Sufficient Statistic of Point Process Model
The canonical sufficient statistic of a point process model is evaluated for a given point pattern.
- A fitted point process model (object of class
- A point pattern (object of class
The canonical sufficient statistic
model is evaluated for the point pattern
This computation is useful for various Monte Carlo methods.
model should be a point process model (object of class
ppm.object), typically obtained
from the model-fitting function
ppm. The argument
X should be a point pattern (object of class
Every point process model fitted by
a probability density of the form
$$f(x) = Z(\theta) \exp(\theta^T S(x))$$
where $x$ denotes a typical realisation (i.e. a point pattern),
$\theta$ is the vector of model coefficients,
$Z(\theta)$ is a normalising constant,
and $S(x)$ is a function of the realisation $x$, called the
``canonical sufficient statistic'' of the model.
For example, the stationary Poisson process has canonical sufficient statistic $S(x)=n(x)$, the number of points in $x$. The stationary Strauss process with interaction range $r$ (and no edge correction) has canonical sufficient statistic $S(x)=(n(x),d(x))$ where $d(x)$ is the number of pairs of points in $x$ which are closer than a distance $r$ to each other.
suffstat(model, X) returns the value of $S(x)$, where $S$ is
the canonical sufficient statistic associated with
evaluated when $x$ is the given point pattern
The result is a numeric vector, with entries which correspond to the
entries of the coefficient vector
The sufficient statistic $S$
does not depend on the fitted coefficients
of the model. However it does depend on the irregular parameters
which are fixed in the original call to
example, the interaction range
r of the Strauss process.
The sufficient statistic also depends on the edge correction that
was used to fit the model.
Non-finite values of the sufficient statistic (
-Inf) may be returned if the point pattern
not a possible realisation of the model (i.e. if
X has zero
probability of occurring under
model for all values of
the canonical coefficients $\theta$).
- A numeric vector of sufficient statistics. The entries
correspond to the model coefficients
data(swedishpines) fitS <- ppm(swedishpines, ~1, Strauss(7)) X <- rpoispp(summary(swedishpines)$intensity, win=swedishpines$window) suffstat(fitS, X)