Performs a test of goodness-of-fit of a point process model. The observed and predicted distributions of the values of a spatial covariate are compared using either the Kolmogorov-Smirnov test, Cramer-von Mises test or Anderson-Darling test. For non-Poisson models, a Monte Carlo test is used.

`cdf.test(...)`# S3 method for ppp
cdf.test(X, covariate, test=c("ks", "cvm", "ad"), ...,
interpolate=TRUE, jitter=TRUE)

An object of class `"htest"`

containing the results of the
test. See `ks.test`

for details. The return value can be
printed to give an informative summary of the test.

The value also belongs to the class `"cdftest"`

for which there is
a plot method.

- X
A point pattern (object of class

`"ppp"`

or`"lpp"`

).- covariate
The spatial covariate on which the test will be based. A function, a pixel image (object of class

`"im"`

), a list of pixel images, or one of the characters`"x"`

or`"y"`

indicating the Cartesian coordinates.- test
Character string identifying the test to be performed:

`"ks"`

for Kolmogorov-Smirnov test,`"cvm"`

for Cramer-von Mises test or`"ad"`

for Anderson-Darling test.- ...
Arguments passed to

`ks.test`

(from the stats package) or`cvm.test`

or`ad.test`

(from the goftest package) to control the test.- interpolate
Logical flag indicating whether to interpolate pixel images. If

`interpolate=TRUE`

, the value of the covariate at each point of`X`

will be approximated by interpolating the nearby pixel values. If`interpolate=FALSE`

, the nearest pixel value will be used.- jitter
Logical flag. If

`jitter=TRUE`

, values of the covariate will be slightly perturbed at random, to avoid tied values in the test.

The outcome of the test involves a small amount of random variability,
because (by default) the coordinates are randomly perturbed to
avoid tied values. Hence, if `cdf.test`

is executed twice, the
\(p\)-values will not be exactly the same. To avoid this behaviour,
set `jitter=FALSE`

.

Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner r.turner@auckland.ac.nz

These functions perform a goodness-of-fit test of a Poisson or Gibbs 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 the Kolmogorov-Smirnov test, the Cramer-von Mises test or the Anderson-Darling test. For Gibbs models, a Monte Carlo test is performed using these test statistics.

The function `cdf.test`

is generic, with methods for
point patterns (`"ppp"`

or `"lpp"`

),
point process models (`"ppm"`

or `"lppm"`

)
and spatial logistic regression models (`"slrm"`

).

If

`X`

is a point pattern dataset (object of class`"ppp"`

), then`cdf.test(X, ...)`

performs a goodness-of-fit test of the uniform Poisson point process (Complete Spatial Randomness, CSR) for this dataset. For a multitype point pattern, the uniform intensity is assumed to depend on the type of point (sometimes called Complete Spatial Randomness and Independence, CSRI).If

`model`

is a fitted point process model (object of class`"ppm"`

or`"lppm"`

) then`cdf.test(model, ...)`

performs a test of goodness-of-fit for this fitted model.If

`model`

is a fitted spatial logistic regression (object of class`"slrm"`

) then`cdf.test(model, ...)`

performs a test of goodness-of-fit for this fitted model.

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, using a classical goodness-of-fit test. Thus, you must nominate a spatial covariate for this test.

If `X`

is a point pattern that does not have marks,
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, or one of the characters `"x"`

or
`"y"`

indicating the Cartesian 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`

.

If `X`

is a multitype point pattern, the argument `covariate`

can be either a `function(x,y,marks)`

,
or a pixel image, or a list of pixel images corresponding to
each possible mark value, or one of the characters `"x"`

or
`"y"`

indicating the Cartesian coordinates.

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. The
A goodness-of-fit test of the uniform distribution is applied
to these numbers using `stats::ks.test`

,
`goftest::cvm.test`

or
`goftest::ad.test`

.

This test was apparently first described (in the context of spatial data, and using Kolmogorov-Smirnov) by Berman (1986). See also Baddeley et al (2005).

If `model`

is not a Poisson process, then
a Monte Carlo test is performed, by generating `nsim`

point patterns which are simulated realisations of the `model`

,
re-fitting the model to each simulated point pattern,
and calculating the test statistic for each fitted model.
The Monte Carlo \(p\) value is determined by comparing
the simulated values of the test statistic
with the value for the original data.

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.

The return value also belongs to the class `"cdftest"`

for which there is a plot method `plot.cdftest`

.
The plot method displays the empirical cumulative distribution
function of the covariate at the data points, and the predicted
cumulative distribution function of the covariate under the model,
plotted against the value of the covariate.

The argument `jitter`

controls whether covariate values are
randomly perturbed, in order to avoid ties.
If the original data contains any ties in the covariate (i.e. points
with equal values of the covariate), and if `jitter=FALSE`

, then
the Kolmogorov-Smirnov test implemented in `ks.test`

will issue a warning that it cannot calculate the exact \(p\)-value.
To avoid this, if `jitter=TRUE`

each value of the covariate will
be perturbed by adding a small random value. The perturbations are
normally distributed with standard deviation equal to one hundredth of
the range of values of the covariate. This prevents ties,
and the \(p\)-value is still correct. There is
a very slight loss of power.

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.

`plot.cdftest`

,
`quadrat.test`

,
`berman.test`

,
`ks.test`

,
`cvm.test`

,
`ad.test`

,
`ppm`

```
op <- options(useFancyQuotes=FALSE)
# test of CSR using x coordinate
cdf.test(nztrees, "x")
cdf.test(nztrees, "x", "cvm")
cdf.test(nztrees, "x", "ad")
# test of CSR using a function of x and y
fun <- function(x,y){2* x + y}
cdf.test(nztrees, fun)
# test of CSR using an image covariate
funimage <- as.im(fun, W=Window(nztrees))
cdf.test(nztrees, funimage)
# multitype point pattern
cdf.test(amacrine, "x")
options(op)
```

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