R: The Connected Component Process Model

Concom {spatstat.core} | R Documentation

## The Connected Component Process Model

### Description

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 κ and interaction parameter γ is a point process with probability density

f(x[1],…,x[n]) = α . κ^n(x) . γ^(-C(x))

for a point pattern x, where x[1],…,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 α 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 γ can be any positive number. If γ = 1 then the model reduces to a Poisson process with intensity κ. If γ < 1 then the process is regular, while if γ > 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,…,x_n) = α . β^n(x) γ^(-U(x))

where β 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 β to the probability density (so the first order trend is β). 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 β * η^δ to the probability density, where δ is the increase in potential.

If desired, the original parameter κ can be recovered from the canonical parameter by κ = β * γ.

The nonstationary connected component process is similar except that the contribution of each individual point x[i] is a function β(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(β) and log(γ) 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.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk

### References

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

### See Also

`ppm`, `pairwise.family`, `ppm.object`

### 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)

[Package spatstat.core version 2.0-0 Index]