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Estimates the empty space function
Fest(X, ..., eps, r=NULL, breaks=NULL,
correction=c("rs", "km", "cs"),
domain=NULL)Fhazard(X, ...)
An object of class "fv"
, see fv.object
,
which can be plotted directly using plot.fv
.
The result of Fest
is
essentially a data frame containing up to seven columns:
the values of the argument
the ``reduced sample'' or ``border correction''
estimator of
the spatial Kaplan-Meier estimator of
the hazard rate
the Chiu-Stoyan estimator of
the uncorrected estimate of X
the theoretical value of
The result of Fhazard
contains only three columns
the values of the argument
the spatial Kaplan-Meier estimate of the
hazard rate
the theoretical value of
The observed point pattern,
from which an estimate of ppp
, or data
in any format acceptable to as.ppp()
.
Extra arguments, passed from Fhazard
to Fest
.
Extra arguments to Fest
are ignored.
Optional. A positive number. The resolution of the discrete approximation to Euclidean distance (see below). There is a sensible default.
Optional. Numeric vector. The values of the argument
This argument is for internal use only.
Optional.
The edge correction(s) to be used to estimate "none"
, "rs"
, "km"
, "cs"
and "best"
.
Alternatively correction="all"
selects all options.
Optional. Calculations will be restricted to this subset of the window. See Details.
The reduced sample (border method)
estimator of
The spatial Kaplan-Meier estimator of
The estimate of hazard rate Smooth.fv
.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au
and Rolf Turner rolfturner@posteo.net
Fest
computes an estimate of the empty space function Fhazard
computes an estimate of its hazard rate
The empty space function
(also called the ``spherical contact distribution''
or the ``point-to-nearest-event'' distribution)
of a stationary point process
An estimate of
This algorithm estimates the empty space function 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
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
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 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,
If the argument domain
is given, the estimate of 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 correction="none"
),
as it is sometimes useful in other contexts.
Care should be taken not to use the uncorrected
empirical
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.
Gest
,
Jest
,
Kest
,
km.rs
,
reduced.sample
,
kaplan.meier
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)))
}
# \testonly{
Fh <- Fhazard(cells)
# }
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