R: Sufficient Statistic of Point Process Model

suffstat {spatstat.core} | 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 `"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 * 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)`.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.

### See Also

`ppm`

### Examples

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

[Package spatstat.core version 1.65-5 Index]