Estimate the Empty Space Function or its Hazard Rate
Estimates the empty space function F(r) or its hazard rate h(r) from a point pattern in a window of arbitrary shape.
Fest(X, ..., eps, r=NULL, breaks=NULL,
correction=c("rs", "km", "cs"),
domain=NULL)
Fhazard(X, ...)X |
The observed point pattern,
from which an estimate of F(r) will be computed.
An object of class |
... |
Extra arguments, passed from |
eps |
Optional. A positive number. The resolution of the discrete approximation to Euclidean distance (see below). There is a sensible default. |
r |
Optional. Numeric vector. The values of the argument r at which F(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. |
correction |
Optional.
The edge correction(s) to be used to estimate F(r).
A vector of character strings selected from
|
domain |
Optional. Calculations will be restricted to this subset of the window. See Details. |
Fest computes an estimate of the empty space function F(r),
and Fhazard computes an estimate of its hazard rate h(r).
The empty space function (also called the “spherical contact distribution” or the “point-to-nearest-event” distribution) of a stationary point process X is the cumulative distribution function F of the distance from a fixed point in space to the nearest point of X.
An estimate of F 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 F is a useful statistic summarising the sizes of gaps in the pattern. For inferential purposes, the estimate of F is usually compared to the true value of F for a completely random (Poisson) point process, which is
F(r) = 1 - exp( - λ * π * r^2)
where λ is the intensity (expected number of points per unit area). Deviations between the empirical and theoretical F curves may suggest spatial clustering or spatial regularity.
This algorithm estimates the empty space function F
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) 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 algorithm uses two discrete approximations which are controlled
by the parameter eps and by the spacing of values of r
respectively. (See below for details.)
First-time users are strongly advised not to specify these arguments.
The estimation of F 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 Chiu-Stoyan estimator (Chiu and Stoyan, 1998).
Our implementation makes essential use of the distance transform algorithm of image processing (Borgefors, 1986). A fine grid of pixels is created in the observation window. The Euclidean distance between two pixels is approximated by the length of the shortest path joining them in the grid, where a path is a sequence of steps between adjacent pixels, and horizontal, vertical and diagonal steps have length 1, 1 and sqrt(2) respectively in pixel units. If the pixel grid is sufficiently fine then this is an accurate approximation.
The parameter eps
is the pixel width of the rectangular raster
used to compute the distance transform (see below). It must not be too
large: the absolute error in distance values due to discretisation is bounded
by eps.
If eps is not specified, the function
checks whether the window Window(X) contains pixel raster
information. If so, then eps is set equal to the
pixel width of the raster; otherwise, eps
defaults to 1/100 of the width of the observation window.
The argument r is the vector of values for the
distance r at which F(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 spacing of successive
r values must be very fine (ideally not greater than eps/4).
The algorithm also returns an estimate of the hazard rate function, h(r) of F(r). The hazard rate is defined by
h(r) = - (d/dr) log(1 - F(r))
The hazard rate of F has been proposed as a useful exploratory statistic (Baddeley and Gill, 1994). The estimate of h(r) given here is a discrete approximation to the hazard rate of the Kaplan-Meier estimator of F. Note that F is absolutely continuous (for any stationary point process X), so the hazard function always exists (Baddeley and Gill, 1997).
If the argument domain is given, the estimate of F(r)
will be based only on the empty space distances
measured from locations inside domain (although their
nearest data points 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 location
in the window to the nearest point of the data pattern, is a biased
estimate of F. However this is also returned by the algorithm
(if correction="none"),
as it is sometimes useful in other contexts.
Care should be taken not to use the uncorrected
empirical F as if it were an unbiased estimator of F.
The result of Fest is
essentially a data frame containing up to seven columns:
r |
the values of the argument r at which the function F(r) has been estimated |
rs |
the “reduced sample” or “border correction” estimator of F(r) |
km |
the spatial Kaplan-Meier estimator of F(r) |
hazard |
the hazard rate lambda(r) of F(r) by the spatial Kaplan-Meier method |
cs |
the Chiu-Stoyan estimator of F(r) |
raw |
the uncorrected estimate of F(r),
i.e. the empirical distribution of the distance from
a random point in the window to the nearest point of
the data pattern |
theo |
the theoretical value of F(r) for a stationary Poisson process of the same estimated intensity. |
The result of Fhazard contains only three columns
r |
the values of the argument r at which the hazard rate h(r) has been estimated |
hazard |
the spatial Kaplan-Meier estimate of the hazard rate h(r) |
theo |
the theoretical value of h(r) for a stationary Poisson process of the same estimated intensity. |
The reduced sample (border method) estimator of F 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 F is always nondecreasing but its maximum value may be less than 1.
The estimate of hazard rate h(r)
returned by the algorithm is an approximately
unbiased estimate for the integral of h()
over the corresponding histogram cell.
It may exhibit oscillations due to discretisation effects.
We recommend modest smoothing, such as kernel smoothing with
kernel width equal to the width of a histogram cell,
using Smooth.fv.
Sizeable amounts of memory may be needed during the calculation.
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. The empty space hazard of a spatial pattern. Research Report 1994/3, Department of Mathematics, University of Western Australia, May 1994.
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.
Borgefors, G. Distance transformations in digital images. Computer Vision, Graphics and Image Processing 34 (1986) 344-371.
Chiu, S.N. and Stoyan, D. (1998) Estimators of distance distributions for spatial patterns. Statistica Neerlandica 52, 239–246.
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.
Fc <- Fest(cells, 0.01)
# Tip: don't use F for the left hand side!
# That's an abbreviation for FALSE
plot(Fc)
# P-P style plot
plot(Fc, cbind(km, theo) ~ theo)
# The empirical F is above the Poisson F
# indicating an inhibited pattern
if(interactive()) {
plot(Fc, . ~ theo)
plot(Fc, asin(sqrt(.)) ~ asin(sqrt(theo)))
}Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.