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shmtvaropt0.1
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shmtvaropt0.1
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.\" ========================================================================
.\"
.IX Title "SHMTVAROPT0 1"
.TH SHMTVAROPT0 1 "2014-09-12" "SHTOOLS 3.0" "SHTOOLS 3.0"
.\" For nroff, turn off justification. Always turn off hyphenation; it makes
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.SH "SHMTVarOpt0"
.IX Header "SHMTVarOpt0"
.IP "SHMTVarOpt0 \-" 14
.IX Item "SHMTVarOpt0 -"
Calculate the minimum variance and corresponding optimal weights of a localized multitaper spectral estimate using zonal tapers.
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
.IP "\s-1SUBROUTINE\s0 SHMTVarOpt0 (" 25
.IX Item "SUBROUTINE SHMTVarOpt0 ("
L, \s-1TAPERS\s0, \s-1LWIN\s0, \s-1KMAX\s0, \s-1SFF\s0, \s-1VAR_OPT\s0, \s-1VAR_UNIT\s0, \s-1WEIGHT_OPT\s0, \s-1UNWEIGHTED_COVAR\s0 \s-1NOCROSS\s0 )
.RS 4
.IP "REAL*8" 19
.IX Item "REAL*8"
\&\s-1TAPERS\s0(\s-1LWIN+1\s0, \s-1KMAX\s0), \s-1SFF\s0(L+WIN+1), \s-1VAR_OPT\s0(\s-1KMAX\s0), \s-1VAR_UNIT\s0(\s-1KMAX\s0)
.IP "\s-1INTEGER\s0" 19
.IX Item "INTEGER"
L, \s-1LWIN\s0, \s-1KMAX\s0
.IP "REAL*8, \s-1OPTIONAL\s0" 19
.IX Item "REAL*8, OPTIONAL"
\&\s-1WEIGHT_OPT\s0(\s-1KMAX\s0, \s-1KMAX\s0), \s-1UNWEIGHTED_COVAR\s0(\s-1KMAX\s0,\ \s-1KMAX\s0)
.IP "\s-1INTEGER\s0, \s-1OPTIONAL\s0" 19
.IX Item "INTEGER, OPTIONAL"
\&\s-1NOCROSS\s0
.RE
.RS 4
.RE
.SH "DESCRIPTION"
.IX Header "DESCRIPTION"
\&\fISHMTVarOpt0\fR will determine the minimum variance that can be achieved by a weighted multitaper spectral analysis using zonal tapers, as is described by Wieczorek and Simons (2006, submitted manuscript). The minimum variance is output as a function of the number of tapers utilized, from 1 to a maximum of \s-1KMAX\s0, 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 \fISHReturnTapersM\fR. The minimum variance and weights are dependent upon the form of the global unwindowed power spectrum, \s-1SFF\s0.
.PP
If the optional argument \s-1WEIGHT_OPT\s0 is specified, then the optimal weights will be returned as a function of the number of tapers employed, from 1 \s-1KMAX\s0. If \s-1UNWEIGHTED_COVAR\s0 is specified, then the unweighted covariance matrix of the \s-1KMAX\s0 tapers (i.e., F_ij) will be output. If the optional argument \s-1NOCROSS\s0 is set to 1, then only the diagnonal terms of Fij will be computed.
.SH "ARGUMENTS"
.IX Header "ARGUMENTS"
.IP "L" 18
.IX Item "L"
(input) \s-1INTEGER\s0
.Sp
The angular degree to determine the minimum variance and optimal weights.
.IP "\s-1TAPERS\s0" 18
.IX Item "TAPERS"
(input) REAL*8, \s-1DIMENSION\s0 (\s-1LWIN+1\s0, \s-1KMAX\s0)
.Sp
A matrix of localization functions obtained from \fISHReturnTapersM\fR.
.IP "\s-1LWIN\s0" 18
.IX Item "LWIN"
(input) \s-1INTEGER\s0
.Sp
The spherical harmonic bandwidth of the localizing windows.
.IP "\s-1KMAX\s0" 18
.IX Item "KMAX"
(input) \s-1INTEGER\s0
.Sp
The maximum number of tapers to be used when calculating the minimum variance and optimal weights. All values from 1 to \s-1KMAX\s0 will be returned.
.IP "\s-1SFF\s0" 18
.IX Item "SFF"
(input) REAL*8, \s-1DIMENSION\s0 (L+LWIN+1)
.Sp
The global unwindowed power spectrum of the function to be localized.
.IP "\s-1VAR_OPT\s0" 18
.IX Item "VAR_OPT"
(output) REAL*8, \s-1DIMENSION\s0 (\s-1KMAX\s0)
.Sp
The minimum variance of the multitaper spectral estimate for degree L using 1 through \s-1KMAX\s0 tapers.
.IP "\s-1VAR_UNIT\s0" 18
.IX Item "VAR_UNIT"
(output) REAL*8, \s-1DIMENSION\s0 (\s-1KMAX\s0)
.Sp
The variance of the multitaper spectral estimate using equal weights for degree L using 1 through \s-1KMAX\s0 tapers.
.IP "\s-1WEIGHT_OPT\s0" 18
.IX Item "WEIGHT_OPT"
(output) REAL*8, \s-1OPTIONAL\s0, \s-1DIMENSION\s0 (\s-1KMAX\s0, \s-1KMAX\s0)
.Sp
The optimal weights (in columns) that minimize the multitaper spectral estimate's variance using 1 through \s-1KMAX\s0 tapers.
.IP "\s-1UNWEIGHTED_COVAR\s0" 18
.IX Item "UNWEIGHTED_COVAR"
(output) REAL*8, \s-1OPTIONAL\s0, \s-1DIMENSION\s0 (\s-1KMAX\s0, \s-1KMAX\s0)
.Sp
The unweighted covariance matrix of the \s-1KMAX\s0 tapers (i.e., Fij in Wieczorek and Simons 2007).
.IP "\s-1NOCROSS\s0" 18
.IX Item "NOCROSS"
(input) \s-1INTEGER\s0, \s-1OPTIONAL\s0
.Sp
If 1, only the diagonal terms of the covariance matrix Fij will be computed. If 0, all terms will be computed.
.SH "SEE ALSO"
.IX Header "SEE ALSO"
\&\fIshreturntapers\fR\|(1), \fIshreturntapersm\fR\|(1), \fIshmtvaropt\fR\|(1)
.PP
<http://shtools.ipgp.fr/>
.SH "REFERENCES"
.IX Header "REFERENCES"
Wieczorek, M. A. and F. J. Simons, Minimum variance multitaper spectral estimation on the sphere, \fIJ. Fourier Anal. Appl.\fR, submitted 2006.
.SH "COPYRIGHT AND LICENSE"
.IX Header "COPYRIGHT AND LICENSE"
Copyright 2012 by Mark Wieczorek <wieczor@ipgp.fr>.
.PP
This is free software; you can distribute and modify it under the terms of the revised \s-1BSD\s0 license.