Gest(X, r=NULL, breaks=NULL, ...)ppp, or data
in any format acceptable to as.ppp().r.
Not normally invoked by the user. See the Details section."fv", see fv.object,
which can be plotted directly using plot.fv.Essentially a data frame containing six columns:
X to the nearest other point of
the patternThe 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 $1$.
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 $$G(r) = 1 - e^{ - \lambda \pi r^2}$$ 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 X$window)
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 two edge corrections implemented here are the border method or ``reduced sample'' estimator, and the spatial Kaplan-Meier estimator (Baddeley and Gill, 1997).
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 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 $G(r)$. The hazard rate is defined as the derivative $$\lambda(r) = - \frac{d}{dr} \log (1 - G(r))$$ This estimate should be used with caution as $G$ 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 $G$.
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 $G$ as if it were an unbiased estimator of $G$.
To simply compute the nearest neighbour distance for each point in the
pattern, use nndist. To determine which point is the
nearest neighbour of a given point, use nnwhich.
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.
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.meierdata(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
plot(G, . ~ r)
plot(G, . ~ theo)
plot(G, asin(sqrt(.)) ~ asin(sqrt(theo)))Run the code above in your browser using DataLab