R: Neighbourhood density function

localK {spatstat.core} | R Documentation

## Neighbourhood density function

### Description

Computes the neighbourhood density function, a local version of the K-function or L-function, defined by Getis and Franklin (1987).

### Usage

      localK(X, ..., rmax = NULL, correction = "Ripley", verbose = TRUE, rvalue=NULL)
      localL(X, ..., rmax = NULL, correction = "Ripley", verbose = TRUE, rvalue=NULL)

### Arguments

`X` | A point pattern (object of class `"ppp"`).
`...` | Ignored.
`rmax` | Optional. Maximum desired value of the argument r.
`correction` | String specifying the edge correction to be applied. Options are `"none"`, `"translate"`, `"translation"`, `"Ripley"`, `"isotropic"` or `"best"`. Only one correction may be specified.
`verbose` | Logical flag indicating whether to print progress reports during the calculation.
`rvalue` | Optional. A single value of the distance argument r at which the function L or K should be computed.

### Details

The command `localL` computes the neighbourhood density function, a local version of the L-function (Besag's transformation of Ripley's K-function) that was proposed by Getis and Franklin (1987). The command `localK` computes the corresponding local analogue of the K-function.

Given a spatial point pattern `X`, the neighbourhood density function L[i](r) associated with the ith point in `X` is computed by

L[i](r) = sqrt( (a/((n-1)* pi)) * sum[j] e[i,j])

where the sum is over all points j != i that lie within a distance r of the ith point, a is the area of the observation window, n is the number of points in `X`, and e[i,j] is an edge correction term (as described in `Kest`). The value of L[i](r) can also be interpreted as one of the summands that contributes to the global estimate of the L function.

By default, the function L[i](r) or K[i](r) is computed for a range of r values for each point i. The results are stored as a function value table (object of class `"fv"`) with a column of the table containing the function estimates for each point of the pattern `X`.

Alternatively, if the argument `rvalue` is given, and it is a single number, then the function will only be computed for this value of r, and the results will be returned as a numeric vector, with one entry of the vector for each point of the pattern `X`.

Inhomogeneous counterparts of `localK` and `localL` are computed by `localKinhom` and `localLinhom`.

### Value

If `rvalue` is given, the result is a numeric vector of length equal to the number of points in the point pattern.

If `rvalue` is absent, the result is 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 neighbourhood density 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

### References

Getis, A. and Franklin, J. (1987) Second-order neighbourhood analysis of mapped point patterns. Ecology 68, 473–477.

### See Also

`Kest`, `Lest`, `localKinhom`, `localLinhom`.

### Examples

      data(ponderosa)
      X <- ponderosa

      # compute all the local L functions
      L <- localL(X)

      # plot all the local L functions against r
      plot(L, main="local L functions for ponderosa", legend=FALSE)

      # plot only the local L function for point number 7
      plot(L, iso007 ~ r)

      # compute the values of L(r) for r = 12 metres
      L12 <- localL(X, rvalue=12)

      # Spatially interpolate the values of L12
      # Compare Figure 5(b) of Getis and Franklin (1987)
      X12 <- X %mark% L12
      Z <- Smooth(X12, sigma=5, dimyx=128)

      plot(Z, col=topo.colors(128), main="smoothed neighbourhood density")
      contour(Z, add=TRUE)
      points(X, pch=16, cex=0.5)

[Package spatstat.core version 1.65-5 Index]