Kcross

From spatstat v1.16-2 by Adrian Baddeley

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 $i$.

Keywords: spatial, nonparametric

Usage

Kcross(X, i, j, r=NULL, breaks=NULL, correction, ...)

Arguments

X: 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.
i: 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).
j: 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).
r: 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
breaks: An alternative to the argument r. Not normally invoked by the user. See the Details section.
correction: A character vector containing any selection of the options "border", "bord.modif", "isotropic", "Ripley", "translate", "none" or "best". It specifie
...: Ignored.

Details

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.

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 $K_{ij}(r)$ 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.

Warnings

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 $[0,1]$.

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.

Examples

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

    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 1.16-2, License: GPL (>= 2)

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