# cdf.test

##### Spatial Distribution Test for Point Pattern or Point Process Model

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,

##### Usage

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

## S3 method for class 'ppm':
cdf.test(model, covariate, test=c("ks", "cvm", "ad"), ...,
jitter=TRUE, nsim=99, verbose=TRUE)

## S3 method for class 'lpp':
cdf.test(X, covariate, test=c("ks", "cvm", "ad"), ..., jitter=TRUE)

## S3 method for class 'lppm':
cdf.test(model, covariate, test=c("ks", "cvm", "ad"),
...,
jitter=TRUE, nsim=99, verbose=TRUE)

## S3 method for class 'slrm':
cdf.test(model, covariate, test=c("ks", "cvm", "ad"), ..., modelname=NULL, covname=NULL)

##### Arguments

- X
- A point pattern (object of class
`"ppp"`

or`"lpp"`

). - model
- A fitted point process model (object of class
`"ppm"`

or`"lppm"`

) or fitted spatial logistic regression (object of class`"slrm"`

). - 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"`

forlatex {Cram'er }{Cramer}-von Mises test or`"ad"`

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

(from thestats package) or`cvm.test`

or`ad.test`

- jitter
- Logical flag. If
`jitter=TRUE`

, values of the covariate will be slightly perturbed at random, to avoid tied values in the test. - modelname,covname
- Character strings giving alternative names for
`model`

and`covariate`

to be used in labelling plot axes. - nsim
- Number of simulated realisations from the
`model`

to be used for the Monte Carlo test, when`model`

is not a Poisson process. - verbose
- Logical value indicating whether to print progress reports when performing a Monte Carlo test.

##### Details

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

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.

`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.

##### Value

- 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.

##### Warning

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`

.

##### References

Baddeley, A., Turner, R.,
*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.
}

# 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)

# fit inhomogeneous Poisson model and test model <- ppm(nztrees ~x) cdf.test(model, "x")

if(interactive()) { # synthetic data: nonuniform Poisson process X <- rpoispp(function(x,y) { 100 * exp(x) }, win=square(1))

# fit uniform Poisson process fit0 <- ppm(X ~1) # fit correct nonuniform Poisson process fit1 <- ppm(X ~x)

# test wrong model cdf.test(fit0, "x") # test right model cdf.test(fit1, "x") }

# multitype point pattern cdf.test(amacrine, "x") yimage <- as.im(function(x,y){y}, W=Window(amacrine)) cdf.test(ppm(amacrine ~marks+y), yimage)

options(op)

##### See Also

`plot.cdftest`

,
`quadrat.test`

,
`berman.test`

,
`ks.test`

,
`cvm.test`

,
`ad.test`

,
`ppm`

*Documentation reproduced from package spatstat, version 1.41-1, License: GPL (>= 2)*