# 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)
##### 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 imp
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
...
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.
##### 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)$from the point pattern X. It assumes that X can be treated as a realisation of a stationary (spatially homogeneous) random spatial point process in the plane, observed through a bounded window. The window (which is specified in X as X$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.

##### 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 $K_{i\bullet}(r)$ has been estimated
• theothe 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 $K_{i\bullet}(r)$ 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$. Its range is always within$[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")
Documentation reproduced from package spatstat, version 1.37-0, License: GPL (>= 2)

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