# qqplot.ppm

##### 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"`

**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"`

`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.

##### See Also

##### 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 1.6-9, License: GPL version 2 or newer*