pcf.ppp
Pair Correlation Function of Point Pattern
Estimates the pair correlation function of a point pattern using kernel methods.
- Keywords
- spatial, nonparametric
Usage
## S3 method for class 'ppp':
pcf(X, \dots, r = NULL, kernel="epanechnikov", bw=NULL, stoyan=0.15,
correction=c("translate", "Ripley"))Arguments
- X
- A point pattern (object of class
"ppp"). - r
- Vector of values for the argument $r$ at which $g(r)$ should be evaluated. There is a sensible default.
- kernel
- Choice of smoothing kernel,
passed to
density. - bw
- Bandwidth for smoothing kernel, passed to
density. - ...
- Other arguments passed to the kernel density estimation
function
density. - stoyan
- Bandwidth coefficient; see Details.
- correction
- Choice of edge correction.
Details
The pair correlation function $g(r)$ is a summary of the dependence between points in a spatial point process. The best intuitive interpretation is the following: the probability $p(r)$ of finding two points at locations $x$ and $y$ separated by a distance $r$ is equal to $$p(r) = \lambda^2 g(r) \,{\rm d}x \, {\rm d}y$$ where $\lambda$ is the intensity of the point process. For a completely random (uniform Poisson) process, $p(r) = \lambda^2$ so $g(r) = 1$.
Formally, the pair correlation function of a stationary point process
is defined by
$$g(r) = \frac{K'(r)}{2\pi r}$$
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.
This routine computes an estimate of $g(r)$ by the kernel smoothing method (Stoyan and Stoyan (1994), pages 284--285). By default, their recommendations are followed exactly.
If correction="translate" then the translation correction
is used. The estimate is given in equation (15.15), page 284 of
Stoyan and Stoyan (1994).
If correction="Ripley" then Ripley's isotropic edge correction
is used; the estimate is given in equation (15.18), page 285
of Stoyan and Stoyan (1994).
If correction=c("translate", "Ripley") then both estimates
will be computed.
The choice of smoothing kernel is controlled by the
argument kernel which is passed to density.
The default is the Epanechnikov kernel, recommended by
Stoyan and Stoyan (1994, page 285).
The bandwidth of the smoothing kernel can be controlled by the
argument bw. Its precise interpretation
is explained in the documentation for density.
For the Epanechnikov kernel, the argument bw is
equivalent to $h/\sqrt{5}$.
Stoyan and Stoyan (1994, page 285) recommend using the Epanechnikov
kernel with support $[-h,h]$ chosen by the rule of thumn
$h = c/\sqrt{\lambda}$,
where $\lambda$ is the (estimated) intensity of the
point process, and $c$ is a constant in the range from 0.1 to 0.2.
See equation (15.16).
If bw is missing, then this rule of thumb will be applied.
The argument stoyan determines the value of $c$.
The argument r is the vector of values for the
distance $r$ at which $g(r)$ should be evaluated.
There is a sensible default.
If it is specified, r must be a vector of increasing numbers
starting from r[1] = 0,
and max(r) must not exceed half the diameter of
the window.
Value
- A function value table
(object of class
"fv"). Essentially a data frame containing the variables r the vector of values of the argument $r$ at which the pair correlation function $g(r)$ has been estimated theo vector of values equal to 1, the theoretical value of $g(r)$ for the Poisson process trans vector of values of $g(r)$ estimated by translation correction iso vector of values of $g(r)$ estimated by Ripley isotropic correction - as required.
References
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.
See Also
Examples
data(simdat)
<testonly>simdat <- simdat[seq(1,simdat$n, by=4)]</testonly>
p <- pcf(simdat)
plot(p, main="pair correlation function for simdat")
# indicates inhibition at distances r < 0.3