Multitype Nearest Neighbour Distance Function (i-to-j)
For a multitype point pattern, estimate the distribution of the distance from a point of type i to the nearest point of type j.
Gcross(X, i, j, r=NULL, breaks=NULL, ..., correction=c("rs", "km", "han"))X |
The observed point pattern, from which an estimate of the cross type distance distribution function Gij(r) will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor). See under Details. |
i |
The type (mark value)
of the points in |
j |
The type (mark value)
of the 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. |
correction |
Optional. Character string specifying the edge correction(s)
to be used. Options are |
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$marks must be a factor.
The arguments i and j will be interpreted as
levels of the factor X$marks. (Warning: this means that
an integer value i=3 will be interpreted as
the number 3, not the 3rd smallest level).
The “cross-type” (type i to type j) nearest neighbour distance distribution function of a multitype point process is the cumulative distribution function Gij(r) of the distance from a typical random point of the process with type i the nearest point of type j.
An estimate of Gij(r) is a useful summary statistic in exploratory data analysis of a multitype point pattern. If the process of type i points were independent of the process of type j points, then Gij(r) would equal Fj(r), the empty space function of the type j points. For a multitype Poisson point process where the type i points have intensity lambda[i], we have
Gij(r) = 1 - exp( - lambda[j] * pi * r^2)
Deviations between the empirical and theoretical Gij curves may suggest dependence between the points of types i and j.
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 (see below). |
The arguments i and j are always interpreted as
levels of the factor X$marks. They are converted to character
strings if they are not already character strings.
The value i=1 does not
refer to the first level of the factor.
The function Gij does not necessarily have a density.
The reduced sample estimator of Gij is pointwise approximately unbiased, but need not be a valid distribution function; it may not be a nondecreasing function of r. Its range is always within [0,1].
The spatial Kaplan-Meier estimator of Gij is always nondecreasing but its maximum value may be less than 1.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au
and Rolf Turner r.turner@auckland.ac.nz
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Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.
Diggle, P. J. (1986). Displaced amacrine cells in the retina of a rabbit : analysis of a bivariate spatial point pattern. J. Neurosci. Meth. 18, 115–125.
Harkness, R.D and Isham, V. (1983) A bivariate spatial point pattern of ants' nests. Applied Statistics 32, 293–303
Lotwick, H. W. and Silverman, B. W. (1982). Methods for analysing spatial processes of several types of points. J. Royal Statist. Soc. Ser. B 44, 406–413.
Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition. Springer Verlag, 1995.
Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511–532.
# amacrine cells data
G01 <- Gcross(amacrine)
# equivalent to:
# G01 <- Gcross(amacrine, "off", "on")
plot(G01)
# empty space function of `on' points
if(interactive()) {
F1 <- Fest(split(amacrine)$on, r = G01$r)
lines(F1$r, F1$km, lty=3)
}
# synthetic example
pp <- runifpoispp(30)
pp <- pp %mark% factor(sample(0:1, npoints(pp), replace=TRUE))
G <- Gcross(pp, "0", "1") # note: "0" not 0Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.