Gest
Nearest Neighbour Distance Function G
Estimates the nearest neighbour distance distribution function $G(r)$ from a point pattern in a window of arbitrary shape.
- Keywords
- spatial, nonparametric
Usage
Gest(X, r=NULL, breaks=NULL, ..., correction=c("rs", "km", "han"))
Arguments
- X
- The observed point pattern,
from which an estimate of $G(r)$ will be computed.
An object of class
ppp
, or data in any format acceptable toas.ppp()
. - 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
- An alternative to the argument
r
. Not normally invoked by the user. See the Details section. - ...
- Ignored.
- correction
- Optional.
The edge correction(s) to be used to estimate $G(r)$.
A vector of character strings selected from
"none"
,"rs"
,"km"
,"Hanisch"
and"best"
.
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 $$G(r) = 1 - e^{ - \lambda \pi r^2}$$ 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 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 $$\lambda(r) = - \frac{d}{dr} \log (1 - G(r))$$ 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 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 $G$.
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
.
Value
- An object of class
"fv"
, seefv.object
, which can be plotted directly usingplot.fv
.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 X
to the nearest other point of the patternhan the Hanisch correction estimator of $G(r)$ theo the theoretical value of $G(r)$ for a stationary Poisson process of the same estimated intensity.
Warnings
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 $1$.
References
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.
See Also
nndist
,
nnwhich
,
Fest
,
Jest
,
Kest
,
km.rs
,
reduced.sample
,
kaplan.meier
Examples
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
plot(G, . ~ r)
plot(G, . ~ theo)
plot(G, asin(sqrt(.)) ~ asin(sqrt(theo)))