Kmulti: Marked K-Function


For a marked point pattern, estimate the multitype K function which counts the expected number of points of subset J within a given distance from a typical point in subset I.


Kmulti(X, I, J, r=NULL, breaks=NULL, correction, ..., ratio=FALSE)



The observed point pattern, from which an estimate of the multitype K function KIJ(r) will be computed. It must be a marked point pattern. See under Details.


Subset index specifying the points of X from which distances are measured. See Details.


Subset index specifying the points in X to which distances are measured. See Details.


numeric vector. The values of the argument r at which the multitype K function KIJ(r) should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument. See below for important conditions on r.


This argument is for internal use only.


A character vector containing any selection of the options "border", "bord.modif", "isotropic", "Ripley", "translate", "translation", "none" or "best". It specifies the edge correction(s) to be applied. Alternatively correction="all" selects all options.




Logical. If TRUE, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.


The function Kmulti generalises Kest (for unmarked point patterns) and Kdot and Kcross (for multitype point patterns) to arbitrary marked point patterns.

Suppose X[I], X[J] are subsets, possibly overlapping, of a marked point process. The multitype K function is defined so that lambda[J] KIJ(r) equals the expected number of additional random points of X[J] within a distance r of a typical point of X[I]. Here lambda[J] is the intensity of X[J] i.e. the expected number of points of X[J] per unit area. The function KIJ is determined by the second order moment properties of X.

The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to as.ppp.

The arguments I and J specify two subsets of the point pattern. They may be any type of subset indices, for example, logical vectors of length equal to npoints(X), or integer vectors with entries in the range 1 to npoints(X), or negative integer vectors.

Alternatively, I and J may be functions that will be applied to the point pattern X to obtain index vectors. If I is a function, then evaluating I(X) should yield a valid subset index. This option is useful when generating simulation envelopes using envelope.

The argument r is the vector of values for the distance r at which KIJ(r) should be evaluated. It is also used to determine the breakpoints (in the sense of hist) for the computation of histograms of distances.

First-time users would be strongly advised not to specify r. However, if it is specified, r must satisfy r[1] = 0, and max(r) must be larger than the radius of the largest disc contained in the window.

This algorithm assumes that X can be treated as a realisation of a stationary (spatially homogeneous) random spatial point process in the plane, observed through a bounded window. The window (which is specified in X as Window(X)) may have arbitrary shape.

Biases due to edge effects are treated in the same manner as in Kest. The edge corrections implemented here are


the border method or “reduced sample” estimator (see Ripley, 1988). This is the least efficient (statistically) and the fastest to compute. It can be computed for a window of arbitrary shape.


Ripley's isotropic correction (see Ripley, 1988; Ohser, 1983). This is currently implemented only for rectangular and polygonal windows.


Translation correction (Ohser, 1983). Implemented for all window geometries.

The pair correlation function pcf can also be applied to the result of Kmulti.


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

Essentially a data frame containing numeric columns


the values of the argument r at which the function KIJ(r) has been estimated


the theoretical value of KIJ(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 KIJ(r) obtained by the edge corrections named.

If ratio=TRUE then the return value also has two attributes called "numerator" and "denominator" which are "fv" objects containing the numerators and denominators of each estimate of K(r).


The function KIJ is not necessarily differentiable.

The border correction (reduced sample) estimator of KIJ used here is pointwise approximately unbiased, but need not be a nondecreasing function of r, while the true KIJ must be nondecreasing.





Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.

Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.

Diggle, P. J. (1986). Displaced amacrine cells in the retina of a rabbit : analysis of a bivariate spatial point pattern. J. Neurosci. Meth. 18, 115–125.

Harkness, R.D and Isham, V. (1983) A bivariate spatial point pattern of ants' nests. Applied Statistics 32, 293–303

Lotwick, H. W. and Silverman, B. W. (1982). Methods for analysing spatial processes of several types of points. J. Royal Statist. Soc. Ser. B 44, 406–413.

Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.

Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition. Springer Verlag, 1995.

Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511–532.

See Also

Kcross, Kdot, Kest, pcf


     # Longleaf Pine data: marks represent diameter
    trees <- longleaf
    K <- Kmulti(trees, marks(trees) <= 15, marks(trees) >= 25)
    # functions determining subsets
    f1 <- function(X) { marks(X) <= 15 }
    f2 <- function(X) { marks(X) >= 15 }
    K <- Kmulti(trees, f1, f2)

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