psstG: Pseudoscore Diagnostic For Fitted Model against Saturation Alternative

Description

Given a point process model fitted to a point pattern dataset, this function computes the pseudoscore diagnostic of goodness-of-fit for the model, against moderately clustered or moderately inhibited alternatives of saturation type.

Usage

psstG(object, r = NULL, breaks = NULL, ...,
      trend = ~1, interaction = Poisson(), rbord = reach(interaction),
      truecoef = NULL, hi.res = NULL)

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").

Optional. Vector of values of the argument $r$ at which the diagnostic 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

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.

Details

This function computes the pseudoscore test statistic which can be used as a diagnostic for goodness-of-fit of a fitted point process model. 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. The alternative hypothesis is a family of hybrid models obtained by combining the fitted model with the Geyer saturation process (see Geyer) with saturation parameter 1. The family of alternatives includes models that are more regular than the fitted model, and others that are more clustered than the fitted model.

For any point pattern $x$, and any $r > 0$, let $S(x,r)$ be the number of points in $x$ whose nearest neighbour (the nearest other point in $x$) is closer than $r$ units. Then 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.

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.

References

Baddeley, A., Rubak, E. and Moller, J. (2011) Score, pseudo-score and residual diagnostics for spatial point process models. Statistical Science 26, 613--646.

Examples

Run this code

X <- rStrauss(200,0.1,0.05)
   plot(psstG(X))
   plot(psstG(X, interaction=Strauss(0.05)))

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