bw.lppl {spatstat.linnet}

Likelihood Cross Validation Bandwidth Selection for Kernel Density on a Linear Network

Description

Uses likelihood cross-validation to select a smoothing bandwidth for the kernel estimation of point process intensity on a linear network.

Usage

   bw.lppl(X, ..., srange=NULL, ns=16, sigma=NULL, weights=NULL,
           distance=c("euclidean", "path"), shortcut=TRUE, warn=TRUE)

Arguments

X

A point pattern on a linear network (object of class "lpp").

srange

Optional numeric vector of length 2 giving the range of values of bandwidth to be searched.

ns

Optional integer giving the number of values of bandwidth to search.

sigma

Optional. Vector of values of the bandwidth to be searched. Overrides the values of ns and srange.

weights

Optional. Numeric vector of weights for the points of X. Argument passed to density.lpp.

distance

Argument passed to density.lpp controlling the type of kernel estimator.

...

Additional arguments passed to density.lpp.

shortcut

Logical value indicating whether to speed up the calculation by omitting the integral term in the cross-validation criterion.

warn

Logical. If TRUE, issue a warning if the maximum of the cross-validation criterion occurs at one of the ends of the search interval.

Details

This function selects an appropriate bandwidth sigma for the kernel estimator of point process intensity computed by density.lpp.

The argument X should be a point pattern on a linear network (class "lpp").

The bandwidth \sigma is chosen to maximise the point process likelihood cross-validation criterion

\mbox{LCV}(\sigma) = \sum_i \log\hat\lambda_{-i}(x_i) - \int_L \hat\lambda(u) \, {\rm d}u

where the sum is taken over all the data points x_i, where \hat\lambda_{-i}(x_i) is the leave-one-out kernel-smoothing estimate of the intensity at x_i with smoothing bandwidth \sigma, and \hat\lambda(u) is the kernel-smoothing estimate of the intensity at a spatial location u with smoothing bandwidth \sigma. See Loader(1999, Section 5.3).

The value of \mbox{LCV}(\sigma) is computed directly, using density.lpp, for ns different values of \sigma between srange[1] and srange[2].

The result is a numerical value giving the selected bandwidth. The result also belongs to the class "bw.optim" which can be plotted to show the (rescaled) mean-square error as a function of sigma.

If shortcut=TRUE, the computation is accelerated by omitting the integral term in the equation above. This is valid because the integral is approximately constant.

Value

A single numerical value giving the selected bandwidth. The result also belongs to the class "bw.optim" (see bw.optim.object) which can be plotted to show the bandwidth selection criterion as a function of sigma.

Author(s)

Greg McSwiggan, Suman Rakshit and Adrian Baddeley [e-mail].

References

Loader, C. (1999) Local Regression and Likelihood. Springer, New York.

McSwiggan, G., Baddeley, A. and Nair, G. (2019) Estimation of relative risk for events on a linear network. Statistics and Computing 30 (2) 469–484.

See Also

density.lpp, bw.scott.

bw.optim.object.

For point patterns in two-dimensional space, use bw.ppl.

Examples

  if(interactive()) {
    b <- bw.lppl(spiders)
    plot(b, main="Likelihood cross validation for spiders")
    plot(density(spiders, b, distance="e"))
  } else {
    b1 <- bw.lppl(spiders, ns=2)
    b2 <- bw.lppl(spiders, ns=2, shortcut=TRUE)
  }