Tests the goodness-of-fit of a Poisson point process model using methods of Berman (1986).
berman.test(...)# S3 method for ppp
berman.test(X, covariate,
which = c("Z1", "Z2"),
alternative = c("two.sided", "less", "greater"), ...)
# S3 method for ppm
berman.test(model, covariate,
which = c("Z1", "Z2"),
alternative = c("two.sided", "less", "greater"), ...)
A point pattern (object of class "ppp" or "lpp").
A fitted point process model (object of class "ppm" or
"lppm").
The spatial covariate on which the test will be based.
An image (object of class "im")
or a function.
Character string specifying the choice of test.
Character string specifying the alternative hypothesis.
Additional arguments controlling the pixel resolution
(arguments dimyx and eps passed to
as.mask)
or other undocumented features.
An object of class "htest" (hypothesis test)
and also of class "bermantest",
containing the results of the test. The return value can be
plotted (by plot.bermantest) or printed
to give an informative summary of the test.
The meaning of a one-sided test must be carefully scrutinised: see the printed output.
These functions perform a goodness-of-fit test of a Poisson point process model fitted to point pattern data. 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 either of two test statistics \(Z_1\) and \(Z_2\) proposed by Berman (1986). The \(Z_1\) test is also known as the Lawson-Waller test.
The function berman.test is generic, with methods for
point patterns ("ppp" or "lpp")
and point process models ("ppm" or "lppm").
If X is a point pattern dataset (object of class
"ppp" or "lpp"), then
berman.test(X, ...) performs a goodness-of-fit test of the
uniform Poisson point process (Complete Spatial Randomness, CSR)
for this dataset.
If model is a fitted point process model
(object of class "ppm" or "lppm")
then berman.test(model, ...) performs
a test of goodness-of-fit for this fitted model. In this case,
model should be a Poisson point process.
The test is performed by comparing the observed distribution of the values of a spatial covariate at the data points, and the predicted distribution of the same covariate under the model. Thus, you must nominate a spatial covariate for this test.
The argument covariate should be either a function(x,y)
or a pixel image (object of class "im" containing the values
of a spatial function.
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.
Next the values of the covariate at all locations in the
observation window are evaluated. The point process intensity
of the fitted model is also evaluated at all locations in the window.
If which="Z1",
the test statistic \(Z_1\) is computed as follows.
The sum \(S\) of the covariate values at all data
points is evaluated. The predicted mean \(\mu\) and variance
\(\sigma^2\) of \(S\) are computed
from the values of the covariate at all locations in the window.
Then we compute \(Z_1 = (S-\mu)/\sigma\).
Closely-related tests were proposed independently
by Waller et al (1993) and Lawson (1993)
so this test is often termed the
Lawson-Waller test in epidemiological literature.
If which="Z2",
the test statistic \(Z_2\) is computed as follows.
The values of the covariate at all locations in the
observation window, weighted by the point process intensity,
are compiled into a cumulative distribution function \(F\).
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.
The standardised sample mean of these numbers is the
statistic \(Z_2\).
In both cases the null distribution of the test statistic is the standard normal distribution, approximately.
The return value is an object of class "htest" containing the
results of the hypothesis test. The print method for this class
gives an informative summary of the test outcome.
Berman, M. (1986) Testing for spatial association between a point process and another stochastic process. Applied Statistics 35, 54--62.
Lawson, A.B. (1993) On the analysis of mortality events around a prespecified fixed point. Journal of the Royal Statistical Society, Series A 156 (3) 363--377.
Waller, L., Turnbull, B., Clark, L.C. and Nasca, P. (1992) Chronic Disease Surveillance and testing of clustering of disease and exposure: Application to leukaemia incidence and TCE-contaminated dumpsites in upstate New York. Environmetrics 3, 281--300.
# NOT RUN {
# Berman's data
data(copper)
X <- copper$SouthPoints
L <- copper$SouthLines
D <- distmap(L, eps=1)
# test of CSR
berman.test(X, D)
berman.test(X, D, "Z2")
# }
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