NS | R Documentation |

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")
)
```

`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 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 of`NS.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`NS`

, or a numeric vector of length 2 (in the order of*[<spatial scale>, <temporal scale>]*) for`NS.spattemp`

.

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`

.

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).

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.

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.

```
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")
```

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