Calculate the theoretical minimum variance of a localized multitaper spectral estimate and the corresponding optimal weights to apply to each localized spectrum.
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 spherical harmonic used degree to determine the minimum variance and optimal weights.
tapers: input, real(dp), dimension (lwin+1,kmax)- A matrix of localization functions obtained from
SHReturnTapersorSHReturnTapersM. taper_order: input, integer(int32), dimension (kmax)- The angular order of the windowing coefficients in
tapers. lwin: input, integer(int32)- The spherical harmonic bandwidth of the localizing windows.
kmax: input, integer(int32)- The maximum number of tapers to be used when calculating the minimum variance and optimal weights.
sff: input, real(dp), dimension (l+lwin+1)- The global unwindowed power spectrum of the function to be localized.
var_opt: output, real(dp), dimension (kmax)- The minimum variance of the multitaper spectral estimate for degree
lusing 1 throughkmaxtapers. var_unit: output, real(dp), dimension (kmax)- The variance of the multitaper spectral estimate using equal weights for degree
lusing 1 throughkmaxtapers. weight_opt: optional, output, real(dp), dimension (kmax,kmax)- The optimal weights (in columns) that minimize the multitaper spectral estimate’s variance using 1 through
kmaxtapers. unweighted_covar: optional, output, real(dp), dimension (kmax,kmax)- The unweighted covariance matrix of the
kmaxtapers (i.e., Fij in Wieczorek and Simons 2007). nocross: optional, input, integer(int32), 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(int32)- 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.
See also
shmtvar, shreturntapers, shreturntapersm, shmultitaperse, shmultitapercse, shlocalizedadmitcorr, shbiasadmitcorr, shbiask, shmtdebias
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