NS function

Normal scale (NS) bandwidth selector

Provides the asymptotically optimal fixed bandwidths for spatial or spatiotemporal normal densities based on a simple expression.

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

NS.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 normal scale formula. Either a character string, "npoints" (default) or "geometric" (only possible for NS), or a positive numeric value. See `Details'.

• scaler: Optional. Controls the value for a scalar representation of the spatial (and temporal for NS.spattemp) scale of the data. Either a character string, "silverman"

  (default), "IQR", "sd", or "var"; or a positive numeric value. 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.

Returns

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

Details

These functions calculate scalar smoothing bandwidths for kernel density estimates of spatial or spatiotemporal data: the optimal values would minimise the asymptotic mean integrated squared error assuming normally distributed data; see pp. 46-48 of Silverman (1986). The NS function returns a single bandwidth for isotropic smoothing of spatial (2D) data. The NS.spattemp function returns two values -- one for the spatial margin and another for the temporal margin, based on independently applying the normal scale 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 NS (not applicable to NS.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 NS.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 ofNS.spattempthe univariate version of the elected scale statistic is applied to the recorded times of the data for the temporal bandwidth. Alternatively, likenstar, the user can specify their own value by simply supplying a single positive numeric value to scalerforNS, or a numeric vector of length 2 (in the order of **[<spatial scale>, <temporal scale>]**) for NS.spattemp`.

Warning

The NS bandwidth is an approximation, and assumes that the target density is normal. This is considered rare in most real-world applications. Nevertheless, it remains a quick and easy `rule-of-thumb' method with which one may obtain a smoothing parameter. Note that a similar expression for the adaptive kernel estimator is not possible (Davies et al., 2018).

Examples

data(pbc)

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

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

References

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

Davies, T.M., Flynn, C.R. and Hazelton, M.L. (2018), On the utility of asymptotic bandwidth selectors for spatially adaptive kernel density estimation, Statistics & Probability Letters [in press].

Silverman, B.W. (1986), Density Estimation for Statistics and Data Analysis, Chapman & Hall, New York.

Wand, M.P. and Jones, C.M., 1995. Kernel Smoothing, Chapman & Hall, London.

Author(s)

T.M. Davies