# ks.test.ppm

##### Kolmogorov-Smirnov Test for Point Process Model

Performs a Kolmogorov-Smirnov test of goodness-of-fit of a Poisson point process model. The test compares the observed and predicted distributions of the values of a spatial covariate.

##### Usage

`ks.test.ppm(model, covariate, ...)`

##### Arguments

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

). - covariate
- The spatial covariate on which the test will be based.
An image (object of class
`"im"`

) or a function. - ...
- Arguments passed to
`smooth.spline`

to control the amount of smoothing.

##### Details

This function performs a goodness-of-fit test of
a fitted point process model. 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
(object of class `"ppm"`

). It should be a Poisson point process.

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.

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
Kolmogorov-Smirnov test of uniformity is applied using
`ks.test`

.

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

##### Value

- An object of class
`"htest"`

containing the results of the test. See`ks.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

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

*Documentation reproduced from package spatstat, version 1.11-2, License: GPL version 2 or newer*