spatstat.core (version 2.1-2)

# Kdot: Multitype K Function (i-to-any)

## Description

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, from)

## Arguments

X

The observed point pattern, from which an estimate of the multitype $$K$$ function $$K_{i\bullet}(r)$$ 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).

r

numeric vector. The values of the argument $$r$$ at which the distribution function $$K_{i\bullet}(r)$$ 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

An alternative way to specify i.

## 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 $$K_{i\bullet}(r)$$ has been estimated

theo

the theoretical value of $$K_{i\bullet}(r)$$ 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_{i\bullet}(r)Ki.(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 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 $$r$$. ## 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 Window(X)) may have arbitrary shape. Biases due to edge effects are treated in the same manner as in Kest, using the chosen edge correction(s).

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 not 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.

## 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 {
# Lansing woods data: 6 types of trees
woods <- lansing

# }
# NOT RUN {
Kh. <- Kdot(woods, "hickory")
# 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")
# }