# 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

```
fitS <- ppm(swedishpines, ~1, Strauss(7))
X <- rpoispp(intensity(swedishpines), win=Window(swedishpines))
suffstat(fitS, X)
```

*Documentation reproduced from package spatstat, version 1.42-2, License: GPL (>= 2)*