OS {sparr}

Oversmoothing (OS) bandwidth selector

Description

Provides fixed bandwidths for spatial or spatiotemporal data based on the maximal smoothing (oversmoothing) principle of Terrell (1990).

Usage

OS(
  pp,
  nstar = c("npoints", "geometric"),
  scaler = c("silverman", "IQR", "sd", "var")
)

OS.spattemp(
  pp,
  tt = NULL,
  nstar = "npoints",
  scaler = c("silverman", "IQR", "sd", "var")
)

Arguments

pp

An object of class ppp giving the observed 2D data to be smoothed.

nstar

Optional. Controls the value to use in place of the number of observations n in the oversmoothing formula. Either a character string, "npoints" (default) or "geometric" (only possible for OS), or a positive numeric value. See ‘Details’.

scaler

Optional. Controls the value for a scalar representation of the spatial (and temporal for OS.spattemp) scale of the data. Either a character string, "silverman" (default), "IQR", "sd", or "var"; or positive numeric value(s). See ‘Details’.

tt

A numeric vector of equal length to the number of points in pp, giving the time corresponding to each spatial observation. If unsupplied, the function attempts to use the values in the marks attribute of the ppp.object in pp.

Details

These functions calculate scalar smoothing bandwidths for kernel density estimates of spatial or spatiotemporal data: the “maximal amount of smoothing compatible with the estimated scale of the observed data”. See Terrell (1990). The OS function returns a single bandwidth for isotropic smoothing of spatial (2D) data. The OS.spattemp function returns two values – one for the spatial margin and another for the temporal margin, based on independently applying Terrell's (1990) rule (in 2D and 1D) to the spatial and temporal margins of the supplied data.

Effective sample size

The formula requires a sample size, and this can be minimally tailored via nstar. By default, the function simply uses the number of observations in pp: nstar = "npoints". Alternatively, the user can specify their own value by simply supplying a single positive numeric value to nstar. For OS (not applicable to OS.spattemp), if pp is a ppp.object with factor-valued marks, then the user has the option of using nstar = "geometric", which sets the sample size used in the formula to the geometric mean of the counts of observations of each mark. This can be useful for e.g. relative risk calculations, see Davies and Hazelton (2010).

Spatial (and temporal) scale

The scaler argument is used to specify spatial (as well as temporal, in use of OS.spattemp) scale. For isotropic smoothing in the spatial margin, one may use the ‘robust’ estimate of standard deviation found by a weighted mean of the interquartile ranges of the x- and y-coordinates of the data respectively (scaler = "IQR"). Two other options are the raw mean of the coordinate-wise standard deviations (scaler = "sd"), or the square root of the mean of the two variances (scaler = "var"). A fourth option, scaler = "silverman" (default), sets the scaling constant to be the minimum of the "IQR" and "sd" options; see Silverman (1986), p. 47. In use of OS.spattemp the univariate version of the elected scale statistic is applied to the recorded times of the data for the temporal bandwidth. Alternatively, like nstar, the user can specify their own value by simply supplying a single positive numeric value to scaler for OS, or a numeric vector of length 2 (in the order of [<spatial scale>, <temporal scale>]) for OS.spattemp.

Value

A single numeric value of the estimated spatial bandwidth for OS, or a named numeric vector of length 2 giving the spatial bandwidth (as h) and the temporal bandwidth (as lambda) for OS.spattemp.

Author(s)

T.M. Davies

References

Davies, T.M. and Hazelton, M.L. (2010), Adaptive kernel estimation of spatial relative risk, Statistics in Medicine, 29(23) 2423-2437.

Terrell, G.R. (1990), The maximal smoothing principle in density estimation, Journal of the American Statistical Association, 85, 470-477.

Examples

data(pbc)

OS(pbc)
OS(pbc,nstar="geometric") # uses case-control marks to replace sample size
OS(pbc,scaler="var") # set different scalar measure of spread

data(burk)
OS.spattemp(burk$cases)
OS.spattemp(burk$cases,scaler="sd")