# pcf.fasp

0th

Percentile

##### Pair Correlation Function obtained from array of K functions

Estimates the (bivariate) pair correlation functions of a point pattern, given an array of (bivariate) K functions.

Keywords
spatial, nonparametric
##### Usage
# S3 method for fasp
pcf(X, …, method="c")
##### Arguments
X

An array of multitype $K$ functions (object of class "fasp").

Arguments controlling the smoothing spline function smooth.spline.

method

Letter "a", "b", "c" or "d" indicating the method for deriving the pair correlation function from the K function.

##### Details

The pair correlation function of a stationary point process is $$g(r) = \frac{K'(r)}{2\pi r}$$ where $K'(r)$ is the derivative of $K(r)$, the reduced second moment function (aka Ripley's $K$ function'') of the point process. See Kest for information about $K(r)$. For a stationary Poisson process, the pair correlation function is identically equal to 1. Values $g(r) < 1$ suggest inhibition between points; values greater than 1 suggest clustering.

We also apply the same definition to other variants of the classical $K$ function, such as the multitype $K$ functions (see Kcross, Kdot) and the inhomogeneous $K$ function (see Kinhom). For all these variants, the benchmark value of $K(r) = \pi r^2$ corresponds to $g(r) = 1$.

This routine computes an estimate of $g(r)$ from an array of estimates of $K(r)$ or its variants, using smoothing splines to approximate the derivatives. It is a method for the generic function pcf.

The argument X should be a function array (object of class "fasp", see fasp.object) containing several estimates of $K$ functions. This should have been obtained from alltypes with the argument fun="K".

The smoothing spline operations are performed by smooth.spline and predict.smooth.spline from the modreg library. Four numerical methods are available:

• "a" apply smoothing to $K(r)$, estimate its derivative, and plug in to the formula above;

• "b" apply smoothing to $Y(r) = \frac{K(r)}{2 \pi r}$ constraining $Y(0) = 0$, estimate the derivative of $Y$, and solve;

• "c" apply smoothing to $Z(r) = \frac{K(r)}{\pi r^2}$ constraining $Z(0)=1$, estimate its derivative, and solve.

• "d" apply smoothing to $V(r) = \sqrt{K(r)}$, estimate its derivative, and solve.

Method "c" seems to be the best at suppressing variability for small values of $r$. However it effectively constrains $g(0) = 1$. If the point pattern seems to have inhibition at small distances, you may wish to experiment with method "b" which effectively constrains $g(0)=0$. Method "a" seems comparatively unreliable.

Useful arguments to control the splines include the smoothing tradeoff parameter spar and the degrees of freedom df. See smooth.spline for details.

##### Value

A function array (object of class "fasp", see fasp.object) representing an array of pair correlation functions. This can be thought of as a matrix Y each of whose entries Y[i,j] is a function value table (class "fv") representing the pair correlation function between points of type i and points of type j.

##### 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.

Kest, Kinhom, Kcross, Kdot, Kmulti, alltypes, smooth.spline, predict.smooth.spline

• pcf.fasp
##### Examples
# NOT RUN {
# multitype point pattern
KK <- alltypes(amacrine, "K")
p <- pcf.fasp(KK, spar=0.5, method="b")
plot(p)
# strong inhibition between points of the same type
# }

Documentation reproduced from package spatstat, version 1.59-0, License: GPL (>= 2)

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