Estimates the spherical contact distribution function of a random set.

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

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`

- X
The observed random set. An object of class

`"ppp"`

,`"psp"`

or`"owin"`

. Alternatively a pixel image (class`"im"`

) with logical values.- 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.- W
Optional. A window (object of class

`"owin"`

) to be taken as the window of observation. The contact distribution function will be estimated from values of the contact distance inside`W`

. The default is`W=Frame(X)`

when`X`

is a window, and`W=Window(X)`

otherwise.- 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"`

. Alternatively`correction="all"`

selects all options.- conditional
Logical value indicating whether to compute the conditional or unconditional distribution. See Details.

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.

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.

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`

```
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)
```

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