The Connected Component Process Model
Creates an instance of the Connected Component point process model which can then be fitted to point pattern data.
- Threshold distance
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.
ppm(), which fits point process models to
point pattern data, requires an argument
"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
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.
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 is fixed, the model becomes an exponential family.
The canonical parameters $\log(\beta)$
are estimated by
ppm(), not fixed in
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
# 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)
- An object of class
"interact"describing the interpoint interaction structure of the connected component process with disc radius $r$.