# psst

##### Pseudoscore Diagnostic For Fitted Model against General Alternative

Given a point process model fitted to a point pattern dataset, and any choice of functional summary statistic, this function computes the pseudoscore test statistic of goodness-of-fit for the model.

##### Usage

```
psst(object, fun, r = NULL, breaks = NULL, ...,
trend = ~1, interaction = Poisson(), rbord = reach(interaction),
truecoef=NULL, hi.res=NULL, funargs = list(correction="best"))
```

##### Arguments

- object
- Object to be analysed.
Either a fitted point process model (object of class
`"ppm"`

) or a point pattern (object of class`"ppp"`

) or quadrature scheme (object of class`"quad"`

). - fun
- Summary function to be applied to each point pattern.
- r
- Optional. Vector of values of the argument $r$ at which the function $S(r)$ should be computed. This argument is usually not specified. There is a sensible default.
- breaks
- Optional alternative to
`r`

for advanced use. - ...
- Ignored.
- trend,interaction,rbord
- Optional. Arguments passed to
`ppm`

to fit a point process model to the data, if`object`

is a point pattern. See`ppm`

for details. - truecoef
- Optional. Numeric vector. If present, this will be treated as
if it were the true coefficient vector of the point process model,
in calculating the diagnostic. Incompatible with
`hi.res`

. - hi.res
- Optional. List of parameters passed to
`quadscheme`

. If this argument is present, the model will be re-fitted at high resolution as specified by these parameters. The coefficients of the re - funargs
- Optional.
List of additional arguments to be passed to
`fun`

.

##### Details

Let $x$ be a point pattern dataset consisting of points $x_1,\ldots,x_n$ in a window $W$. Consider a point process model fitted to $x$, with conditional intensity $\lambda(u,x)$ at location $u$. For the purpose of testing goodness-of-fit, we regard the fitted model as the null hypothesis. Given a functional summary statistic $S$, consider a family of alternative models obtained by exponential tilting of the null model by $S$. The pseudoscore for the null model is $$V(r) = \sum_i \Delta S(x_i, x, r) - \int_W \Delta S(u,x, r) \lambda(u,x) {\rm d} u$$ where the $\Delta$ operator is $$\Delta S(u,x, r) = S(x\cup{u}, r) - S(x\setminus u, r)$$ the difference between the values of $S$ for the point pattern with and without the point $u$.

According to the Georgii-Nguyen-Zessin formula, $V(r)$ should have mean zero if the model is correct (ignoring the fact that the parameters of the model have been estimated). Hence $V(r)$ can be used as a diagnostic for goodness-of-fit.

This algorithm computes $V(r)$ by direct evaluation of the sum and integral. It is computationally intensive, but it is available for any summary statistic $S(r)$.

The diagnostic $V(r)$ is also called
the **pseudoresidual** of $S$. On the right
hand side of the equation for $V(r)$ given above,
the sum over points of $x$ is called the
**pseudosum** and the integral is called the **pseudocompensator**.

##### Value

- A function value table (object of class
`"fv"`

), essentially a data frame of function values.Columns in this data frame include

`dat`

for the pseudosum,`com`

for the compensator and`res`

for the pseudoresidual. There is a plot method for this class. See`fv.object`

.

##### References

Baddeley, A., Rubak, E. and Moller, J. (2011)
Score, pseudo-score and residual
diagnostics for spatial point process models.
To appear in *Statistical Science*.

##### See Also

##### Examples

```
data(cells)
fit0 <- ppm(cells, ~1) # uniform Poisson
<testonly>fit0 <- ppm(cells, ~1, nd=8)</testonly>
G0 <- psst(fit0, Gest)
G0
if(interactive()) plot(G0)
```

*Documentation reproduced from package spatstat, version 1.27-0, License: GPL (>= 2)*