Kdot

Multitype K Function (i-to-any)

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

Usage

Kdot(X, i, r=NULL, breaks=NULL, correction, ..., ratio=FALSE)

Details

This function Kdot and its companions Kcross 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 argument i will be interpreted as a level of the factor X$marks. If i is missing, it defaults to the first level of the marks factor, i = levels(X$marks)[1]. The ``type $i$ to any type'' multitype $K$ function of a stationary multitype point process $X$ is defined so that $\lambda K_{i\bullet}(r)$ equals the expected number of additional random points within a distance $r$ of a typical point of type $i$ in the process $X$. Here $\lambda$ is the intensity of the process, i.e. the expected number of points of $X$ per unit area. The function $K_{i\bullet}$ is determined by the second order moment properties of $X$.

An estimate of $K_{i\bullet}(r)$ is a useful summary statistic in exploratory data analysis of a multitype point pattern. If the subprocess of type $i$ points were independent of the subprocess of points of all types not equal to $i$, then $K_{i\bullet}(r)$ would equal $\pi r^2$. Deviations between the empirical $K_{i\bullet}$ curve and the theoretical curve $\pi r^2$ may suggest dependence between types.

This algorithm estimates the distribution function $K_{i\bullet}(r)$ 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 $K_{i\bullet}(r)$ 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 pair correlation function can also be applied to the result of Kdot; see pcf.

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.

The reduced sample estimator of $K_{i\bullet}$ 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 $[0,1]$.

References

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

Aliases

• Kdot

Examples

# Lansing woods data: 6 types of trees
    data(lansing)

    Kh. <- Kdot(lansing, "hickory")
    <testonly>sub <- lansing[seq(1,lansing$n, by=80), ]
    Kh. <- Kdot(sub, "hickory")</testonly>
    # diagnostic plot for independence between hickories and other trees
    plot(Kh.)

    # synthetic example with two marks "a" and "b"
    pp <- runifpoispp(50)
    pp <- pp %mark% factor(sample(c("a","b"), npoints(pp), replace=TRUE))
    K <- Kdot(pp, "a")