R: Local pair correlation function

localpcf {spatstat} | R Documentation

## 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"`) 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.
`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 estimation of `lambda`.

### 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 ith point in `X` is computed by

g[i](r) = (a/(2 * pi * n) * sum[j] k(d[i,j] - r)

where the sum is over all points j != i, a is the area of the observation window, n is the number of points in `X`, and d[i,j] is the distance between points `i` and `j`. Here `k` is the Epanechnikov kernel,

k(t) = (3/(4*delta)) * max(0, 1 - 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/lambda where 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`.

### 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 `i`th point. The last two columns contain the `r` and `theo` values.

### Author(s)

Adrian Baddeley Adrian.Baddeley@curtin.edu.au

and Rolf Turner r.turner@auckland.ac.nz

### 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)

[Package spatstat version 1.54-0 Index]