# 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 **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

```
# 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
ppm(swedishpines, ~polynom(x/10,y/10,3), Concom(r=7), rbord=7)
```

*Documentation reproduced from package spatstat, version 1.31-2, License: GPL (>= 2)*