psstA: Pseudoscore Diagnostic For Fitted Model against Area-Interaction 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 area-interaction type.

Usage

psstA(object, r = NULL, breaks = NULL, ...,
        trend = ~1, interaction = Poisson(),
        rbord = reach(interaction), ppmcorrection = "border",
        correction = "all",
        truecoef = NULL, hi.res = NULL,
        nr=spatstat.options("psstA.nr"),
        ngrid=spatstat.options("psstA.ngrid"))

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.

ppmcorrection

Optional. Character string specifying the edge correction for the pseudolikelihood to be used in fitting the point process model. Passed to ppm.

correction

Optional. Character string specifying which diagnostic quantities will be computed. Options are "all" and "best". The default is to compute all diagnostic quantities.

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

Optional. Number of r values to be used if r is not specified.

ngrid

Integer. Number of points in the square grid used to compute the approximate area.

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.

Warning

This computation can take a very long time.

To shorten the computation time, choose smaller values of the arguments nr and ngrid, or reduce the values of their defaults spatstat.options("psstA.nr") and spatstat.options("psstA.ngrid").

Computation time is roughly proportional to nr * npoints * ngrid^2 where npoints is the number of points in the point pattern.

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. 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. The alternative hypothesis is a family of hybrid models obtained by combining the fitted model with the area-interaction process (see AreaInter). The family of alternatives includes models that are slightly more regular than the fitted model, and others that are slightly more clustered than the fitted model.

The pseudoscore, evaluated at the null model, is $$V(r) = \sum_i A(x_i, x, r) - \int_W A(u,x, r) \lambda(u,x) {\rm d} u$$ where $$A(u,x,r) = B(x\cup{u},r) - B(x\setminus u, r)$$ where $B(x,r)$ is the area of the union of the discs of radius $r$ centred at the points of $x$ (i.e. $B(x,r)$ is the area of the dilation of $x$ by a distance $r$). Thus $A(u,x,r)$ is the unclaimed area associated with $u$, that is, the area of that part of the disc of radius $r$ centred at the point $u$ that is not covered by any of the discs of radius $r$ centred at points of $x$.

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

pso <- spatstat.options(psstA.ngrid=16,psstA.nr=10)
   X <- rStrauss(200,0.1,0.05)
   plot(psstA(X))
   plot(psstA(X, interaction=Strauss(0.05)))
   spatstat.options(pso)

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