AreaInter(r)"interact"
describing the interpoint interaction
structure of the area-interaction process with disc radius $r$.2 * r. Two discs of radius r overlap if their centres
are closer than 2 * r units apart. 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 area interaction structure is
yielded by the function AreaInter(). See the examples below.
In standard form, the area-interaction process (Widom and Rowlinson, 1970; Baddeley and Van Lieshout, 1995) with disc radius $r$, intensity parameter $\kappa$ and interaction parameter $\gamma$ is a point process with probability density $$f(x_1,\ldots,x_n) = \alpha \kappa^{n(x)} \gamma^{-A(x)}$$ where $x_1,\ldots,x_n$ represent the points of the pattern, $n(x)$ is the number of points in the pattern, and $A(x)$ is the area of the region formed by the union of discs of radius $r$ centred at the points $x_1,\ldots,x_n$. Here $\alpha$ is a normalising constant.
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, an area interaction process can be used to model either clustered or regular point patterns. Two points interact if the distance between them is less than $2r$.
The standard form of the model, shown above, is a little
complicated to interpret in practical applications.
For example, each isolated point of the pattern $x$ contributes a factor
$\kappa \gamma^{-\pi r^2}$
to the probability density.
In
The old parameters $\kappa,\gamma$ of the standard form are related to the new parameters $\beta,\eta$ of the canonical scale-free form, by $$\beta = \kappa \gamma^{-\pi r^2} = \kappa /\eta$$ and $$\eta = \gamma^{\pi r^2}$$ provided $\gamma$ and $\kappa$ are positive and finite.
In the canonical scale-free form, the parameter $\eta$
can take any nonnegative value. The value $\eta = 1$
again corresponds to a Poisson process, with intensity $\beta$.
If $\eta < 1$ then the process is regular,
while if $\eta > 1$ the process is clustered.
The value $\eta = 0$ corresponds to a hard core process
with hard core radius $r$ (interaction distance $2r$).
The nonstationary area interaction 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 AreaInter() is the disc radius r.
When r is fixed, the model becomes an exponential family.
The canonical parameters $\log(\beta)$
and $\log(\eta)$
are estimated by ppm(), not fixed in
AreaInter().
Widom, B. and Rowlinson, J.S. (1970). New model for the study of liquid-vapor phase transitions. The Journal of Chemical Physics 52 (1970) 1670--1684.
ppm,
pairwise.family,
ppm.object# prints a sensible description of itself
AreaInter(r=0.1)
# Note the reach is twice the radius
reach(AreaInter(r=1))
# Fit the stationary area interaction process to Swedish Pines data
data(swedishpines)
ppm(swedishpines, ~1, AreaInter(r=7))
# Fit the stationary area interaction process to `cells'
data(cells)
ppm(cells, ~1, AreaInter(r=0.06))
# eta=0 indicates hard core process.
# Fit a nonstationary area interaction with log-cubic polynomial trend
ppm(swedishpines, ~polynom(x/10,y/10,3), AreaInter(r=7))Run the code above in your browser using DataLab