# Kest.fft

##### 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

##### 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 r the vector of values of the argument $r$ at which the function $K$ has been estimated border the estimates of $K(r)$ for these values of $r$ theo the 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.

##### See Also

##### Examples

```
pp <- runifpoint(10000)
spatstat.options(npixel=512)
<testonly>spatstat.options(npixel=256)</testonly>
Kpp <- Kest.fft(pp, 0.01)
plot(Kpp)
```

*Documentation reproduced from package spatstat, version 1.5-5, License: GPL version 2 or newer*