Cross Validated Bandwidth Selection for Relative Risk Estimation
Uses cross-validation to select a smoothing bandwidth for the estimation of relative risk.
bw.relrisk(X, method = "likelihood", nh = spatstat.options("n.bandwidth"),
hmin=NULL, hmax=NULL, warn=TRUE)X |
A multitype point pattern (object of class |
method |
Character string determining the cross-validation method.
Current options are |
nh |
Number of trial values of smoothing bandwith |
hmin, hmax |
Optional. Numeric values.
Range of trial values of smoothing bandwith |
warn |
Logical. If |
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 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 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.
A numerical value giving the selected bandwidth.
The result also belongs to the class "bw.optim"
which can be plotted.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner r.turner@auckland.ac.nz.
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.
data(urkiola) b <- bw.relrisk(urkiola) b plot(b) b <- bw.relrisk(urkiola, hmax=20) plot(b)
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