# Kdot

0th

Percentile

##### 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 $i$.

Keywords
spatial, nonparametric
##### 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.

##### 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). 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 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. ##### 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). ##### 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.

Kdot, Kest, Kmulti, pcf

• Kdot
##### 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.)

# }
# NOT RUN {
# 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")

# }

Documentation reproduced from package spatstat, version 1.59-0, License: GPL (>= 2)

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