# 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.

##### 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
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).
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
This argument is for internal use only.
correction
A character vector containing any selection of the options "border", "bord.modif", "isotropic", "Ripley", "translate", "translation", "none" or
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 $j\neq i$.

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$marks must be a factor. The argument i will be interpreted as a level of the factor X$marks. (Warning: this means that an integer value i=3 will be interpreted as the number 3, not the 3rd smallest level). If i is missing, it defaults to the first level of the marks factor, i = levels(X$marks)[1]. The argument lambdaI supplies the values of the intensity of the sub-process of points of type i. It may be either [object Object],[object Object],[object Object],[object Object] If lambdaI is omitted, then it will be estimated using a leave-one-out' kernel smoother, as described in Baddeley, latex{Mller{Moller} and Waagepetersen (2000). The estimate of lambdaI for a given point is computed by removing the point from the point pattern, applying kernel smoothing to the remaining points using density.ppp, and evaluating the smoothed intensity at the point in question. The smoothing kernel bandwidth is controlled by the arguments sigma and varcov, which are passed to density.ppp along with any extra arguments. Similarly the argument lambdadot should contain estimated values of the intensity of the entire point process. It may be either a pixel image, a numeric vector of length equal to the number of points in X, a function, or omitted. For advanced use only, the optional argument lambdaIdot is a matrix containing estimated values of the products of these two intensities for each pair of points, the first point of type i and the second of any type. The argument r is the vector of values for the distance$r$at which$K_{i\bullet}(r)$should be evaluated. The values of$r$must be increasing nonnegative numbers and the maximum$r$value must exceed the radius of the largest disc contained in the window. The argument correction chooses the edge correction as explained e.g. in Kest. The pair correlation function can also be applied to the result of Kcross.inhom; see pcf. } latex{Mller{Moller}, J. and Waagepetersen, R. Statistical Inference and Simulation for Spatial Point Processes Chapman and Hall/CRC Boca Raton, 2003. } Warnings { The argument i is interpreted as a level of the factor X$marks. It is converted to a character string if it is not already a character string. The value i=1 does not refer to the first level of the factor. } Kdot, Kinhom, Kcross.inhom, Kmulti.inhom, pcf # 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") lambdadot <- density.ppp(lg, sigma=0.15, at="points") K <- Kdot.inhom(lansing, "maple", lambdaI=lambdaM, lambdadot=lambdadot)

# 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) [object Object],[object Object],[object Object] spatial nonparametric

##### Value

• An object of class "fv" (see fv.object).

Essentially a data frame containing numeric columns

• rthe values of the argument $r$ at which the function $K_{i\bullet}(r)$ has been estimated
• theothe theoretical value of $K_{i\bullet}(r)$ 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 $K_{i\bullet}(r)$ obtained by the edge corrections named.

##### Aliases
• Kdot.inhom
Documentation reproduced from package spatstat, version 1.36-0, License: GPL (>= 2)

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