Fest

0th

Percentile

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.

Keywords
spatial, nonparametric
Usage
Fest(X, ..., eps, r=NULL, breaks=NULL,
correction=c("rs", "km", "cs"),
domain=NULL)Fhazard(X, ...)
Arguments
X
The observed point pattern, from which an estimate of $F(r)$ will be computed. An object of class ppp, or data in any format acceptable to as.ppp().
...
Extra arguments, passed from Fhazard to Fest. Extra arguments to Fest are ignored.
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 "none", "rs", "km", "cs" and "best".
domain
Optional. Calculations will be restricted to this subset of the window. See Details.
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 $$F(r) = 1 - e^{ - \lambda \pi r^2}$$ where $\lambda$ 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.

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. 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

• Fest
• Fhazard
Examples
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

plot(Fc, . ~ theo)
plot(Fc, asin(sqrt(.)) ~ asin(sqrt(theo)))
<testonly>Fh <- Fhazard(cells)</testonly>
Documentation reproduced from package spatstat, version 1.41-1, License: GPL (>= 2)

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