"cdf.test"(model, covariate, test=c("ks", "cvm", "ad"), ..., nsim=19, verbose=TRUE, interpolate=FALSE, fast=TRUE, jitter=TRUE)"mppm" representing a point process model
fitted to multiple spatial point patterns.
model, or one of the strings
"x" or "y".
"ks" for Kolmogorov-Smirnov test,
"cvm" for Cramer-von Mises test
or "ad" for Anderson-Darling test.
cdf.test
to control the test.
TRUE, values of the covariate
are only sampled at the original quadrature points used to
fit the model. If FALSE, values of the covariate
are sampled at all pixels, which can be slower by three orders of
magnitude.
TRUE, observed values of the covariate
are perturbed by adding small random values, to avoid
tied observations.
"cdftest" and "htest"
containing the results of the
test. See cdf.test for details.
cdf.test for the class mppm. This function performs a goodness-of-fit test of
a point process model that has been fitted to multiple point patterns.
The observed distribution
of the values of a spatial covariate at the data points,
and the predicted distribution of the same values under the model,
are compared using the Kolmogorov-Smirnov,
Cramer-von Mises
or Anderson-Darling test of goodness-of-fit.
These are exact tests if the model is Poisson;
otherwise, for a Gibbs model, a Monte Carlo p-value is computed by
generating simulated realisations of the model and applying the
selected goodness-of-fit test to each simulation.
The argument model should be a fitted point process model
fitted to multiple point patterns
(object of class "mppm").
The argument covariate contains the values of a spatial
function. It can be
function(x,y)
"im"
function(x,y), one for each point pattern
hyperframe)
of which the first
column will be taken as containing the covariate
model
"x" or "y",
indicating the spatial coordinates.
If covariate is an image, it should have numeric values,
and its domain should cover the observation window of the
model. If covariate is a function, it should expect
two arguments x and y which are vectors of coordinates,
and it should return a numeric vector of the same length
as x and y.
First the original data point pattern is extracted from model.
The values of the covariate at these data points are
collected.
The predicted distribution of the values of the covariate
under the fitted model is computed as follows.
The values of the covariate at all locations in the
observation window are evaluated,
weighted according to the point process intensity of the fitted model,
and compiled into a cumulative distribution function $F$ using
ewcdf.
The probability integral transformation is then applied:
the values of the covariate at the original data points
are transformed by the predicted cumulative distribution function
$F$ into numbers between 0 and 1. If the model is correct,
these numbers are i.i.d. uniform random numbers.
A goodness-of-fit test of the uniform distribution is applied
to these numbers using ks.test,
cvm.test or ad.test.
The argument interpolate determines
how pixel values will be handled when codecovariate is a pixel image.
The value of the covariate at a data point is obtained
by looking up the value of the nearest pixel if
interpolate=FALSE, or by linearly interpolating
between the values of the four nearest pixels
if interpolate=TRUE. Linear interpolation is slower,
but is sometimes necessary to avoid tied values of the covariate
arising when the pixel grid is coarse.
If model is a Poisson point process, then the
Kolmogorov-Smirnov,
Cramer-von Mises
and Anderson-Darling tests are theoretically exact.
This test was apparently first described (in the context of
spatial data, and for Kolmogorov-Smirnov) by Berman (1986).
See also Baddeley et al (2005).
If model is not a Poisson point process, then the
Kolmogorov-Smirnov,
Cramer-von Mises
and Anderson-Darling tests are biased.
Instead they are used as the basis of a Monte Carlo test.
First nsim simulated realisations of the model will be generated.
Each simulated realisation consists of a list of simulated point
patterns, one for each of the original data patterns. This
can take a very long time. The model is then re-fitted to each
simulation, and the refitted model is subjected to the goodness-of-fit
test described above. A Monte Carlo p-value is then computed by
comparing the p-value of the original test with the
p-values obtained from the simulations.
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.
Berman, M. (1986) Testing for spatial association between a point process and another stochastic process. Applied Statistics 35, 54--62.
cdf.test,
quadrat.test,
mppm
# three i.i.d. realisations of nonuniform Poisson process
lambda <- as.im(function(x,y) { 300 * exp(x) }, square(1))
dat <- hyperframe(X=list(rpoispp(lambda), rpoispp(lambda), rpoispp(lambda)))
# fit uniform Poisson process
fit0 <- mppm(X~1, dat)
# fit correct nonuniform Poisson process
fit1 <- mppm(X~x, dat)
# test wrong model
cdf.test(fit0, "x")
# test right model
cdf.test(fit1, "x")
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