spatstat (version 1.55-0)

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, method = "likelihood", nh = spatstat.options("n.bandwidth"),
   hmin=NULL, hmax=NULL, warn=TRUE)

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.

Value

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

Details

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

Consider the indicators \(y_{ij}\) 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 likelihood, the squared error, or the approximately standardised squared error, of the indicators \(y_{ij}\) relative to the fitted values \(\hat p_j(x_i)\). See Diggle (2003).

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

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.

See Also

relrisk, bw.stoyan

Examples

# NOT RUN {
  data(urkiola)
  
# }
# NOT RUN {
  b <- bw.relrisk(urkiola)
  b
  plot(b)
  b <- bw.relrisk(urkiola, hmax=20)
  plot(b)
  
# }