suffstat {spatstat.model} | R Documentation |
Sufficient Statistic of Point Process Model
Description
The canonical sufficient statistic of a point process model is evaluated for a given point pattern.
Usage
suffstat(model, X=data.ppm(model))
Arguments
model |
A fitted point process model (object of class
|
X |
A point pattern (object of class |
Details
The canonical sufficient statistic
of model
is evaluated for the point pattern X
.
This computation is useful for various Monte Carlo methods.
Here model
should be a point process model (object of class
"ppm"
, see ppm.object
), typically obtained
from the model-fitting function ppm
. The argument
X
should be a point pattern (object of class "ppp"
).
Every point process model fitted by ppm
has
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 fitted with no edge correction) has canonical sufficient statistic
S(x)=(n(x),s(x))
where s(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 model
,
evaluated when x
is the given point pattern X
.
The result is a numeric vector, with entries which correspond to the
entries of the coefficient vector coef(model)
.
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 ppm
, for
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. For example in a Strauss process,
-
If the model is fitted with
correction="none"
, the sufficient statistic isS(x) = (n(x), s(x))
wheren(x)
is the number of points ands(x)
is the number of pairs of points which are closer thanr
units apart. -
If the model is fitted with
correction="periodic"
, the sufficient statistic is the same as above, except that distances are measured in the periodic sense. -
If the model is fitted with
correction="translate"
, thenn(x)
is unchanged buts(x)
is replaced by a weighted sum (the sum of the translation correction weights for all pairs of points which are closer thanr
units apart). -
If the model is fitted with
correction="border"
(the default), then points lying less thanr
units from the boundary of the observation window are treated as fixed. Thusn(x)
is replaced by the numbern_r(x)
of points lying at leastr
units from the boundary of the observation window, ands(x)
is replaced by the numbers_r(x)
of pairs of points, which are closer thanr
units apart, and at least one of which lies more thanr
units from the boundary of the observation window.
Non-finite values of the sufficient statistic (NA
or
-Inf
) may be returned if the point pattern X
is
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
).
Value
A numeric vector of sufficient statistics. The entries
correspond to the model coefficients coef(model)
.
Author(s)
Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner rolfturner@posteo.net and Ege Rubak rubak@math.aau.dk.
See Also
Examples
fitS <- ppm(swedishpines~1, Strauss(7))
suffstat(fitS)
X <- rpoispp(intensity(swedishpines), win=Window(swedishpines))
suffstat(fitS, X)