psst
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.
Usage
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)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
rfor advanced use. - ...
- Ignored.
- model
- Optional. A fitted point process model (object of
class
"ppm") to be re-fitted to the data usingupdate.ppm, ifobjectis a point pattern. Overrides the arguments - trend,interaction,rbord
- Optional. Arguments passed to
ppmto fit a point process model to the data, ifobjectis a point pattern. Seeppmfor 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
- List of additional arguments to be passed to
fun. - verbose
- Logical value determining whether to print progress reports during the computation.
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
datfor the pseudosum,comfor the compensator andresfor the pseudoresidual. There is a plot method for this class. Seefv.object.
References
Baddeley, A., Rubak, E. and