# Hest

0th

Percentile

##### Spherical Contact Distribution Function

Estimates the spherical contact distribution function of a random set.

Keywords
spatial, nonparametric
##### Usage
Hest(X, r=NULL, breaks=NULL, ...,
correction=c("km", "rs", "han"),
conditional=TRUE)
##### Arguments
X
The observed random set. An object of class "ppp", "psp" or "owin".
r
Optional. Vector of values for the argument $r$ at which $H(r)$ should be evaluated. Users are advised not to specify this argument; there is a sensible default.
breaks
This argument is for internal use only.
...
Arguments passed to as.mask to control the discretisation.
correction
Optional. The edge correction(s) to be used to estimate $H(r)$. A vector of character strings selected from "none", "rs", "km", "han" and "best".
conditional
Logical value indicating whether to compute the conditional or unconditional distribution. See Details.
##### Details

The spherical contact distribution function of a stationary random set $X$ is the cumulative distribution function $H$ of the distance from a fixed point in space to the nearest point of $X$, given that the point lies outside $X$. That is, $H(r)$ equals the probability that X lies closer than $r$ units away from the fixed point $x$, given that X does not cover $x$.

Let $D = d(x,X)$ be the shortest distance from an arbitrary point $x$ to the set X. Then the spherical contact distribution function is $$H(r) = P(D \le r \mid D > 0)$$ For a point process, the spherical contact distribution function is the same as the empty space function $F$ discussed in Fest.

The argument X may be a point pattern (object of class "ppp"), a line segment pattern (object of class "psp") or a window (object of class "owin"). It is assumed to be a realisation of a stationary random set.

The algorithm first calls distmap to compute the distance transform of X, then computes the Kaplan-Meier and reduced-sample estimates of the cumulative distribution following Hansen et al (1999). If conditional=TRUE (the default) the algorithm returns an estimate of the spherical contact function $H(r)$ as defined above. If conditional=FALSE, it instead returns an estimate of the cumulative distribution function $H^\ast(r) = P(D \le r)$ which includes a jump at $r=0$ if X has nonzero area.

Accuracy depends on the pixel resolution, which is controlled by the arguments eps, dimyx and xy passed to as.mask. For example, use eps=0.1 to specify square pixels of side 0.1 units, and dimyx=256 to specify a 256 by 256 grid of pixels.

##### Value

• An object of class "fv", see fv.object, which can be plotted directly using plot.fv.

Essentially a data frame containing up to six columns:

• rthe values of the argument $r$ at which the function $H(r)$ has been estimated
• rsthe reduced sample'' or border correction'' estimator of $H(r)$
• kmthe spatial Kaplan-Meier estimator of $H(r)$
• hazardthe hazard rate $\lambda(r)$ of $H(r)$ by the spatial Kaplan-Meier method
• hanthe spatial Hanisch-Chiu-Stoyan estimator of $H(r)$
• rawthe uncorrected estimate of $H(r)$, i.e. the empirical distribution of the distance from a fixed point in the window to the nearest point of X

##### 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.

Hansen, M.B., Baddeley, A.J. and Gill, R.D. First contact distributions for spatial patterns: regularity and estimation. Advances in Applied Probability 31 (1999) 15-33.

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.

Fest

• Hest
##### Examples
X <- runifpoint(42)
H <- Hest(X)
Y <- rpoisline(10)
H <- Hest(Y)
H <- Hest(Y, dimyx=256)
data(heather)
H <- Hest(heather$coarse) H <- Hest(heather$coarse, conditional=FALSE)
Documentation reproduced from package spatstat, version 1.42-2, License: GPL (>= 2)

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