# 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, 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
Optional. Vector of values for the argument \(r\) at which \(K(r)\) should be evaluated. There is a sensible default.

- …
Arguments passed to

`as.mask`

determining the spatial resolution for the FFT calculation.- breaks
This argument is for internal use only.

##### 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 the arguments `…`

, or by setting the parameter `npixel`

in
`spatstat.options`

.

##### Value

An object of class `"fv"`

(see `fv.object`

).

Essentially a data frame containing columns

the vector of values of the argument \(r\) at which the function \(K\) has been estimated

the estimates of \(K(r)\) for these values of \(r\)

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

```
# NOT RUN {
pp <- runifpoint(10000)
# }
# NOT RUN {
Kpp <- Kest.fft(pp, 0.01)
plot(Kpp)
# }
```

*Documentation reproduced from package spatstat, version 1.55-1, License: GPL (>= 2)*