R: Spherical Contact Distribution Function

Hest {spatstat.core} | R Documentation

## 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)

### Arguments

`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.

### 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.

### 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`

### Author(s)

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.

### 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)

[Package spatstat.core version 2.3-1 Index]