spatstat (version 1.23-1)

localKinhom: Inhomogeneous Neighbourhood Density Function

Description

Computes spatially-weighted versions of the the local $K$-function or $L$-function.

Usage

localKinhom(X, lambda, ...,
              correction = "Ripley", verbose = TRUE, rvalue=NULL,
              sigma = NULL, varcov = NULL)
  localLinhom(X, lambda, ..., 
              correction = "Ripley", verbose = TRUE, rvalue=NULL,
              sigma = NULL, varcov = NULL)

Arguments

X

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

lambda

Optional. Values of the estimated intensity function. 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 locatio

...

Extra arguments. Ignored if lambda is present. Passed to density.ppp if lambda is omitted.

correction

String specifying the edge correction to be applied. Options are "none", "translate", "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.

sigma, varcov

Optional arguments passed to density.ppp to control the kernel smoothing procedure for estimating lambda, if lambda is missing.

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 ith point. The last two columns contain the r and theo values.

Details

The functions localKinhom and localLinhom are inhomogeneous or weighted versions of the neighbourhood density function implemented in localK and localL.

Given a spatial point pattern X, the inhomogeneous neighbourhood density function $L_i(r)$ associated with the $i$th point in X is computed by $$L_i(r) = \sqrt{\frac 1 \pi \sum_j \frac{e_{ij}}{\lambda_j}}$$ where the sum is over all points $j \neq i$ that lie within a distance $r$ of the $i$th point, $\lambda_j$ is the estimated intensity of the point pattern at the point $j$, and $e_{ij}$ 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 inhomogeneous L function (see Linhom).

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.

See Also

Kinhom, Linhom, localK, localL.

Examples

data(ponderosa)
  X <- ponderosa

  # compute all the local L functions
  L <- localLinhom(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)