# Concom

##### The Connected Component Process Model

Creates an instance of the Connected Component point process model which can then be fitted to point pattern data.

##### Usage

`Concom(r)`

##### Arguments

- r
Threshold distance

##### Details

This function defines the interpoint interaction structure of a point process called the connected component process. It can be used to fit this model to point pattern data.

The function `ppm()`

, which fits point process models to
point pattern data, requires an argument
of class `"interact"`

describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the connected component interaction is
yielded by the function `Concom()`

. See the examples below.

In **standard form**, the connected component process
(Baddeley and Moller, 1989) with disc radius \(r\),
intensity parameter \(\kappa\) and interaction parameter
\(\gamma\) is a point process with probability density
$$
f(x_1,\ldots,x_n) =
\alpha \kappa^{n(x)} \gamma^{-C(x)}
$$
for a point pattern \(x\), where
\(x_1,\ldots,x_n\) represent the
points of the pattern, \(n(x)\) is the number of points in the
pattern, and \(C(x)\) is defined below.
Here \(\alpha\) is a normalising constant.

To define the term `C(x)`

, suppose that we construct a planar
graph by drawing an edge between
each pair of points \(x_i,x_j\) which are less than
\(r\) units apart. Two points belong to the same connected component
of this graph if they are joined by a path in the graph.
Then \(C(x)\) is the number of connected components of the graph.

The interaction parameter \(\gamma\) can be any positive number. If \(\gamma = 1\) then the model reduces to a Poisson process with intensity \(\kappa\). If \(\gamma < 1\) then the process is regular, while if \(\gamma > 1\) the process is clustered. Thus, a connected-component interaction process can be used to model either clustered or regular point patterns.

In spatstat, the model is parametrised in a different form,
which is easier to interpret.
In **canonical form**, the probability density is rewritten as
$$
f(x_1,\ldots,x_n) =
\alpha \beta^{n(x)} \gamma^{-U(x)}
$$
where \(\beta\) is the new intensity parameter and
\(U(x) = C(x) - n(x)\) is the interaction potential.
In this formulation, each isolated point of the pattern contributes a
factor \(\beta\) to the probability density (so the
first order trend is \(\beta\)). The quantity
\(U(x)\) is a true interaction potential, in the sense that
\(U(x) = 0\) if the point pattern \(x\) does not contain any
points that lie close together.

When a new point \(u\) is added to an existing point pattern \(x\), the rescaled potential \(-U(x)\) increases by zero or a positive integer. The increase is zero if \(u\) is not close to any point of \(x\). The increase is a positive integer \(k\) if there are \(k\) different connected components of \(x\) that lie close to \(u\). Addition of the point \(u\) contributes a factor \(\beta \eta^\delta\) to the probability density, where \(\delta\) is the increase in potential.

If desired, the original parameter \(\kappa\) can be recovered from the canonical parameter by \(\kappa = \beta\gamma\).

The *nonstationary* connected component process is similar except that
the contribution of each individual point \(x_i\)
is a function \(\beta(x_i)\)
of location, rather than a constant beta.

Note the only argument of `Concom()`

is the threshold distance `r`

.
When `r`

is fixed, the model becomes an exponential family.
The canonical parameters \(\log(\beta)\)
and \(\log(\gamma)\)
are estimated by `ppm()`

, not fixed in
`Concom()`

.

##### Value

An object of class `"interact"`

describing the interpoint interaction
structure of the connected component process with disc radius \(r\).

##### Edge correction

The interaction distance of this process is infinite.
There are no well-established procedures for edge correction
for fitting such models, and accordingly the model-fitting function
`ppm`

will give an error message saying that the user must
specify an edge correction. A reasonable solution is
to use the border correction at the same distance `r`

, as shown in the
Examples.

##### References

Baddeley, A.J. and Moller, J. (1989)
Nearest-neighbour Markov point processes and random sets.
*International Statistical Review* **57**, 89--121.

##### See Also

##### Examples

```
# NOT RUN {
# prints a sensible description of itself
Concom(r=0.1)
# Fit the stationary connected component process to redwood data
ppm(redwood, ~1, Concom(r=0.07), rbord=0.07)
# Fit the stationary connected component process to `cells' data
ppm(cells, ~1, Concom(r=0.06), rbord=0.06)
# eta=0 indicates hard core process.
# Fit a nonstationary connected component model
# with log-cubic polynomial trend
# }
# NOT RUN {
ppm(swedishpines, ~polynom(x/10,y/10,3), Concom(r=7), rbord=7)
# }
```

*Documentation reproduced from package spatstat, version 1.59-0, License: GPL (>= 2)*