spatstat (version 1.33-0)

localpcf: Local pair correlation function

Description

Computes individual contributions to the pair correlation function from each data point.

Usage

localpcf(X, ..., delta=NULL, rmax=NULL, nr=512, stoyan=0.15)
  localpcfinhom(X, ..., delta=NULL, rmax=NULL, nr=512, stoyan=0.15,
         lambda=NULL, sigma=NULL, varcov=NULL)

Arguments

X

A point pattern (object of class "ppp").

delta

Smoothing bandwidth for pair correlation. The halfwidth of the Epanechnikov kernel.

rmax

Optional. Maximum value of distance $r$ for which pair correlation values $g(r)$ should be computed.

nr

Optional. Number of values of distance $r$ for which pair correlation $g(r)$ should be computed.

stoyan

Optional. The value of the constant $c$ in Stoyan's rule of thumb for selecting the smoothing bandwidth delta.

lambda

Optional. Values of the estimated intensity function, for the inhomogeneous pair correlation. Either a vector giving the intensity values at the points of the pattern X, a pixel image (object of class "im") gi

sigma,varcov,...

These arguments are ignored by localpcf but are passed by localpcfinhom (when lambda=NULL) to the function density.ppp to control the kernel smoothing

Value

• An object of class "fv", see fv.object, which can be plotted directly using plot.fv. Essentially a data frame containing columns
• r
  the vector of values of the argument $r$ at which the function $K$ has been estimated
• theo
  the theoretical value $K(r) = \pi r^2$ or $L(r)=r$ for a stationary Poisson process
• together with columns containing the values of the local pair correlation function for each point in the pattern. Column i corresponds to the ith point. The last two columns contain the r and theo values.

Details

localpcf computes the contribution, from each individual data point in a point pattern X, to the empirical pair correlation function of X. These contributions are sometimes known as LISA (local indicator of spatial association) functions based on pair correlation. localpcfinhom computes the corresponding contribution to the inhomogeneous empirical pair correlation function of X. Given a spatial point pattern X, the local pcf $g_i(r)$ associated with the $i$th point in X is computed by $$g_i(r) = \frac a {2 \pi n} \sum_j k(d_{i,j} - r)$$ where the sum is over all points $j \neq i$, $a$ is the area of the observation window, $n$ is the number of points in X, and $d_{ij}$ is the distance between points i and j. Here k is the Epanechnikov kernel, $$k(t) = \frac 3 { 4\delta} \max(0, 1 - \frac{t^2}{\delta^2}).$$ Edge correction is performed using the border method (for the sake of computational efficiency): the estimate $g_i(r)$ is set to NA if $r > b_i$, where $b_i$ is the distance from point $i$ to the boundary of the observation window.

The smoothing bandwidth $\delta$ may be specified. If not, it is chosen by Stoyan's rule of thumb $\delta = c/\hat\lambda$ where $\hat\lambda = n/a$ is the estimated intensity and $c$ is a constant, usually taken to be 0.15. The value of $c$ is controlled by the argument stoyan.

For localpcfinhom, the optional argument lambda specifies the values of the estimated intensity function. If lambda is given, it should be either a numeric vector giving the intensity values at the points of the pattern X, a pixel image (object of class "im") giving the intensity values at all locations, a fitted point process model (object of class "ppm") or a function(x,y) which can be evaluated to give the intensity value at any location. If lambda is not given, then it will be estimated using a leave-one-out kernel density smoother as described in pcfinhom.

See Also

localK, localKinhom, pcf, pcfinhom

Examples

data(ponderosa)
  X <- ponderosa

  g <- localpcf(X, stoyan=0.5)
  colo <- c(rep("grey", npoints(X)), "blue")
  a <- plot(g, main=c("local pair correlation functions", "Ponderosa pines"),
          legend=FALSE, col=colo, lty=1)

  # plot only the local pair correlation function for point number 7
  plot(g, est007 ~ r)

  gi <- localpcfinhom(X, stoyan=0.5)
  a <- plot(gi, main=c("inhomogeneous local pair correlation functions",
                       "Ponderosa pines"),
                legend=FALSE, col=colo, lty=1)