Kdot(X, i, r=NULL, breaks=NULL, correction, ..., ratio=FALSE)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.
Not normally invoked by the user. See the Details section."border", "bord.modif",
"isotropic", "Ripley", "translate",
"translation",
"none" or TRUE, the numerator and denominator of
each edge-corrected estimate will also be saved,
for use in analysing replicated point patterns."fv" (see fv.object).Essentially a data frame containing numeric columns
"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 $K(r)$.
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 $[0,1]$.
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 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 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.
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# 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")Run the code above in your browser using DataLab