Reduced Second Moment Measure
Estimates the reduced second moment measure $\kappa$ from a point pattern in a window of arbitrary shape.
Kmeasure(X, sigma, edge=TRUE, ..., varcov=NULL)
- 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
- Standard deviation $\sigma$ of the Gaussian
smoothing kernel. Incompatible with
- logical value indicating whether an edge correction should be applied.
- Variance-covariance matrix of the Gaussian smoothing kernel.
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
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
to form a density estimate of $\kappa$. The calculation
corresponds to the edge correction known as the ``translation
The Gaussian smoothing kernel may be specified by either of the
sigma is a single
number, this specifies an isotropic Gaussian kernel
with standard deviation
sigma on each coordinate axis.
sigma is a vector of two numbers, this specifies a Gaussian
kernel with standard deviation
sigma on the $x$ axis,
sigma on the $y$ axis, and zero
correlation between the $x$ and $y$ axes. If
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
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.
- A real-valued pixel image (an object of class
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).
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.
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