Gest {spatstat}R Documentation

Nearest Neighbour Distance Function G

Description

Estimates the nearest neighbour distance distribution function G(r) from a point pattern in a window of arbitrary shape.

Usage

Gest(X, r=NULL, breaks=NULL, ...)

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 to as.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.

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 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) = - (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.

Value

An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
Essentially a data frame containing six 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 pattern
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.

Author(s)

Adrian Baddeley adrian@maths.uwa.edu.au http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner rolf@math.unb.ca http://www.math.unb.ca/~rolf

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.

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

  ## Not run: 
     plot(G, . ~ r)
     plot(G, . ~ theo)
     plot(G, asin(sqrt(.)) ~ asin(sqrt(theo)))
  
## End(Not run)

[Package spatstat version 1.10-3 Index]