psst

0th

Percentile

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.

Keywords
models, spatial
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

Special cases: psstA, psstG.

Alternative functions: Kres, Gres.

Aliases
  • psst
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)

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