Calculate the theoretical minimum variance of a localized multitaper spectral estimate and the corresponding optimal weights to apply to each localized spectrum.

## Usage

`var_opt`

, `var_unit`

, `weight_opt`

= SHMTVarOpt (`l`

, `tapers`

, `taper_order`

, `sff`

, [`lwin`

, `kmax`

, `nocross`

])

## Returns

`var_opt`

: float, dimension (`kmax`

)- The minimum variance of the multitaper spectral estimate for degree
`l`

using 1 through`kmax`

tapers. `var_unit`

: float, dimension (`kmax`

)- The variance of the multitaper spectral estimate using equal weights for degree
`l`

using 1 through`kmax`

tapers. `weight_opt`

: float, dimension (`kmax`

,`kmax`

)- The optimal weights (in columns) that minimize the multitaper spectral estimate’s variance using 1 through
`kmax`

tapers.

## Parameters

`l`

: integer- The spherical harmonic degree used to calculate the theoretical minimum variance and optimal weights.
`tapers`

: float, dimension (`lwinin`

+1,`kmaxin`

)- A matrix of localization functions obtained from
`SHReturnTapers`

or`SHReturnTapersM`

. `taper_order`

: integer, dimension (`kmaxin`

)- The angular order of the windowing coefficients in
`tapers`

. `sff`

: float, dimension (`l`

+`lwinin`

+1)- The global unwindowed power spectrum of the function to be localized.
`lwin`

: optional, integer, default =`lwinin`

- The spherical harmonic bandwidth of the localizing windows.
`kmax`

: optional, integer, default =`kmaxin`

- The maximum number of tapers to be used when calculating the minimum variance and optimal weights.
`nocross`

: optional, integer, default = 0- If 1, only the diagonal terms of the covariance matrix Fij will be computed. If 0, all terms will be computed.

## Description

`SHMTVarOpt`

will determine the minimum variance that can be achieved by a weighted multitaper spectral analysis, as is described by Wieczorek and Simons (2007). The minimum variance is output as a function of the number of tapers utilized, from 1 to a maximum of `kmax`

, and the corresponding variance using equal weights is output for comparison. The windowing functions are assumed to be solutions to the spherical-cap concentration problem, as determined by a call to `SHReturnTapers`

or `SHReturnTapersM`

. The minimum variance and weights are dependent upon the form of the global unwindowed power spectrum, `Sff`

.

If the optional argument `nocross`

is set to 1, then only the diagnonal terms of `Fij`

will be computed.

## References

Wieczorek, M. A. and F. J. Simons, Minimum-variance multitaper spectral estimation on the sphere, J. Fourier Anal. Appl., 13, doi:10.1007/s00041-006-6904-1, 665-692, 2007.

## See also

shmtvar, shreturntapers, shreturntapersm, shmultitaperse, shmultitapercse; shlocalizedadmitcorr, shbiasadmitcorr, shbiask, shmtdebias

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