spatstat (version 1.48-0)

pcf.ppp: Pair Correlation Function of Point Pattern

Description

Estimates the pair correlation function of a point pattern using kernel methods.

Usage

"pcf"(X, ..., r = NULL, kernel="epanechnikov", bw=NULL, stoyan=0.15, correction=c("translate", "Ripley"), divisor = c("r", "d"), var.approx = FALSE, domain=NULL, ratio=FALSE)

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.

divisor

Choice of divisor in the estimation formula: either "r" (the default) or "d". See Details.

var.approx

Logical value indicating whether to compute an analytic approximation to the variance of the estimated pair correlation.

domain

Optional. Calculations will be restricted to this subset of the window. See Details.

ratio

Logical. If TRUE, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.

Value

A function value table (object of class "fv"). Essentially a data frame containing the variablesas required.If ratio=TRUE then the return value also has two attributes called "numerator" and "denominator" which are "fv" objects containing the numerators and denominators of each estimate of $g(r)$.The return value also has an attribute "bw" giving the smoothing bandwidth that was used.

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, $p(r) = lambda^2 dx dy$ 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 kernel smoothing.

• If divisor="r" (the default), then the standard kernel estimator (Stoyan and Stoyan, 1994, pages 284--285) is used. By default, the recommendations of Stoyan and Stoyan (1994) are followed exactly.
• If divisor="d" then a modified estimator is used: the contribution from an interpoint distance $d[ij]$ to the estimate of $g(r)$ is divided by $d[ij]$ instead of dividing by $r$. This usually improves the bias of the estimator when $r$ is close to zero.

There is also a choice of spatial edge corrections (which are needed to avoid bias due to edge effects associated with the boundary of the spatial window):

• If correction="translate" or correction="translation" then the translation correction is used. For divisor="r" the translation-corrected 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. For divisor="r" the isotropic-corrected 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.

Alternatively correction="all" selects all options. 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.default. 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 smoothing bandwidth that was used in the calculation is returned as an attribute of the final result. 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.

If the argument domain is given, estimation will be restricted to this region. That is, the estimate of $g(r)$ will be based on pairs of points in which the first point lies inside domain and the second point is unrestricted. The argument domain should be a window (object of class "owin") or something acceptable to as.owin. It must be a subset of the window of the point pattern X.

To compute a confidence band for the true value of the pair correlation function, use lohboot.

If var.approx = TRUE, the variance of the estimate of the pair correlation will also be calculated using an analytic approximation (Illian et al, 2008, page 234) which is valid for stationary point processes which are not too clustered. This calculation is not yet implemented when the argument domain is given.

References

Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. (2008) Statistical Analysis and Modelling of Spatial Point Patterns. Wiley.

Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.

See Also

Kest, pcf, density, lohboot.

Examples

  X <- simdat
  
  p <- pcf(X)
  plot(p, main="pair correlation function for X")
  # indicates inhibition at distances r < 0.3

  pd <- pcf(X, divisor="d")

  # compare estimates
  plot(p, cbind(iso, theo) ~ r, col=c("blue", "red"),
         ylim.covers=0, main="", lwd=c(2,1), lty=c(1,3), legend=FALSE)
  plot(pd, iso ~ r, col="green", lwd=2, add=TRUE)
  legend("center", col=c("blue", "green"), lty=1, lwd=2,
         legend=c("divisor=r","divisor=d"))

  # calculate approximate variance and show POINTWISE confidence bands
  pv <- pcf(X, var.approx=TRUE)
  plot(pv, cbind(iso, iso+2*sqrt(v), iso-2*sqrt(v)) ~ r)