# SHMultiTaperMaskSE()

Perform a localized multitaper spectral analysis using arbitrary windows derived from a mask.

# Usage

mtse, sd = SHMultiTaperMaskSE (sh, tapers, [lmax,  lmaxt, k, taper_wt, norm, csphase])

# Returns

mtse : float dimension (lmax-lmaxt+1)
:   The localized multitaper power spectrum estimate. 

sd : float, dimension (lmax-lmaxt+1)
:   The standard error of the localized multitaper power spectral estimates.

# Parameters

sh : float, dimension (2, lmaxin+1, lmaxin+1)
:   The spherical harmonic coefficients of the function to be localized.

tapers : float, dimension ((lmaxtin+1)**2, kin)
:   An array of the k windowing functions, arranged in columns, obtained from a call to SHReturnTapersMap. The spherical harmonic coefficients are packed according to the conventions in SHCilmToVector.

lmax : optional, integer, default = lmaxin
:   The spherical harmonic bandwidth of sh. This must be less than or equal to lmaxin.

lmaxt : optional, integer, default = lmaxtin
:   The spherical harmonic bandwidth of the windowing functions in the array tapers.

k : optional, integer, default = kin
:   The number of tapers to be utilized in performing the multitaper spectral analysis.

taper_wt : optional, float, dimension (kin), default = -1
:   The weights used in calculating the multitaper spectral estimates and standard error. Optimal values of the weights (for a known global power spectrum) can be obtained from the routine SHMTVarOpt. The default value specifies not to use taper_wt.

norm : optional, integer, default = 1
:   1 (default) = 4-pi (geodesy) normalized harmonics; 2 = Schmidt semi-normalized harmonics; 3 = unnormalized harmonics; 4 = orthonormal harmonics.

csphase : optional, integer, default = 1
:   1 (default) = do not apply the Condon-Shortley phase factor to the associated Legendre functions; -1 = append the Condon-Shortley phase factor of (-1)^m to the associated Legendre functions.

# Description

SHMultiTaperMaskSE will perform a localized multitaper spectral analysis of an input function expressed in spherical harmonics using an arbitrary set of windows derived from a mask. The maximum degree of the localized multitaper cross-power spectrum estimate is lmax-lmaxt. The matrix tapers contains the spherical harmonic coefficients of the windows and can be obtained by a call to SHReturnTapersMap. The coefficients of each window are stored in a single column, ordered according to the conventions used in SHCilmToVector.

If the optional array taper_wt is specified, these weights will be used in calculating a weighted average of the individual k tapered estimates mtse and the corresponding standard error of the estimates sd. If not present, the weights will all be assumed to be equal. When taper_wt is not specified, the mutltitaper spectral estimate for a given degree is calculated as the average obtained from the k individual tapered estimates. The standard error of the multitaper estimate at degree l is simply the population standard deviation, S = sqrt(sum (Si - mtse)^2 / (k-1)), divided by sqrt(k). See Wieczorek and Simons (2007) for the relevant expressions when weighted estimates are used.

The employed spherical harmonic normalization and Condon-Shortley phase convention can be set by the optional arguments norm and csphase; if not set, the default is to use geodesy 4-pi normalized harmonics that exclude the Condon-Shortley phase of (-1)^m.

# 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.
