Spherical Contact Distribution Function
Estimates the spherical contact distribution function of a random set.
Hest(X, r=NULL, breaks=NULL, ..., correction=c("km", "rs", "han"), conditional=TRUE)
- The observed random set.
An object of class
- 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.
- This argument is for internal use only.
- Arguments passed to
as.maskto control the discretisation.
The edge correction(s) to be used to estimate $H(r)$.
A vector of character strings selected from
- Logical value indicating whether to compute the conditional or unconditional distribution. See 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
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
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).
conditional=TRUE (the default) the algorithm
returns an estimate of the spherical contact function
$H(r)$ as defined above.
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
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.
- An object of class
fv.object, which can be plotted directly using
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
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.
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)