Detect high-density features in a spatial point pattern using the (unrestricted) Allard-Fraley estimator.

```
clusterset(X, what=c("marks", "domain"),
..., verbose=TRUE,
fast=FALSE,
exact=!fast)
```

If `what="marks"`

, a multitype point pattern (object of class

`"ppp"`

).

If `what="domain"`

, a window (object of class

`"owin"`

).

If `what=c("marks", "domain")`

(the default),
a list consisting of a multitype point pattern and a window.

- X
A dimensional spatial point pattern (object of class

`"ppp"`

).- what
Character string or character vector specifying the type of result. See Details.

- verbose
Logical value indicating whether to print progress reports.

- fast
Logical. If

`FALSE`

(the default), the Dirichlet tile areas will be computed exactly using polygonal geometry, so that the optimal choice of tiles will be computed exactly. If`TRUE`

, the Dirichlet tile areas will be approximated using pixel counting, so the optimal choice will be approximate.- exact
Logical. If

`TRUE`

, the Allard-Fraley estimator of the domain will be computed exactly using polygonal geometry. If`FALSE`

, the Allard-Fraley estimator of the domain will be approximated by a binary pixel mask. The default is initially set to`FALSE`

.- ...
Optional arguments passed to

`as.mask`

to control the pixel resolution if`exact=FALSE`

.

Adrian Baddeley Adrian.Baddeley@curtin.edu.au

and Rolf Turner r.turner@auckland.ac.nz

Allard and Fraley (1997) developed a technique for recognising features of high density in a spatial point pattern in the presence of random clutter.

This algorithm computes the *unrestricted* Allard-Fraley estimator.
The Dirichlet (Voronoi) tessellation of the point pattern `X`

is
computed. The smallest `m`

Dirichlet cells are selected,
where the number `m`

is determined by a maximum likelihood
criterion.

If

`fast=FALSE`

(the default), the areas of the tiles of the Dirichlet tessellation will be computed exactly using polygonal geometry. This ensures that the optimal selection of tiles is computed exactly.If

`fast=TRUE`

, the Dirichlet tile areas will be approximated by counting pixels. This is faster, and is usually correct (depending on the pixel resolution, which is controlled by the arguments`...`

).

The type of result depends on the character vector `what`

.

If

`what="marks"`

the result is the point pattern`X`

with a vector of marks labelling each point with a value`yes`

or`no`

depending on whether the corresponding Dirichlet cell is selected by the Allard-Fraley estimator. In other words each point of`X`

is labelled as either a cluster point or a non-cluster point.If

`what="domain"`

, the result is the Allard-Fraley estimator of the cluster feature set, which is the union of all the selected Dirichlet cells, represented as a window (object of class`"owin"`

).If

`what=c("marks", "domain")`

the result is a list containing both of the results described above.

Computation of the Allard-Fraley set estimator depends on
the argument `exact`

.

If

`exact=TRUE`

(the default), the Allard-Fraley set estimator will be computed exactly using polygonal geometry. The result is a polygonal window.If

`exact=FALSE`

, the Allard-Fraley set estimator will be approximated by a binary pixel mask. This is faster than the exact computation. The result is a binary mask.

Allard, D. and Fraley, C. (1997)
Nonparametric maximum likelihood estimation of features in
spatial point processes using Voronoi tessellation.
*Journal of the American Statistical Association*
**92**, 1485--1493.

`nnclean`

,
`sharpen`

```
opa <- par(mfrow=c(1,2))
W <- grow.rectangle(as.rectangle(letterR), 1)
X <- superimpose(runifpoint(300, letterR),
runifpoint(50, W), W=W)
plot(W, main="clusterset(X, 'm')")
plot(clusterset(X, "marks", fast=TRUE), add=TRUE, chars=c(1, 3), cols=1:2)
plot(letterR, add=TRUE)
plot(W, main="clusterset(X, 'd')")
plot(clusterset(X, "domain", exact=FALSE), add=TRUE)
plot(letterR, add=TRUE)
par(opa)
```

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