Marked Nearest Neighbour Distance Function
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"))X |
The observed point pattern, from which an estimate of the multitype distance distribution function GIJ(r) will be computed. It must be a marked point pattern. See under Details. |
I |
Subset of points of |
J |
Subset of points in |
r |
Optional. Numeric vector. The values of the argument r at which the distribution function GIJ(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. |
breaks |
This argument is for internal use only. |
... |
Ignored. |
disjoint |
Optional flag indicating whether
the subsets |
correction |
Optional. Character string specifying the edge correction(s)
to be used. Options are |
Suppose X[I], X[J] are subsets, possibly overlapping, of a marked point process. This function computes an estimate of the cumulative distribution function GIJ(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 GIJ(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 GIJ(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 GIJ(r). This estimate should be used with caution as GIJ(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 GIJ.
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 GIJ as if it were an unbiased estimator of
GIJ.
An object of class "fv" (see fv.object).
Essentially a data frame containing six numeric columns
r |
the values of the argument r at which the function GIJ(r) has been estimated |
rs |
the “reduced sample” or “border correction” estimator of GIJ(r) |
han |
the Hanisch-style estimator of GIJ(r) |
km |
the spatial Kaplan-Meier estimator of GIJ(r) |
hazard |
the hazard rate lambda(r) of GIJ(r) by the spatial Kaplan-Meier method |
raw |
the uncorrected estimate of GIJ(r), i.e. the empirical distribution of the distances from each point of type i to the nearest point of type j |
theo |
the theoretical value of GIJ(r) for a marked Poisson process with the same estimated intensity |
The function GIJ does not necessarily have a density.
The reduced sample estimator of GIJ 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 GIJ 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.
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.
trees <- longleaf
# Longleaf Pine data: marks represent diameter
Gm <- Gmulti(trees, marks(trees) <= 15, marks(trees) >= 25)
plot(Gm)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.