# 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
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=TRUE),
saveall=FALSE)
##### 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.
##### 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. As a shortcut, the argument nrep determines the number of Metropolis-Hastings iterations used to generate each simulated realisation, if control is absent. 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". 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"{ The order statistics of the residual field for the data are plotted against the sample means, over the nsim simulations, of the corresponding order statistics of the residual field for the simulated datasets. Dotted lines show the 2.5 and 97.5 percentiles, over the nsim simulations, of each order statistic. }

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.

##### 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.

##### 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. to appear.

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

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

• 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, 5)</testonly>

diagnose.ppm(fit, type="pearson")
qqplot.ppm(fit, type="pearson")
<testonly>qqplot.ppm(fit, 5, type="pearson")</testonly>

###########################################
## oops, I need the plot coordinates
mypreciousdata <- .Last.value
mypreciousdata <- qqplot.ppm(fit, type="pearson")
<testonly>mypreciousdata <- qqplot.ppm(fit, 5, 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, 5, 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, 10)</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, 5, expr)</testonly>
Documentation reproduced from package spatstat, version 1.6-11, License: GPL version 2 or newer

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