qqplot.ppm

0th

Percentile

Q-Q Plot of Residuals from Fitted Point Process Model

Given a point process model fitted to a point pattern, produce a Q-Q plot based on residuals from the model.

Keywords
models, hplot, spatial
Usage
qqplot.ppm(fit, nsim=100, expr=NULL, ..., type="raw",
             style="mean", fast=TRUE, verbose=TRUE, plot.it=TRUE,
             dimyx=NULL, nrep=if(fast) 5e4 else 1e5,
             control=list(nrep=nrep,expand=default.expand(fit),
                          periodic=(as.owin(fit)$type=="rectangle")),
             saveall=FALSE,
             monochrome=FALSE,
             limcol=if(monochrome) "black" else "red",
             maxerr=max(100, ceiling(nsim/10)),
             check=TRUE, repair=TRUE)
Arguments
fit
The fitted point process model, which is to be assessed using the Q-Q plot. An object of class "ppm". Smoothed residuals obtained from this fitted model will provide the ``data'' quantiles for the Q-Q plot.
nsim
The number of simulations from the ``reference'' point process model.
expr
Determines the simulation mechanism which provides the ``theoretical'' quantiles for the Q-Q plot. See Details.
...
Arguments passed to diagnose.ppm influencing the computation of residuals.
type
String indicating the type of residuals or weights to be used. Current options are "eem" for the Stoyan-Grabarnik exponential energy weights, "raw" for the raw residuals, "inverse" for the inverse-lam
style
Character string controlling the type of Q-Q plot. Options are "classical" and "mean". See Details.
fast
Logical flag controlling the speed and accuracy of computation. Use fast=TRUE for interactive use and fast=FALSE for publication standard plots. See Details.
verbose
Logical flag controlling whether the algorithm prints progress reports during long computations.
plot.it
Logical flag controlling whether the function produces a plot or simply returns a value (silently).
dimyx
Dimensions of the pixel grid on which the smoothed residual field will be calculated. A vector of two integers.
nrep
If control is absent, then nrep gives the number of iterations of the Metropolis-Hastings algorithm that should be used to generate one simulation of the fitted point process.
control
List of parameters controlling the Metropolis-Hastings algorithm rmh which generates each simulated realisation from the model (unless the model is Poisson). This list becomes the argument con
saveall
Logical flag indicating whether to save all the intermediate calculations.
monochrome
Logical flag indicating whether the plot should be in black and white (monochrome=TRUE), or in colour (monochrome=FALSE).
limcol
String. The colour to be used when plotting the 95-percent limit curves.
maxerr
Maximum number of failures tolerated while generating simulated realisations. See Details.
check
Logical value indicating whether to check the internal format of fit. If there is any possibility that this object has been restored from a dump file, or has otherwise lost track of the environment where it was originally computed
repair
Logical value indicating whether to repair the internal format of fit, if it is found to be damaged.
Details

This function generates a Q-Q plot of the residuals from a fitted point process model. It is an addendum to the suite of diagnostic plots produced by the function diagnose.ppm, kept separate because it is computationally intensive. The quantiles of the theoretical distribution are estimated by simulation.

In classical statistics, a Q-Q plot of residuals is a useful diagnostic for checking the distributional assumptions. Analogously, in spatial statistics, a Q-Q plot of the (smoothed) residuals from a fitted point process model is a useful way to check the interpoint interaction part of the model (Baddeley et al, 2005). The systematic part of the model (spatial trend, covariate effects, etc) is assessed using other plots made by diagnose.ppm.

The argument fit represents the fitted point process model. It must be an object of class "ppm" (typically produced by the maximum pseudolikelihood fitting algorithm ppm). Residuals will be computed for this fitted model using residuals.ppm, and the residuals will be kernel-smoothed to produce a ``residual field''. The values of this residual field will provide the ``data'' quantiles for the Q-Q plot.

The argument expr is not usually specified. It provides a way to modify the ``theoretical'' or ``reference'' quantiles for the Q-Q plot.

In normal usage we set expr=NULL. The default is to generate nsim simulated realisations of the fitted model fit, re-fit this model to each of the simulated patterns, evaluate the residuals from these fitted models, and use the kernel-smoothed residual field from these fitted models as a sample from the reference distribution for the Q-Q plot.

In advanced use, expr may be an expression. It will be re-evaluated nsim times, and should include random computations so that the results are not identical each time. The result of evaluating expr should be either a point pattern (object of class "ppp") or a fitted point process model (object of class "ppm"). If the value is a point pattern, then the original fitted model fit will be fitted to this new point pattern using update.ppm, to yield another fitted model. Smoothed residuals obtained from these nsim fitted models will yield the ``theoretical'' quantiles for the Q-Q plot.

Simulation is performed (if expr=NULL) using the Metropolis-Hastings algorithm rmh. Each simulated realisation is the result of running the Metropolis-Hastings algorithm from an independent random starting state each time. The iterative and termination behaviour of the Metropolis-Hastings algorithm are governed by the argument control. See rmhcontrol for information about this argument. As a shortcut, the argument nrep determines the number of Metropolis-Hastings iterations used to generate each simulated realisation, if control is absent.

By default, simulations are generated in an expanded window. Use the argument control to change this, as explained in the section on Warning messages. The argument type selects the type of residual or weight that will be computed. For options, see diagnose.ppm.

The argument style determines the type of Q-Q plot. It is highly recommended to use the default, style="mean". [object Object],[object Object]

The argument fast is a simple way to control the accuracy and speed of computation. If fast=FALSE, the residual field is computed on a fine grid of pixels (by default 100 by 100 pixels, see below) and the Q-Q plot is based on the complete set of order statistics (usually 10,000 quantiles). If fast=TRUE, the residual field is computed on a coarse grid (at most 40 by 40 pixels) and the Q-Q plot is based on the percentiles only. This is about 7 times faster. It is recommended to use fast=TRUE for interactive data analysis and fast=FALSE for definitive plots for publication.

The argument dimyx gives full control over the resolution of the pixel grid used to calculate the smoothed residuals. Its interpretation is the same as the argument dimyx to the function as.mask. Note that dimyx[1] is the number of pixels in the $y$ direction, and dimyx[2] is the number in the $x$ direction. If dimyx is not present, then the default pixel grid dimensions are controlled by spatstat.options("npixel").

Since the computation is so time-consuming, qqplot.ppm returns a list containing all the data necessary to re-display the Q-Q plot. It is advisable to assign the result of qqplot.ppm to something (or use .Last.value if you forgot to.) The return value is an object of class "qqppm". There are methods for plot.qqppm and print.qqppm. See the Examples.

The argument saveall is usually set to FALSE. If saveall=TRUE, then the intermediate results of calculation for each simulated realisation are saved and returned. The return value includes a 3-dimensional array sim containing the smoothed residual field images for each of the nsim realisations. When saveall=TRUE, the return value is an object of very large size, and should not be saved on disk.

Errors may occur during the simulation process, because random data are generated. For example:

  • one of the simulated patterns may be empty.
  • one of the simulated patterns may cause an error in the code that fits the point process model.
  • the user-supplied argumentexprmay have a bug.
Empty point patterns do not cause a problem for the code, but they are reported. Other problems that would lead to a crash are trapped; the offending simulated data are discarded, and the simulation is retried. The argument maxerr determines the maximum number of times that such errors will be tolerated (mainly as a safeguard against an infinite loop).

Value

  • An object of class "qqppm" containing the information needed to reproduce the Q-Q plot. Entries x and y are numeric vectors containing quantiles of the simulations and of the data, respectively.

Side Effects

Produces a Q-Q plot if plot.it is TRUE.

Warning messages

A warning message will be issued if any of the simulations trapped an error (a potential crash). A warning message will be issued if all, or many, of the simulated point patterns are empty. This usually indicates a problem with the simulation procedure. The default behaviour of qqplot.ppm is to simulate patterns on an expanded window (specified through the argument control) in order to avoid edge effects. The model's trend is extrapolated over this expanded window. If the trend is strongly inhomogeneous, the extrapolated trend may have very large (or even infinite) values. This can cause the simulation algorithm to produce empty patterns.

The only way to suppress this problem entirely is to prohibit the expansion of the window, by setting the control argument to something like control=list(nrep=1e6, expand=1). Here expand=1 means there will be no expansion. See rmhcontrol for more information about the argument control.

References

Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617--666.

Stoyan, D. and Grabarnik, P. (1991) Second-order characteristics for stochastic structures connected with Gibbs point processes. Mathematische Nachrichten, 151:95--100.

See Also

diagnose.ppm, lurking, residuals.ppm, eem, ppm.object, ppm, rmh, rmhcontrol

Aliases
  • qqplot.ppm
Examples
data(cells)

    fit <- ppm(cells, ~1, Poisson())
    diagnose.ppm(fit)  # no suggestion of departure from stationarity
    qqplot.ppm(fit, 80)  # strong evidence of non-Poisson interaction
    <testonly>qqplot.ppm(fit, 4)</testonly>

    diagnose.ppm(fit, type="pearson")  
     qqplot.ppm(fit, type="pearson")
    <testonly>qqplot.ppm(fit, 4, type="pearson")</testonly>

    ###########################################
    ## oops, I need the plot coordinates
    mypreciousdata <- .Last.value
    mypreciousdata <- qqplot.ppm(fit, type="pearson")
    <testonly>mypreciousdata <- qqplot.ppm(fit, 4, type="pearson")</testonly>
    plot(mypreciousdata)

    ######################################################
    # Q-Q plots based on fixed n
    # The above QQ plots used simulations from the (fitted) Poisson process.
    # But I want to simulate conditional on n, instead of Poisson
    # Do this by setting rmhcontrol(p=1)
    fixit <- list(p=1)
    qqplot.ppm(fit, 100, control=fixit)
    <testonly>qqplot.ppm(fit, 4, control=fixit)</testonly>

    ######################################################
    # Inhomogeneous Poisson data
    X <- rpoispp(function(x,y){1000 * exp(-3*x)}, 1000)
    plot(X)
    # Inhomogeneous Poisson model
    fit <- ppm(X, ~x, Poisson())
    qqplot.ppm(fit, 100)
    <testonly>qqplot.ppm(fit, 4)</testonly>
    # conclusion: fitted inhomogeneous Poisson model looks OK

    ######################################################
    # Advanced use of 'expr' argument
    # 
    # set the initial conditions in Metropolis-Hastings algorithm
    # 
    expr <- expression(rmh(fit, start=list(n.start=42), verbose=FALSE))
    qqplot.ppm(fit, 100, expr)
    <testonly>qqplot.ppm(fit, 4, expr)</testonly>
Documentation reproduced from package spatstat, version 1.18-4, License: GPL (>= 2)

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