bw.relrisk: Cross Validated Bandwidth Selection for Relative Risk Estimation

Description

Uses cross-validation to select a smoothing bandwidth for the estimation of relative risk.

Usage

bw.relrisk(X, ...)
   # S3 method for ppp
bw.relrisk(X, method = "likelihood", ...,
           nh = spatstat.options("n.bandwidth"),
           hmin=NULL, hmax=NULL, warn=TRUE)

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.

Arguments

X: A multitype point pattern (object of class "ppp" which has factor valued marks).
method: Character string determining the cross-validation method. Current options are "likelihood", "leastsquares" or "weightedleastsquares".
nh: Number of trial values of smoothing bandwith sigma to consider. The default is 32.
hmin, hmax: Optional. Numeric values. Range of trial values of smoothing bandwith sigma to consider. There is a sensible default.
warn: Logical. If TRUE, issue a warning if the minimum of the cross-validation criterion occurs at one of the ends of the search interval.
...: Additional arguments passed to density.ppp or to other methods for bw.relrisk.

Author

Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner r.turner@auckland.ac.nz.

Details

This function selects an appropriate bandwidth for the nonparametric estimation of relative risk using relrisk.

Consider the indicators $y_{i j}$ which equal $1$ when data point $x_{i}$ belongs to type $j$ , and equal $0$ otherwise. For a particular value of smoothing bandwidth, let ${\hat{p}}_{j} (u)$ be the estimated probabilities that a point at location $u$ will belong to type $j$ . Then the bandwidth is chosen to minimise either the negative likelihood, the squared error, or the approximately standardised squared error, of the indicators $y_{i j}$ relative to the fitted values ${\hat{p}}_{j} (x_{i})$ . See Diggle (2003) or Baddeley et al (2015).

The result is a numerical value giving the selected bandwidth sigma. The result also belongs to the class "bw.optim" allowing it to be printed and plotted. The plot shows the cross-validation criterion as a function of bandwidth.

The range of values for the smoothing bandwidth sigma is set by the arguments hmin, hmax. There is a sensible default, based on multiples of Stoyan's rule of thumb bw.stoyan.

If the optimal bandwidth is achieved at an endpoint of the interval [hmin, hmax], the algorithm will issue a warning (unless warn=FALSE). If this occurs, then it is probably advisable to expand the interval by changing the arguments hmin, hmax.

Computation time depends on the number nh of trial values considered, and also on the range [hmin, hmax] of values considered, because larger values of sigma require calculations involving more pairs of data points.

References

Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

Diggle, P.J. (2003) Statistical analysis of spatial point patterns, Second edition. Arnold.

Kelsall, J.E. and Diggle, P.J. (1995) Kernel estimation of relative risk. Bernoulli 1, 3--16.

Examples

Run this code

  op <- spatstat.options(n.bandwidth=8)
  b <- bw.relrisk(urkiola)
  b
  plot(b)
  b <- bw.relrisk(urkiola, hmax=20)
  plot(b)
  spatstat.options(op)

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