spatstat.explore (version 3.2-7)

bw.ppl: Likelihood Cross Validation Bandwidth Selection for Kernel Density

Description

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

Usage

bw.ppl(X, ..., srange=NULL, ns=16, sigma=NULL, varcov1=NULL,
          weights=NULL, shortcut=FALSE, warn=TRUE)

Value

A numerical value giving the selected bandwidth. The result also belongs to the class "bw.optim"

which can be plotted.

Arguments

X

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

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.

varcov1

Optional. Variance-covariance matrix matrix of the kernel with bandwidth \(h=1\). See section on Anisotropic Smoothing.

weights

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

...

Additional arguments passed to density.ppp.

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.

Anisotropic Smoothing

Anisotropic kernel smoothing is available in density.ppp using the argument varcov to specify the variance-covariance matrix of the anisotropic kernel. In order to choose the matrix varcov, the user can call bw.ppl using the argument varcov1 to specify a ‘template’ matrix. Scalar multiples of varcov1 will be considered and the optimal scale factor will be determined. That is, bw.ppl will try smoothing the data using varcov = h^2 * varcov1 for different values of h. The result of bw.ppl will be the optimal value of h.

Author

Adrian Baddeley , Rolf Turner and Ege Rubak .

Details

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

The bandwidth \(\sigma\) is chosen to maximise the point process likelihood cross-validation criterion $$ \mbox{LCV}(\sigma) = \sum_i \log\hat\lambda_{-i}(x_i) - \int_W \hat\lambda(u) \, {\rm d}u $$ 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.ppp, 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.

References

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

See Also

density.ppp, bw.diggle, bw.scott, bw.CvL, bw.frac.

Examples

  if(interactive()) {
    b <- bw.ppl(redwood)
    plot(b, main="Likelihood cross validation for redwoods")
    plot(density(redwood, b))
  }
  # \testonly{
    b1 <- bw.ppl(redwood, srange=c(0.03, 0.07), ns=2)
    b2 <- bw.ppl(redwood, srange=c(0.03, 0.07), ns=2, shortcut=TRUE)
  # }