spatstat.explore (version 3.5-3)

Hest: Spherical Contact Distribution Function

Description

Estimates the spherical contact distribution function of a random set.

Usage

Hest(X, r=NULL, breaks=NULL, ...,
     W,
     correction=c("km", "rs", "han"),
     conditional=TRUE)

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:

r

the values of the argument \(r\) at which the function \(H(r)\) has been estimated

rs

the ``reduced sample'' or ``border correction'' estimator of \(H(r)\)

km

the spatial Kaplan-Meier estimator of \(H(r)\)

hazard

the hazard rate \(\lambda(r)\) of \(H(r)\) by the spatial Kaplan-Meier method

han

the spatial Hanisch-Chiu-Stoyan estimator of \(H(r)\)

raw

the 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

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.

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.

See Also

Fest

Examples

   X <- runifpoint(42)
   H <- Hest(X)
   Y <- rpoisline(10)
   H <- Hest(Y)
   H <- Hest(Y, dimyx=256)
   X <- heather$coarse
   plot(Hest(X))
   H <- Hest(X, conditional=FALSE)

   P <- owin(poly=list(x=c(5.3, 8.5, 8.3, 3.7, 1.3, 3.7),
                       y=c(9.7, 10.0, 13.6, 14.4, 10.7, 7.2)))
   plot(X)
   plot(P, add=TRUE, col="red")
   H <- Hest(X, W=P)
   Z <- as.im(FALSE, Frame(X))
   Z[X] <- TRUE
   Z <- Z[P, drop=FALSE]
   plot(Z)
   H <- Hest(Z)