R: Model Compensator of K Function

Kcom {spatstat.core} | R Documentation

## Model Compensator of K Function

### Description

Given a point process model fitted to a point pattern dataset, this function computes the compensator of the K function based on the fitted model (as well as the usual nonparametric estimates of K based on the data alone). Comparison between the nonparametric and model-compensated K functions serves as a diagnostic for the model.

### Usage

    Kcom(object, r = NULL, breaks = NULL, ...,
         correction = c("border", "isotropic", "translate"),
         conditional = !is.poisson(object),
         restrict = FALSE,
         model = NULL,
         trend = ~1, interaction = Poisson(), rbord = reach(interaction),
         compute.var = TRUE,
         truecoef = NULL, hi.res = NULL)

### 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 function K(r) should be computed. This argument is usually not specified. There is a sensible default.
`breaks` | This argument is for advanced use only.
`...` | Ignored.
`correction` | Optional vector of character strings specifying the edge correction(s) to be used. See `Kest` for options.
`conditional` | Optional. Logical value indicating whether to compute the estimates for the conditional case. See Details.
`restrict` | Logical value indicating whether to compute the restriction estimator (`restrict=TRUE`) or the reweighting estimator (`restrict=FALSE`, the default). Applies only if `conditional=TRUE`. See Details.
`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`.
`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.
`compute.var` | Logical value indicating whether to compute the Poincare variance bound for the residual K function (calculation is only implemented for the isotropic correction).
`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.

### Details

This command provides a diagnostic for the goodness-of-fit of a point process model fitted to a point pattern dataset. It computes an estimate of the K function of the dataset, together with a model compensator of the K function, which should be approximately equal if the model is a good fit to the data.

The first argument, `object`, is usually a fitted point process model (object of class `"ppm"`), obtained from the model-fitting function `ppm`.

For convenience, `object` can also be a point pattern (object of class `"ppp"`). In that case, a point process model will be fitted to it, by calling `ppm` using the arguments `trend` (for the first order trend), `interaction` (for the interpoint interaction) and `rbord` (for the erosion distance in the border correction for the pseudolikelihood). See `ppm` for details of these arguments.

The algorithm first extracts the original point pattern dataset (to which the model was fitted) and computes the standard nonparametric estimates of the K function. It then also computes the model compensator of the K function. The different function estimates are returned as columns in a data frame (of class `"fv"`).

The argument `correction` determines the edge correction(s) to be applied. See `Kest` for explanation of the principle of edge corrections. The following table gives the options for the `correction` argument, and the corresponding column names in the result:

`correction` | description of correction | nonparametric | compensator
`"isotropic"` | Ripley isotropic correction | `iso` | `icom`
`"translate"` | Ohser-Stoyan translation correction | `trans` | `tcom`
`"border"` | border correction | `border` | `bcom`

The nonparametric estimates can all be expressed in the form

K(r) = sum[i] sum[j < i] e(x[i], x[j], r, x) I( d(x[i],x[j]) <= r )

where x[i] is the i-th data point, d(x[i],x[j]) is the distance between x[i] and x[j], and e(x[i],x[j],r,x) is a term that serves to correct edge effects and to re-normalise the sum. The corresponding model compensator is

C K(r) = integral[u] lambda(u,x) sum[j] e(u, x[j], r, x+u) I( d(u,x[j]) <= r )

where the integral is over all locations u in the observation window, lambda(u,x) denotes the conditional intensity of the model at the location u, and x+u denotes the data point pattern x augmented by adding the extra point u.

If the fitted model is a Poisson point process, then the formulae above are exactly what is computed. If the fitted model is not Poisson, the formulae above are modified slightly to handle edge effects.

The modification is determined by the arguments `conditional` and `restrict`. The value of `conditional` defaults to `FALSE` for Poisson models and `TRUE` for non-Poisson models. If `conditional=FALSE` then the formulae above are not modified. If `conditional=TRUE`, then the algorithm calculates the restriction estimator if `restrict=TRUE`, and calculates the reweighting estimator if `restrict=FALSE`. See Appendix D of Baddeley, Rubak and Moller (2011). Thus, by default, the reweighting estimator is computed for non-Poisson models.

The nonparametric estimates of K(r) are approximately unbiased estimates of the K-function, assuming the point process is stationary. The model compensators are unbiased estimates of the mean values of the corresponding nonparametric estimates, assuming the model is true. Thus, if the model is a good fit, the mean value of the difference between the nonparametric estimates and model compensators is approximately zero.

### Value

A function value table (object of class `"fv"`), essentially a data frame of function values. There is a plot method for this class. See `fv.object`.

### 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

Related functions: `Kres`, `Kest`.

Alternative functions: `Gcom`, `psstG`, `psstA`, `psst`.

Point process models: `ppm`.

### Examples

        fit0 <- ppm(cells, ~1) # uniform Poisson

        if(interactive()) {
          plot(Kcom(fit0))
    # compare the isotropic-correction estimates
          plot(Kcom(fit0), cbind(iso, icom) ~ r)
    # uniform Poisson is clearly not correct
        }

        fit1 <- ppm(cells, ~1, Strauss(0.08))

        K1 <- Kcom(fit1)
        K1
        if(interactive()) {
          plot(K1)
          plot(K1, cbind(iso, icom) ~ r)
          plot(K1, cbind(trans, tcom) ~ r)
    # how to plot the difference between nonparametric estimates and compensators
          plot(K1, iso - icom ~ r)
    # fit looks approximately OK; try adjusting interaction distance
        }
        fit2 <- ppm(cells, ~1, Strauss(0.12))

        K2 <- Kcom(fit2)
        if(interactive()) {
          plot(K2)
          plot(K2, cbind(iso, icom) ~ r)
          plot(K2, iso - icom ~ r)
        }

[Package spatstat.core version 2.4-4 Index]