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

`Kest.fft(X, sigma, r=NULL, ..., breaks=NULL)`

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

- 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.

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk

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`

.

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`

```
pp <- runifpoint(10000)
# \testonly{
op <- spatstat.options(npixel=125)
# }
Kpp <- Kest.fft(pp, 0.01)
plot(Kpp)
spatstat.options(op)
```

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