Calculate the minimum variance and corresponding optimal weights of a localized multitaper spectral estimate.
var_opt
, var_unit
, weight_opt
= SHMTVarOpt (l
, tapers
, taper_order
, sff
, [lwin
, kmax
, nocross
])
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.
l
: integer
: The angular degree to determine the 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. If this matrix was created using SHReturnTapersM
, then this array must be composed of zeros.
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.
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.
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.
shreturntapers, shreturntapersm, shmultitaperse, shmultitapercse; shlocalizedadmitcorr, shbiasadmitcorr, shbiask, shmtdebias