Kdot(X, i=1)
Kdot(X, i=1, correction=c("border", "isotropic", "Ripley", "translate"))
Kdot(X, i=1, r, correction)
Kdot(X, i=1, breaks)X from which distances are measured.r.
Not normally invoked by the user. See the Details section."border", "bord.modif",
"isotropic", "Ripley" or "translate".
It specifies the edge correction(s) to be applied."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.i is interpreted as
a level of the factor X$marks. Beware of the usual
trap with factors: numerical values are not
interpreted in the same way as character values. See the first example.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 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. (Warning: this means that
an integer value i=3 will be interpreted as the 3rd smallest level,
not the number 3).
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$n, by=80), ]
Kh. <- Kdot(sub, "hickory")</testonly>
# diagnostic plot for independence between hickories and other trees
plot(Kh.)
# synthetic example with two marks "a" and "b"
pp <- runifpoispp(50)
pp <- pp %mark% sample(c("a","b"), pp$n, replace=TRUE)
K <- Kdot(pp, "a")Run the code above in your browser using DataLab