Given a point process model fitted to a point pattern on a linear network, produce a Q-Q plot based on residuals from the model.
qqplot.lppm(fit, nsim=100, expr=NULL, ..., type="raw",
style="mean", fast=TRUE, verbose=TRUE, plot.it=TRUE,
probs=NULL,
saveall=FALSE,
monochrome=FALSE,
limcol=if(monochrome) "black" else "red",
maxerr=max(100, ceiling(nsim/10)),
envir.expr)An object of class "qqlppm" 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.
The fitted point process model on a network, which is to be assessed
using the Q-Q plot. An object of class "lppm".
Smoothed residuals obtained from this fitted model will provide the
“data” quantiles for the Q-Q plot.
The number of simulations from the “reference” point process model.
Determines the simulation mechanism which provides the “theoretical” quantiles for the Q-Q plot. See Details.
Arguments passed to diagnose.lppm influencing the
computation of residuals.
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-lambda residuals,
and "pearson" for the Pearson residuals.
A partial match is adequate.
Character string controlling the type of Q-Q plot.
Options are "classical" and "mean".
See Details.
Logical flag controlling the speed and accuracy of computation.
Use fast=TRUE for interactive use and fast=FALSE
for publication standard plots. See Details.
Logical value controlling whether the algorithm prints progress reports during long computations.
Logical value controlling whether the function produces a plot or simply returns a value (silently).
Probabilities for which the quantiles should be calculated and plotted. A numeric vector of values between 0 and 1.
Logical flag indicating whether to save all the intermediate calculations.
Logical flag indicating whether the plot should be
in black and white (monochrome=TRUE), or in colour
(monochrome=FALSE).
String. The colour to be used when plotting the 95% limit curves.
Maximum number of failures tolerated while generating simulated realisations. See Details.
Optional. An environment in which the expression
expr should be evaluated.
Produces a Q-Q plot if plot.it=TRUE (the default).
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.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner rolfturner@posteo.net and Ege Rubak rubak@math.aau.dk.
This function generates a Q-Q plot of the residuals from a
fitted point process model on a linear network.
It is an addendum to the suite of
diagnostic plots produced by the function diagnose.lppm,
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.lppm.
The argument fit represents the fitted point process
model. It must be an object of class "lppm" (typically produced
by the maximum pseudolikelihood fitting algorithm lppm).
Residuals will be computed for this fitted model using
residuals.lppm,
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 using
simulate.lppm,
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 on a network (object of class
"lpp") or a fitted point process model on a network (object of class
"lppm"). If the value is a point pattern, then the
original fitted model fit will be re-fitted to this new point
pattern using update.lppm, to yield another fitted
model. Smoothed residuals obtained from these nsim fitted
models will yield the “theoretical” quantiles for the Q-Q plot.
Alternatively expr can be a list of point patterns,
or an envelope object that contains a list of point patterns
(typically generated by calling envelope.lpp
or envelope.lppm with savepatterns=TRUE).
These point patterns will be used as the simulated patterns.
The argument type selects the type of residual or weight
that will be computed. For options, see diagnose.lppm.
The argument style determines the type of Q-Q plot. It is
highly recommended to use the default, style="mean".
style="classical"The quantiles of the residual field for the data (on the \(y\) axis) are plotted against the quantiles of the pooled simulations (on the \(x\) axis). This plot is biased, and therefore difficult to interpret, because of strong autocorrelations in the residual field and the large differences in sample size.
style="mean"Quantiles of the residual field for the original data are plotted
against the sample means, over the nsim simulations,
of the corresponding quantiles of the residual field
for the simulated datasets.
Dotted lines show the 2.5 and 97.5 percentiles, over the
nsim simulations, of each of these quantiles.
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
(up to 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.
Since the computation is so time-consuming, qqplot.lppm returns
a list containing all the data necessary to re-display the Q-Q plot.
It is advisable to assign the result of qqplot.lppm to something
(or use .Last.value if you forgot to.)
The return value is an object of class "qqlppm". There are methods for
plot and print. 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 argument expr may 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).
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.
diagnose.lppm,
lurking.lppm,
residuals.lppm,
lppm
if(online <- interactive()) {
X <- unmark(chicago)
fit <- lppm(X ~ x + y)
ns <- 100
} else {
X <- runiflpp(20, simplenet)
fit <- lppm(X ~ y)
ns <- 4
}
qqplot.lppm(fit, nsim=ns)
qqplot.lppm(fit, nsim=ns, type="pearson")
# \donttest{
# capture the plot coordinates
mypreciousdata <- qqplot.lppm(fit, ns, type="pearson")
## or use the idiom .Last.value if you forgot to assign them
qqplot.lppm(fit, ns, type="pearson")
mypreciousdata <- .Last.value
plot(mypreciousdata)
# }
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