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.
psst(object, fun, r = NULL, breaks = NULL, ...,
model=NULL,
trend = ~1, interaction = Poisson(), rbord = reach(interaction),
truecoef=NULL, hi.res=NULL, funargs = list(correction="best"),
verbose=TRUE)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").
Summary function to be applied to each point pattern.
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.
Optional alternative to r for advanced use.
Ignored.
Optional. A fitted point process model (object of
class "ppm") to be re-fitted to the data
using update.ppm, if object is a point pattern.
Overrides the arguments trend,interaction,rbord.
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.
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 resulting fitted model will be taken as the true coefficients.
Then the diagnostic will be computed for the default
quadrature scheme, but using the high resolution coefficients.
List of additional arguments to be passed to fun.
Logical value determining whether to print progress reports during the computation.
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.
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.
Baddeley, A., Rubak, E. and Moller, J. (2011) Score, pseudo-score and residual diagnostics for spatial point process models. Statistical Science 26, 613--646.
# NOT RUN {
data(cells)
fit0 <- ppm(cells, ~1) # uniform Poisson
# }
# NOT RUN {
G0 <- psst(fit0, Gest)
G0
if(interactive()) plot(G0)
# }
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