I.Kmulti(X, I, J, r=NULL, breaks=NULL, correction, ...)X
from which distances are measured.X to which
distances are measured.r.
Not normally invoked by the user. See the Details section."border", "bord.modif",
"isotropic", "Ripley", "translate",
"none" or "best".
It specifie"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 $K_{IJ}(r)$
obtained by the edge corrections named.The border correction (reduced sample) estimator of $K_{IJ}$ used here is pointwise approximately unbiased, but need not be a nondecreasing function of $r$, while the true $K_{IJ}$ must be nondecreasing.
Kmulti
generalises Kest (for unmarked point
patterns) and Kdot and Kcross (for
multitype point patterns) to arbitrary marked point patterns.Suppose $X_I$, $X_J$ are subsets, possibly overlapping, of a marked point process. The multitype $K$ function is defined so that $\lambda_J K_{IJ}(r)$ equals the expected number of additional random points of $X_J$ within a distance $r$ of a typical point of $X_I$. Here $\lambda_J$ is the intensity of $X_J$ i.e. the expected number of points of $X_J$ per unit area. The function $K_{IJ}$ is determined by the second order moment properties of $X$.
The argument X must be a point pattern (object of class
"ppp") or any data that are acceptable to as.ppp.
The arguments I and J specify two subsets of the
point pattern. They may be logical vectors of length equal to
X$n, or integer vectors with entries in the range 1 to
X$n, etc.
The argument r is the vector of values for the
distance $r$ at which $K_{IJ}(r)$ should be evaluated.
It is also used to determine the breakpoints
(in the sense of hist)
for the computation of histograms of distances.
First-time users would be strongly advised not to specify r.
However, if it is specified, r must satisfy r[1] = 0,
and max(r) must be larger than the radius of the largest disc
contained in the window.
This algorithm 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$window)
may have arbitrary shape.
Biases due to edge effects are
treated in the same manner as in Kest.
The edge corrections implemented here are
[object Object],[object Object],[object Object]
The pair correlation function pcf can also be applied to the
result of Kmulti.
Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.
Diggle, P. J. (1986). Displaced amacrine cells in the retina of a rabbit : analysis of a bivariate spatial point pattern. J. Neurosci. Meth. 18, 115--125. Harkness, R.D and Isham, V. (1983) A bivariate spatial point pattern of ants' nests. Applied Statistics 32, 293--303 Lotwick, H. W. and Silverman, B. W. (1982). Methods for analysing spatial processes of several types of points. J. Royal Statist. Soc. Ser. B 44, 406--413.
Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition. Springer Verlag, 1995.
Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511--532.
Kcross,
Kdot,
Kest,
pcfdata(longleaf)
# Longleaf Pine data: marks represent diameter
<testonly>longleaf <- longleaf[seq(1,longleaf$n, by=50), ]</testonly>
K <- Kmulti(longleaf, longleaf$marks <= 15, longleaf$marks >= 25)
plot(K)Run the code above in your browser using DataLab