# kstest.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

```
kstest(...)
## S3 method for class 'ppp':
kstest(X, covariate, ..., jitter=TRUE)
## S3 method for class 'ppm':
kstest(model, covariate, ..., jitter=TRUE)
```

##### Arguments

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

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

). - 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. - ...
- Arguments passed to
`ks.test`

to control the test. - jitter
- Logical flag. If
`jitter=TRUE`

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

##### Details

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 the Kolmogorov-Smirnov test.

The function `kstest`

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

) and point process models (`"ppm"`

).

- If
`X`

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

), then`kstest(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"`

) then`kstest(model, ...)`

performs a test of goodness-of-fit for this fitted model. In this case,`model`

should be a Poisson point process.

`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
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).

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 `"kstest"`

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

.
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

`"kstest"`

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 `kstest`

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

.

##### 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

```
# real data: NZ trees
data(nztrees)
# test of CSR using x coordinate
kstest(nztrees, "x")
# test of CSR using a function of x and y
fun <- function(x,y){2* x + y}
kstest(nztrees, fun)
# test of CSR using an image covariate
funimage <- as.im(fun, W=as.owin(nztrees))
kstest(nztrees, funimage)
# fit inhomogeneous Poisson model and test
model <- ppm(nztrees, ~x)
kstest(model, "x")
# 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
kstest(fit0, "x")
# test right model
kstest(fit1, "x")
# multitype point pattern
data(amacrine)
kstest(amacrine, "x")
yimage <- as.im(function(x,y){y}, W=as.owin(amacrine))
kstest(ppm(amacrine, ~marks+y), yimage)
```

*Documentation reproduced from package spatstat, version 1.20-2, License: GPL (>= 2)*