- Optional. Numeric vector giving the values of the argument $r$ at which the cross K function $K_{ij}(r)$ should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument. Se
- breaks
- Optional. An alternative to the argument
`r`

. Not normally invoked by the user. See the**Details**section. - correction
- A character vector containing any selection of the
options
`"border"`

,`"bord.modif"`

,`"isotropic"`

,`"Ripley"`

,`"translate"`

,`"none"`

or`"best"`

. It specifie - ...
- Ignored.
- sigma
- Standard deviation of isotropic Gaussian smoothing kernel,
used in computing leave-one-out kernel estimates of
`lambdaI`

,`lambdadot`

if they are omitted. - varcov
- Variance-covariance matrix of anisotropic Gaussian kernel,
used in computing leave-one-out kernel estimates of
`lambdaI`

,`lambdadot`

if they are omitted. Incompatible with`sigma`

. - lambdaIJ
- Optional. A matrix containing estimates of the
product of the intensities
`lambdaI`

and`lambdaJ`

for each pair of points of types`i`

and`j`

respectively.

##### Details

This is a generalisation of the function `Kcross`

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

.

The inhomogeneous cross-type $K$ function is described by Moller and Waagepetersen (2003, pages 48-49 and 51-53). Briefly, given a multitype point process, suppose the sub-process of points of type $j$ has intensity function $\lambda_j(u)$ at spatial locations $u$. Suppose we place a mass of $1/\lambda_j(\zeta)$ at each point $\zeta$ of type $j$. Then the expected total mass per unit area is 1. The inhomogeneous ``cross-type'' $K$ function $K_{ij}^{\mbox{inhom}}(r)$ equals the expected total mass within a radius $r$ of a point of the process of type $i$. If the process of type $i$ points were independent of the process of type $j$ points, then $K_{ij}^{\mbox{inhom}}(r)$ would equal $\pi r^2$. Deviations between the empirical $K_{ij}$ curve and the theoretical curve $\pi r^2$ suggest dependence between the points of types $i$ and $j$.

The argument `X`

must be a point pattern (object of class
`"ppp"`

) or any data that are acceptable to `as.ppp`

.
It must be a marked point pattern, and the mark vector
`X$marks`

must be a factor.

The arguments `i`

and `j`

will be interpreted as
levels of the factor `X$marks`

. (Warning: this means that
an integer value `i=3`

will be interpreted as the 3rd smallest level,
not the number 3).
If `i`

and `j`

are missing, they default to the first
and second level of the marks factor, respectively.

The argument `lambdaI`

supplies the values
of the intensity of the sub-process of points of type `i`

.
It may be either
[object Object],[object Object],[object Object],[object Object]
If `lambdaI`

is omitted, then it will be estimated using
a `leave-one-out' kernel smoother, as described in Baddeley, Moller
and Waagepetersen (2000). The estimate of `lambdaI`

for a given
point is computed by removing the point from the
point pattern, applying kernel smoothing to the remaining points using
`density.ppp`

, and evaluating the smoothed intensity
at the point in question. The smoothing kernel bandwidth is controlled
by the arguments `sigma`

and `varcov`

, which are passed to
`density.ppp`

along with any extra arguments.

Similarly `lambdaJ`

should contain
estimated values of the intensity of the sub-process of points of
type `j`

. It may be either a pixel image, a function,
a numeric vector, or omitted.
The optional argument `lambdaIJ`

is for advanced use only.
It is a matrix containing estimated
values of the products of these two intensities for each pair of
data points of types `i`

and `j`

respectively.
The argument `r`

is the vector of values for the
distance $r$ at which $K_{ij}(r)$ should be evaluated.
The values of $r$ must be increasing nonnegative numbers
and the maximum $r$ value must exceed the radius of the
largest disc contained in the window.

The argument `correction`

chooses the edge correction
as explained e.g. in `Kest`

.

The pair correlation function can also be applied to the
result of `Kcross.inhom`

; see `pcf`

.

##### 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 $K_{ij}(r)$ has been estimated theo the theoretical value of $K_{ij}(r)$ for a marked Poisson process, namely $\pi r^2$ - 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 $K_{ij}(r)$ 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)$whiteoak
# method (1): estimate intensities by nonparametric smoothing
lambdaM <- density.ppp(ma, sigma=0.15, at="points")
lambdaW <- density.ppp(wh, sigma=0.15, at="points")
K <- Kcross.inhom(lansing, "whiteoak", "maple", lambdaW, lambdaM)
# method (2): leave-one-out
K <- Kcross.inhom(lansing, "whiteoak", "maple", sigma=0.15)
# method (3): fit parametric intensity model
fit <- ppm(lansing, ~marks * polynom(x,y,2))
# evaluate fitted intensities at data points
# (these are the intensities of the sub-processes of each type)
inten <- fitted(fit, dataonly=TRUE)
# split according to types of points
lambda <- split(inten, lansing$marks)
K <- Kcross.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))
K <- Kcross.inhom(X, "A", "B",
lambdaI=as.im(50, X$window), lambdaJ=lamB)
```

*Documentation reproduced from package spatstat, version 1.23-3, License: GPL (>= 2)*