spatstat.core (version 2.1-2)

# Kcross: Multitype K Function (Cross-type)

## Description

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 $$i$$.

## Usage

Kcross(X, i, j, r=NULL, breaks=NULL, correction,
…, ratio=FALSE, from, to )

## 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

The type (mark value) of the points in X from which distances are measured. A character string (or something that will be converted to a character string). Defaults to the first level of marks(X).

j

The type (mark value) of the points in X to which distances are measured. A character string (or something that will be converted to a character string). 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 conditions on $$r$$.

breaks

This argument is for internal use only.

correction

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.

Ignored.

ratio

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

from,to

An alternative way to specify i and j respectively.

## Value

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

Essentially a data frame containing numeric columns

r

the values of the argument $$r$$ at which the function $$K_{ij}(r)$$ has been estimated

theo

the 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)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 arguments i and j are always interpreted as levels of the factor X$marks. They are converted to character strings if they are not already character strings. The value i=1 does not refer to the first level of the factor. ## Details This function Kcross and its companions Kdot and Kmulti are generalisations of the function Kest to multitype point patterns. A multitype point pattern is a spatial pattern of points classified into a finite number of possible colours'' or types''. In the spatstat package, a multitype pattern is represented as a single point pattern object in which the points carry marks, and the mark value attached to each point determines the type of that point. The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to as.ppp. It must be a marked point pattern, and the mark vector X$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 Window(X)) 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 not 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.

## 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.

Kdot, Kest, Kmulti, pcf

## Examples

# NOT RUN {
# amacrine cells data
K01 <- Kcross(amacrine, "off", "on")
plot(K01)

# }
# NOT RUN {
## K10 <- Kcross(amacrine, "on", "off")

# synthetic example: point pattern with marks 0 and 1
## pp <- runifpoispp(50)
## pp <- pp %mark% factor(sample(0:1, npoints(pp), replace=TRUE))
## K <- Kcross(pp, "0", "1")
## K <- Kcross(pp, 0, 1) # equivalent
# }