Spherical Contact Distribution Function
Estimates the spherical contact distribution function of a random set.
Hest(X, r=NULL, breaks=NULL, ...,
W,
correction=c("km", "rs", "han"),
conditional=TRUE)X |
The observed random set.
An object of class |
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 |
W |
Optional. A window (object of class |
correction |
Optional.
The edge correction(s) to be used to estimate H(r).
A vector of character strings selected from
|
conditional |
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 <= r | 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*(r) = P(D <= 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.
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 |
Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk with contributions from Kassel Hingee.
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)
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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.