# Hest

##### 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
- Optional. An alternative to the argument
`r`

. Not normally invoked by the user. - ...
- 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.

##### 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`

##### 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

##### Examples

```
X <- runifpoint(42)
H <- Hest(X)
Y <- rpoisline(10)
H <- Hest(Y)
data(heather)
H <- Hest(heather$coarse)
H <- Hest(heather$coarse, conditional=FALSE)
```

*Documentation reproduced from package spatstat, version 1.18-4, License: GPL (>= 2)*