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shmtvaropt.doc
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shmtvaropt.doc
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Calculate the theoretical minimum variance of a localized multitaper spectral
estimate and the corresponding optimal weights to apply to each localized
spectrum. This routine only works using the tapers of the spherical cap
concentration problem.
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.