# cdf.test.mppm

##### Spatial Distribution Test for Multiple Point Process Model

Performs a spatial distribution test
of a Poisson point process model fitted to multiple spatial point
patterns. The test compares the observed
and predicted distributions of the values of a spatial covariate,
using either the Kolmogorov-Smirnov,

##### Usage

```
## S3 method for class 'mppm':
cdf.test(model, covariate, test=c("ks", "cvm", "ad"), ...,
verbose=TRUE, interpolate=FALSE, fast=TRUE, jitter=TRUE)
```

##### Arguments

- model
- An object of class
`"mppm"`

representing a point process model fitted to multiple spatial point patterns. - covariate
- The spatial covariate on which the test will be based.
A function, a pixel image, a list of functions, a list of pixel
images, a hyperframe, or a character string containing the name
of one of the covariates in
`model`

. - 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
`cdf.test`

to control the test. - verbose
- Logical flag indicating whether to print progress reports.
- interpolate
- Logical flag indicating whether to interpolate between
pixel values when code{covariate} is a pixel image.
See
*Details*. - fast
- Logical flag. If
`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 - jitter
- Logical flag. If
`TRUE`

, observed values of the covariate are perturbed by adding small random values, to avoid tied observations.

##### Details

This function is a method for the generic function
`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 test.
The argument `model`

should be a fitted point process model
fitted to multiple point patterns
(object of class `"mppm"`

). It should be a Poisson point process.
The argument `covariate`

contains the values of a spatial
function. It can be

- a
`function(x,y)`

- a pixel image (object of class
`"im"`

- a list of
`function(x,y)`

, one for each point pattern - a list of pixel images, one for each point pattern
- a hyperframe (see
`hyperframe`

) of which the first column will be taken as containing the covariate - a character string giving the name of one of the covariates
in
`model`

.

`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`

.

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).

The argument `interpolate`

determines
how pixel values will be handled when code{covariate} 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.

##### Value

- An object of class
`"cdftest"`

and`"htest"`

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

for details.

##### 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*
**67**, 617--666.

Berman, M. (1986)
Testing for spatial association between a point process
and another stochastic process.
*Applied Statistics* **35**, 54--62.

##### See Also

##### Examples

```
# 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 covariate = x coordinate
xcoord <- function(x,y) { x }
# test wrong model
cdf.test(fit0, xcoord)
# test right model
cdf.test(fit1, xcoord)
```

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