Reduced Second Moment Measure

Description

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

Usage

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

Arguments

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).

Author(s)

Adrian Baddeley adrian@maths.uwa.edu.au http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner rolf@math.unb.ca http://www.math.unb.ca/~rolf

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

Kest, spatstat.options, im.object

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