For a marked point pattern,
estimate the distribution of the distance
from a typical point in subset I
to the nearest point of subset \(J\).
Gmulti(X, I, J, r=NULL, breaks=NULL, ...,
disjoint=NULL, correction=c("rs", "km", "han"))An object of class "fv" (see fv.object).
Essentially a data frame containing six numeric columns
the values of the argument \(r\) at which the function \(G_{IJ}(r)\) has been estimated
the ``reduced sample'' or ``border correction'' estimator of \(G_{IJ}(r)\)
the Hanisch-style estimator of \(G_{IJ}(r)\)
the spatial Kaplan-Meier estimator of \(G_{IJ}(r)\)
the hazard rate \(\lambda(r)\) of \(G_{IJ}(r)\) by the spatial Kaplan-Meier method
the uncorrected estimate of \(G_{IJ}(r)\), i.e. the empirical distribution of the distances from each point of type \(i\) to the nearest point of type \(j\)
the theoretical value of \(G_{IJ}(r)\) for a marked Poisson process with the same estimated intensity
The observed point pattern, from which an estimate of the multitype distance distribution function \(G_{IJ}(r)\) will be computed. It must be a marked point pattern. See under Details.
Subset of points of X from which distances are
measured.
Subset of points in X to which distances are measured.
Optional. Numeric vector. The values of the argument \(r\) at which the distribution function \(G_{IJ}(r)\) 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 \(r\).
This argument is for internal use only.
Ignored.
Optional flag indicating whether
the subsets I and J are disjoint.
If missing, this value will be computed by inspecting the
vectors I and J.
Optional. Character string specifying the edge correction(s)
to be used. Options are "none", "rs", "km",
"hanisch" and "best".
Alternatively correction="all" selects all options.
The function \(G_{IJ}\) does not necessarily have a density.
The reduced sample estimator of \(G_{IJ}\) 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 \([0,1]\).
The spatial Kaplan-Meier estimator of \(G_{IJ}\) is always nondecreasing but its maximum value may be less than \(1\).
Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.
The function Gmulti
generalises Gest (for unmarked point
patterns) and Gdot and Gcross (for
multitype point patterns) to arbitrary marked point patterns.
Suppose \(X_I\), \(X_J\) are subsets, possibly overlapping, of a marked point process. This function computes an estimate of the cumulative distribution function \(G_{IJ}(r)\) of the distance from a typical point of \(X_I\) to the nearest distinct point of \(X_J\).
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 any type of subset indices, for example,
logical vectors of length equal to npoints(X),
or integer vectors with entries in the range 1 to
npoints(X), or negative integer vectors.
Alternatively, I and J may be functions
that will be applied to the point pattern X to obtain
index vectors. If I is a function, then evaluating
I(X) should yield a valid subset index. This option
is useful when generating simulation envelopes using
envelope.
This algorithm estimates the distribution function \(G_{IJ}(r)\)
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 Gest.
The argument r is the vector of values for the
distance \(r\) at which \(G_{IJ}(r)\) should be evaluated.
It is also used to determine the breakpoints
(in the sense of hist)
for the computation of histograms of distances. The reduced-sample and
Kaplan-Meier estimators are computed from histogram counts.
In the case of the Kaplan-Meier estimator this introduces a discretisation
error which is controlled by the fineness of the breakpoints.
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. Furthermore, the successive entries of r
must be finely spaced.
The algorithm also returns an estimate of the hazard rate function, \(\lambda(r)\), of \(G_{IJ}(r)\). This estimate should be used with caution as \(G_{IJ}(r)\) is not necessarily differentiable.
The naive empirical distribution of distances from each point of
the pattern X to the nearest other point of the pattern,
is a biased estimate of \(G_{IJ}\).
However this is also returned by the algorithm, as it is sometimes
useful in other contexts. Care should be taken not to use the uncorrected
empirical \(G_{IJ}\) as if it were an unbiased estimator of
\(G_{IJ}\).
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.
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.
Gcross,
Gdot,
Gest
trees <- longleaf
# Longleaf Pine data: marks represent diameter
# \testonly{
trees <- trees[seq(1, npoints(trees), by=50), ]
# }
Gm <- Gmulti(trees, marks(trees) <= 15, marks(trees) >= 25)
plot(Gm)
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