# Kdot.inhom

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##### Inhomogeneous Multitype K Dot Function

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

Keywords
spatial, nonparametric
##### Usage
Kdot.inhom(X, i, lambdaI=NULL, lambdadot=NULL, ..., r=NULL, breaks=NULL,
correction = c("border", "isotropic", "Ripley", "translate"),
sigma=NULL, varcov=NULL, lambdaIdot=NULL)
##### Arguments
X
The observed point pattern, from which an estimate of the inhomogeneous cross type $K$ function $K_{i\bullet}(r)$ 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).
lambdaI
Optional. Values of 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
Optional. Values of the estimated intensity of the entire point process, Either a pixel image (object of class "im"), a numeric vector containing the intensity values at each of the points in X, or a functi
...
Ignored.
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
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.
lambdaIdot
Optional. A matrix containing estimates of the product of the intensities lambdaI and lambdadot for each pair of points, the first point of type i and the second of any type.
##### Details

This is a generalisation of the function Kdot to include an adjustment for spatially inhomogeneous intensity, in a manner similar to the function Kinhom.

Briefly, given a multitype point process, consider the points without their types, and suppose this unmarked point process has intensity function $\lambda(u)$ at spatial locations $u$. Suppose we place a mass of $1/\lambda(\zeta)$ at each point $\zeta$ of the process. Then the expected total mass per unit area is 1. The inhomogeneous dot-type'' $K$ function $K_{i\bullet}^{\mbox{inhom}}(r)$ equals the expected total mass within a radius $r$ of a point of the process of type $i$, discounting this point itself. If the process of type $i$ points were independent of the points of other types, then $K_{i\bullet}^{\mbox{inhom}}(r)$ would equal $\pi r^2$. Deviations between the empirical $K_{i\bullet}$ curve and the theoretical curve $\pi r^2$ suggest dependence between the points of types $i$ and $j$ for $j\neq i$.

##### References

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

Kdot, Kinhom, Kcross.inhom, pcf

• Kdot.inhom
##### Examples
# Lansing Woods data
data(lansing)
lansing <- lansing[seq(1,lansing$n, by=10)] ma <- split(lansing)$maple
lg <- unmark(lansing)

# Estimate intensities by nonparametric smoothing
lambdaM <- density.ppp(ma, sigma=0.15, at="points")
K <- Kdot.inhom(lansing, "maple", lambdaI=lambdaM,

# Equivalent
K <- Kdot.inhom(lansing, "maple", sigma=0.15)

# synthetic example: type A points have intensity 50,
#                    type B points have intensity 50 + 100 * x
lamB <- as.im(function(x,y){50 + 100 * x}, owin())
lamdot <- as.im(function(x,y) { 100 + 100 * x}, owin())
X <- superimpose(A=runifpoispp(50), B=rpoispp(lamB))
K <- Kdot.inhom(X, "B",  lambdaI=lamB,     lambdadot=lamdot)`
Documentation reproduced from package spatstat, version 1.17-0, License: GPL (>= 2)

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