# Kest.fft

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##### K-function using FFT

Estimates the reduced second moment function $K(r)$ from a point pattern in a window of arbitrary shape, using the Fast Fourier Transform.

Keywords
spatial, nonparametric
##### Usage
Kest.fft(X, sigma, r=NULL, breaks=NULL)
##### Arguments
X
The observed point pattern, from which an estimate of $K(r)$ will be computed. An object of class "ppp", or data in any format acceptable to as.ppp().
sigma
standard deviation of the isotropic Gaussian smoothing kernel.
r
vector of values for the argument $r$ at which $K(r)$ should be evaluated. There is a sensible default.
breaks
An alternative to the argument r. Not normally invoked by the user. See Details.
##### Details

This is an alternative to the function Kest for estimating the $K$ function. It may be useful for very large patterns of points.

Whereas Kest computes the distance between each pair of points analytically, this function discretises the point pattern onto a rectangular pixel raster and applies Fast Fourier Transform techniques to estimate $K(t)$. The hard work is done by the function Kmeasure.

The result is an approximation whose accuracy depends on the resolution of the pixel raster. The resolution is controlled by setting the parameter npixel in spatstat.options.

##### Value

• An object of class "fv" (see fv.object). Essentially a data frame containing columns
• rthe vector of values of the argument $r$ at which the function $K$ has been estimated
• borderthe estimates of $K(r)$ for these values of $r$
• theothe theoretical value $K(r) = \pi r^2$ for a stationary Poisson process

##### References

Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.

Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.

Ohser, J. (1983) On estimators for the reduced second moment measure of point processes. Mathematische Operationsforschung und Statistik, series Statistics, 14, 63 -- 71. Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.

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.

Kest, Kmeasure, spatstat.options

• Kest.fft
##### Examples
pp <- runifpoint(10000)
spatstat.options(npixel=512)
<testonly>spatstat.options(npixel=125)</testonly>
Kpp <- Kest.fft(pp, 0.01)
plot(Kpp)
Documentation reproduced from package spatstat, version 1.16-2, License: GPL (>= 2)

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