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.

spatial, nonparametric
Kdot.inhom(X, i=1, lambdaI, lambdadot, r, breaks, correction, ..., lambdaIdot=NULL)
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.
Number or character string identifying the type (mark value) of the points in X from which distances are measured.
Values of the the estimated intensity of the sub-process of points of type i. Either a pixel image (object of class "im"), or a numeric vector containing the type i intensity values at each of the typ
Values of the the estimated intensity of the entire point process, Either a pixel image (object of class "im"), or a numeric vector containing the intensity values at each point in X.
Optional. Numeric vector giving the values of the argument $r$ at which the dot K function $K_{i\bullet}(r)$ should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument.
Optional. An alternative to the argument r. Not normally invoked by the user. See the Details section.
A character vector containing any selection of the options "border", "bord.modif", "isotropic", "Ripley" or "translate". It specifies the edge correction(s) to be applied.
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.

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 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).

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] The argument lambdadot should contain estimated values of the intensity of the entire point process. It may be either a pixel image, or a numeric vector of length equal to the number of points in X.

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.


  • 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.


Kdot.inhom(X, i=1, lambdaI, lambdadot, ..., lambdaIdot=NULL)


The argument i is interpreted as a level of the factor X$marks. Beware of the usual trap with factors: numerical values are not interpreted in the same way as character values.


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

See Also

Kdot, Kinhom, Kcross.inhom, pcf

  • Kdot.inhom
# Lansing Woods data
    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)
    lambdadot <- density.ppp(lg, sigma=0.15)
    K <- Kdot.inhom(lansing, "maple", lambdaI=lambdaM[ma],
    K <- Kdot.inhom(lansing, "maple", lambdaI=lambdaM,

    # 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))
    XB <- split(X)$B
    K <- Kdot.inhom(X, "B",  lambdaI=lamB[XB], lambdadot=lamdot[X])
    K <- Kdot.inhom(X, "B",  lambdaI=lamB,     lambdadot=lamdot)
Documentation reproduced from package spatstat, version 1.9-3, License: GPL version 2 or newer

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