Calculate the minimum variance and corresponding optimal weights of a localized multitaper spectral estimate.

## Usage

call SHMTVarOpt (l, tapers, taper_order, lwin, kmax, sff, var_opt, var_unit, weight_opt, unweighted_covar, nocross, exitstatus)

## Parameters

l : input, integer
The angular degree to determine the minimum variance and optimal weights.
tapers : input, real*8, dimension (lwin+1, kmax)
A matrix of localization functions obtained from SHReturnTapers or SHReturnTapersM.
taper_order : input, integer, dimension (kmax)
The angular order of the windowing coefficients in TAPERS. If this matrix was created using SHReturnTapersM, then this array must be composed of zeros.
lwin : input, integer
The spherical harmonic bandwidth of the localizing windows.
kmax : input, integer
The maximum number of tapers to be used when calculating the minimum variance and optimal weights.
sff : input, real*8, dimension (l+lwin+1)
The global unwindowed power spectrum of the function to be localized.
var_opt : output, real*8, dimension (kmax)
The minimum variance of the multitaper spectral estimate for degree l using 1 through kmax tapers.
var_unit : output, real*8, dimension (kmax)
The variance of the multitaper spectral estimate using equal weights for degree l using 1 through kmax tapers.
weight_opt : optional, output, real*8, dimension (kmax, kmax)
The optimal weights (in columns) that minimize the multitaper spectral estimate’s variance using 1 through kmax tapers.
unweighted_covar : optional, output, real*8, dimension (kmax, kmax)
The unweighted covariance matrix of the kmax tapers (i.e., Fij in Wieczorek and Simons 2007).
nocross : optional, input, integer, default = 0
If 1, only the diagonal terms of the covariance matrix Fij will be computed. If 0, all terms will be computed.
exitstatus : output, optional, integer
If present, instead of executing a STOP when an error is encountered, the variable exitstatus will be returned describing the error. 0 = No errors; 1 = Improper dimensions of input array; 2 = Improper bounds for input variable; 3 = Error allocating memory; 4 = File IO error.

## 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 weight_opt is specified, then the optimal weights will be returned as a function of the number of tapers employed, from 1 to kmax. If unweighted_covar is specified, then the unweighted covariance matrix of the kmax tapers (i.e., Fij) will be output. 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.