Description Usage Arguments Details Value Author(s) References Examples

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

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`pp` |
An object of class |

`nstar` |
Optional. Controls the value to use in place of the number of
observations |

`scaler` |
Optional. Controls the value for a scalar representation of
the spatial (and temporal for |

`tt` |
A numeric vector of equal length to the number of points in |

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`

.

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`

.

T.M. Davies

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.

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