Multitype K Function (Cross-type)

For a multitype point pattern, estimate the multitype $K$ function which counts the expected number of points of type $j$ within a given distance of a point of type $i$.

spatial, nonparametric
Kcross(X, i, j, r=NULL, breaks=NULL, correction, ...)
The observed point pattern, from which an estimate of the 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.
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).
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).
numeric vector. The values of the argument $r$ at which the distribution function $K_{ij}(r)$ should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument. See below for important
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", "translate", "none" or "best". It specifie

This function Kcross and its companions Kdot and Kmulti are generalisations of the function Kest to multitype point patterns.

A multitype point pattern is a spatial pattern of points classified into a finite number of possible ``colours'' or ``types''. In the spatstat package, a multitype pattern is represented as a single point pattern object in which the points carry marks, and the mark value attached to each point determines the type of that point. 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. If i and j are missing, they default to the first and second level of the marks factor, respectively. The ``cross-type'' (type $i$ to type $j$) $K$ function of a stationary multitype point process $X$ is defined so that $\lambda_j K_{ij}(r)$ equals the expected number of additional random points of type $j$ within a distance $r$ of a typical point of type $i$ in the process $X$. Here $\lambda_j$ is the intensity of the type $j$ points, i.e. the expected number of points of type $j$ per unit area. The function $K_{ij}$ is determined by the second order moment properties of $X$.

An estimate of $K_{ij}(r)$ is a useful summary statistic in exploratory data analysis of a multitype point pattern. If the process of type $i$ points were independent of the process of type $j$ points, then $K_{ij}(r)$ would equal $\pi r^2$. Deviations between the empirical $K_{ij}$ curve and the theoretical curve $\pi r^2$ may suggest dependence between the points of types $i$ and $j$.

This algorithm estimates the distribution function $K_{ij}(r)$ from the point pattern X. It 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 X$window) may have arbitrary shape. Biases due to edge effects are treated in the same manner as in Kest, using the border correction.

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 pair correlation function can also be applied to the result of Kcross; 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 $K_{ij}(r)$ has been estimated
  • theothe 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.


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. See the first example.

The reduced sample estimator of $K_{ij}$ is pointwise approximately unbiased, but need not be a valid distribution function; it may not be a nondecreasing function of $r$. Its range is always within $[0,1]$.


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.

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.

See Also

Kdot, Kest, Kmulti, pcf

  • Kcross
     # cat retina data
    K01 <- Kcross(betacells, "off", "on") 

    K10 <- Kcross(betacells, "on", "off")

    # synthetic example    
    pp <- runifpoispp(50)
    pp <- pp %mark% factor(sample(0:1, pp$n, replace=TRUE))
    K <- Kcross(pp, "0", "1") # note: "0" not 0
Documentation reproduced from package spatstat, version 1.19-2, License: GPL (>= 2)

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