R: Pseudoscore Diagnostic For Fitted Model against...

psstA {spatstat.core} | R Documentation

## 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, ...,
          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"))

### 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"`).
`r` | 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` | This argument is for internal use only.
`...` | Extra arguments passed to `quadscheme` to determine the quadrature scheme, if `object` is a point pattern.
`model` | 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`.
`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 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.
`nr` | 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.

### 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],...,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( A(x[i], x, r)) - integral( A(u,x,r) lambda(u,x) du)

where

A(u,x,r) = B(x union 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.

### 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.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Ege Rubak rubak@math.aau.dk and Jesper Moller.

### 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.

### See Also

Alternative functions: `psstG`, `psst`, `Gres`, `Kres`.

Point process models: `ppm`.

Options: `spatstat.options`

### Examples

       if(live <- interactive()) {
         X <- rStrauss(200,0.1,0.05)
       } else {
         pso <- spatstat.options(psstA.ngrid=16,psstA.nr=10,
                ndummy.min=16,npixel=32)
         X <- cells
       }

       plot(psstA(X))
       plot(psstA(X, interaction=Strauss(0.05)))

       if(!live) spatstat.options(pso)

[Package spatstat.core version 2.3-0 Index]