Estimation of the spatio-temporal anisotropy

Estimation of the spatio-temporal anisotropy without an underlying spatio-temporal model. Different methods are implemented using a linear model to predict the temporal gamma values or the ratio of the ranges of a spatial and temporal variogram model or a spatial variogram model to predict the temporal gamma values or the spatio-temporal anisotropy value as used in a metric spatio-temporal variogram.

Usage

estiStAni(empVgm, interval, method = "linear", spatialVgm, 
          temporalVgm, s.range=NA, t.range=NA)

Arguments

empVgm: An empirical spatio-temporal variogram.
interval: A search interval for the optimisation of the spatio-temporal anisotropy parameter
method: A character string determining the method to be used (one of linear, range, vgm or metric, see below for details)
spatialVgm: A spatial variogram definition from the call to vgm. The model is optimised based on the pure spatial values in empVgm.
temporalVgm: A temporal variogram definition from the call to vgm. The model is optimised based on the pure temporal values in empVgm.
s.range: A spatial cutoff value applied to the empirical variogram empVgm.
t.range: A temporal cutoff value applied to the empirical variogram empVgm.

Details

linear: A linear model is fitted to the pure spatial gamma values based on the spatial distances. An optimal scaling is searched to stretch the temporal distances such that the linear model explains best the pure temporal gamma values. This assumes (on average) a linear relationship between distance and gamma, hence it is advisable to use only those pairs of pure spatial (pure temporal) distance and gamma value that show a considerable increase (i.e. drop all values beyond the range by setting values for s.range and t.range).
range: A spatial and temporal variogram model is fitted to the pure spatial and temporal gamma values respectively. The spatio-temporal anisotropy estimate is the ratio of the spatial range over the temporal range.
vgm: A spatial variogram model is fitted to the pure spatial gamma values. An optimal scaling is used to stretch the temporal distances such that the spatial variogram model explains best the pure temporal gamma values.
metric: A metric spatio-temporal variogram model is fitted with joint component according to the defined spatial variogram spatialVgm. The starting value of stAni is the mean of the interval parameter (see vgmST for the metric variogram definition). The spatio-temporal anisotropy as estimated in the spatio-temporal variogram is returned. Note that the parameter interval is only used to set the starting value. Hence, the estimate might exceed the given interval.

Value

A scalar representing the spatio-temporal anisotropy estimate.

Note

Different methods might lead to very different estimates. All but the linear approach are sensitive to the variogram model selection.

Author

Benedikt Graeler

Examples

data(vv)

estiStAni(vv, c(10, 150))
#> [1] 83.57463
estiStAni(vv, c(10, 150), "vgm", vgm(80, "Sph", 120, 20))
#> [1] 62.83509