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.
psstA(object, r = NULL, breaks = NULL, …,
model = 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"))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.
This argument is for internal use only.
Extra arguments passed to quadscheme to determine
the quadrature scheme, if object is a point pattern.
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,ppmcorrection.
Optional. Character string specifying the edge correction
for the pseudolikelihood to be used
in fitting the point process model. Passed to ppm.
Optional. Character string specifying which diagnostic quantities
will be computed. Options are "all" and "best".
The default is to compute all diagnostic quantities.
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.
Optional. Number of r values to be used
if r is not specified.
Integer. Number of points in the square grid used to compute the approximate area.
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.
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.
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.
Baddeley, A., Rubak, E. and Moller, J. (2011) Score, pseudo-score and residual diagnostics for spatial point process models. Statistical Science 26, 613--646.
Alternative functions:
psstG,
psst,
Gres,
Kres.
Point process models: ppm.
Options: spatstat.options
# NOT RUN {
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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