# suffstat

##### Sufficient Statistic of Point Process Model

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

`"ppm"`

).- X
A point pattern (object of class

`"ppp"`

).

##### 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 is \(S(x) = (n(x), s(x))\) where \(n(x)\) is the number of points and \(s(x)\) is the number of pairs of points which are closer than \(r\) 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"`

, then \(n(x)\) is unchanged but \(s(x)\) is replaced by a weighted sum (the sum of the translation correction weights for all pairs of points which are closer than \(r\) units apart).If the model is fitted with

`correction="border"`

(the default), then points lying less than \(r\) units from the boundary of the observation window are treated as fixed. Thus \(n(x)\) is replaced by the number \(n_r(x)\) of points lying at least \(r\) units from the boundary of the observation window, and \(s(x)\) is replaced by the number \(s_r(x)\) of pairs of points, which are closer than \(r\) units apart, and at least one of which lies more than \(r\) 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)`

.

##### See Also

##### Examples

```
# NOT RUN {
fitS <- ppm(swedishpines~1, Strauss(7))
suffstat(fitS)
X <- rpoispp(intensity(swedishpines), win=Window(swedishpines))
suffstat(fitS, X)
# }
```

*Documentation reproduced from package spatstat, version 1.56-1, License: GPL (>= 2)*