Nearest Neighbour Distance Function G
Estimates the nearest neighbour distance distribution function G(r) from a point pattern in a window of arbitrary shape.
Gest(X, r=NULL, breaks=NULL, ...,
correction=c("rs", "km", "han"),
domain=NULL)X |
The observed point pattern,
from which an estimate of G(r) will be computed.
An object of class |
r |
Optional. Numeric vector. The values of the argument r at which G(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.
The edge correction(s) to be used to estimate G(r).
A vector of character strings selected from
|
domain |
Optional. Calculations will be restricted to this subset of the window. See Details. |
The nearest neighbour distance distribution function (also called the “event-to-event” or “inter-event” distribution) of a point process X is the cumulative distribution function G of the distance from a typical random point of X to the nearest other point of X.
An estimate of G derived from a spatial point pattern dataset can be used in exploratory data analysis and formal inference about the pattern (Cressie, 1991; Diggle, 1983; Ripley, 1988). In exploratory analyses, the estimate of G is a useful statistic summarising one aspect of the “clustering” of points. For inferential purposes, the estimate of G is usually compared to the true value of G for a completely random (Poisson) point process, which is
G(r) = 1 - exp( - lambda * pi * r^2)
where lambda is the intensity (expected number of points per unit area). Deviations between the empirical and theoretical G curves may suggest spatial clustering or spatial regularity.
This algorithm estimates the nearest neighbour distance distribution
function G
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.
The argument X is interpreted as a point pattern object
(of class "ppp", see ppp.object) and can
be supplied in any of the formats recognised
by as.ppp().
The estimation of G is hampered by edge effects arising from the unobservability of points of the random pattern outside the window. An edge correction is needed to reduce bias (Baddeley, 1998; Ripley, 1988). The edge corrections implemented here are the border method or “reduced sample” estimator, the spatial Kaplan-Meier estimator (Baddeley and Gill, 1997) and the Hanisch estimator (Hanisch, 1984).
The argument r is the vector of values for the
distance r at which G(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
estimators are computed from histogram counts.
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(r). The hazard rate is defined as the derivative
lambda(r) = - (d/dr) log(1 - G(r))
This estimate should be used with caution as G is not necessarily differentiable.
If the argument domain is given, the estimate of G(r)
will be based only on the nearest neighbour distances
measured from points falling inside domain (although their
nearest neighbours may lie outside domain).
This is useful in bootstrap techniques. The argument domain
should be a window (object of class "owin") or something acceptable to
as.owin. It must be a subset of the
window of the point pattern X.
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. However it is sometimes useful.
It can be returned by the algorithm, by selecting correction="none".
Care should be taken not to use the uncorrected
empirical G as if it were an unbiased estimator of G.
Essentially a data frame containing some or all of the following columns:
r |
the values of the argument r at which the function G(r) has been estimated |
rs |
the “reduced sample” or “border correction” estimator of G(r) |
km |
the spatial Kaplan-Meier estimator of G(r) |
hazard |
the hazard rate lambda(r) of G(r) by the spatial Kaplan-Meier method |
raw |
the uncorrected estimate of G(r),
i.e. the empirical distribution of the distances from
each point in the pattern |
han |
the Hanisch correction estimator of G(r) |
theo |
the theoretical value of G(r) for a stationary Poisson process of the same estimated intensity. |
The function G does not necessarily have a density. Any valid c.d.f. may appear as the nearest neighbour distance distribution function of a stationary point process.
The reduced sample estimator of G 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 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
Baddeley, A.J. Spatial sampling and censoring. In O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. van Lieshout (eds) Stochastic Geometry: Likelihood and Computation. Chapman and Hall, 1998. Chapter 2, pages 37-78.
Baddeley, A.J. and Gill, R.D. Kaplan-Meier estimators of interpoint distance distributions for spatial point processes. Annals of Statistics 25 (1997) 263-292.
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.
Hanisch, K.-H. (1984) Some remarks on estimators of the distribution function of nearest-neighbour distance in stationary spatial point patterns. Mathematische Operationsforschung und Statistik, series Statistics 15, 409–412.
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.
nndist,
nnwhich,
Fest,
Jest,
Kest,
km.rs,
reduced.sample,
kaplan.meier
data(cells)
G <- Gest(cells)
plot(G)
# P-P style plot
plot(G, cbind(km,theo) ~ theo)
# the empirical G is below the Poisson G,
# indicating an inhibited pattern
if(interactive()) {
plot(G, . ~ r)
plot(G, . ~ theo)
plot(G, asin(sqrt(.)) ~ asin(sqrt(theo)))
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