# Kmeasure

##### Reduced Second Moment Measure

Estimates the reduced second moment measure $\kappa$ from a point pattern in a window of arbitrary shape.

- Keywords
- spatial, nonparametric

##### Usage

`Kmeasure(X, sigma, edge=TRUE, ..., varcov=NULL)`

##### Arguments

- X
- The observed point pattern,
from which an estimate of $\kappa$ will be computed.
An object of class
`"ppp"`

, or data in any format acceptable to`as.ppp()`

. - sigma
- Standard deviation $\sigma$ of the Gaussian
smoothing kernel. Incompatible with
`varcov`

. - edge
- logical value indicating whether an edge correction should be applied.
- ...
- Ignored.
- varcov
- Variance-covariance matrix of the Gaussian smoothing kernel.
Incompatible with
`sigma`

.

##### Details

The reduced second moment measure $\kappa$ of a stationary point process $X$ is defined so that, for a `typical' point $x$ of the process, the expected number of other points $y$ of the process such that the vector $y - x$ lies in a region $A$, equals $\lambda \kappa(A)$. Here $\lambda$ is the intensity of the process, i.e. the expected number of points of $X$ per unit area.

The more familiar K-function $K(t)$ is just the value of the reduced second moment measure for each disc centred at the origin; that is, $K(t) = \kappa(b(0,t))$.

An estimate of $\kappa$ derived from a spatial point pattern dataset can be useful in exploratory data analysis. Its advantage over the K-function is that it is also sensitive to anisotropy and directional effects.

This function computes an estimate of $\kappa$
from a point pattern dataset `X`

,
which is assumed to be a realisation of a stationary point process,
observed inside a known, bounded window. Marks are ignored.

The algorithm approximates the point pattern and its window by binary pixel
images, introduces a Gaussian smoothing kernel
and uses the Fast Fourier Transform `fft`

to form a density estimate of $\kappa$. The calculation
corresponds to the edge correction known as the ``translation
correction''.

The Gaussian smoothing kernel may be specified by either of the
arguments `sigma`

or `varcov`

. If `sigma`

is a single
number, this specifies an isotropic Gaussian kernel
with standard deviation `sigma`

on each coordinate axis.
If `sigma`

is a vector of two numbers, this specifies a Gaussian
kernel with standard deviation `sigma[1]`

on the $x$ axis,
standard deviation `sigma[2]`

on the $y$ axis, and zero
correlation between the $x$ and $y$ axes. If `varcov`

is
given, this specifies the variance-covariance matrix of the
Gaussian kernel. There do not seem to be any well-established rules
for selecting the smoothing kernel in this context.
The density estimate of $\kappa$
is returned in the form of a real-valued pixel image.
Pixel values are estimates of the
integral of the second moment density over the pixel.
(The uniform Poisson process would have values identically equal to
$a$ where $a$ is the area of a pixel.)
Sums of pixel values over a desired region $A$ are estimates of the
value of $\kappa(A)$. The image `x`

and `y`

coordinates are on the same scale as vector displacements in the
original point pattern window. The point `x=0, y=0`

corresponds
to the `typical point'.
A peak in the image near `(0,0)`

suggests clustering;
a dip in the image near `(0,0)`

suggests inhibition;
peaks or dips at other positions suggest possible periodicity.

##### Value

- A real-valued pixel image (an object of class
`"im"`

, see`im.object`

) whose pixel values are estimates of the value of the reduced second moment measure for each pixel (i.e. estimates of the integral of the second moment density over each pixel).

##### References

Stoyan, D, Kendall, W.S. and Mecke, J. (1995)
*Stochastic geometry and its applications*.
2nd edition. Springer Verlag.

Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.

##### See Also

##### Examples

```
data(cells)
image(Kmeasure(cells, 0.05))
# shows pronounced dip around origin consistent with strong inhibition
data(redwood)
image(Kmeasure(redwood, 0.03), col=grey(seq(1,0,length=32)))
# shows peaks at several places, reflecting clustering and ?periodicity
```

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