This module provides routines for performing global spectral analyses, the construction of spatiospectral localization windows, and localized multitaper spectral analyses.

Global spectral analysis

Function name Description
spectrum Calculate the spectrum of a real or complex function.
cross_spectrum Calculate the cross-spectrum of a real or complex function.
SHAdmitCorr Calculate the admittance and correlation spectra of two functions.
SHConfidence Compute the probability that two sets of spherical harmonic coefficients are correlated at a given degree and for a given correlation coefficient.

Multitaper spectral estimation (spherical cap domain)

Function name Description
SHMultiTaperSE Perform a localized multitaper spectral analysis.
SHMultiTaperCSE Perform a localized multitaper cross-spectral analysis.
SHLocalizedAdmitCorr Calculate the localized admittance and correlation spectra of two functions at a given location.
SHReturnTapers Calculate the eigenfunctions of the spherical-cap concentration problem.
SHReturnTapersM Calculate the eigenfunctions of the spherical-cap concentration problem for a single angular order.
ComputeDm Compute the space-concentration kernel of a spherical cap.
ComputeDG82 Compute the tridiagonal matrix of Gr├╝nbaum et al. (1982) that commutes with the space-concentration kernel of a spherical cap.
SHFindLWin Determine the spherical-harmonic bandwidth that is necessary to achieve a certain concentration factor.
SHBiasK Calculate the multitaper (cross-)power spectrum expectation of a windowed function.
SHMTCouplingMatrix Calculate the multitaper coupling matrix for a given set of localization windows.
SHBiasAdmitCorr Calculate the expected multitaper admittance and correlation spectra associated with the input global cross-power spectra of two functions.
SHMTDebias Invert for the global power spectrum given a localized multitaper spectrum estimate.
SHMTVarOpt Calculate the theoretical minimum variance of a localized multitaper spectral estimate and the corresponding optimal weights to apply to each localized spectrum.
SHMTVar Calculate the theoretical variance of a multitaper spectral estimate for a given input power spectrum.
SHSjkPG Calculate the expectation of the product of two functions, each multiplied by a different data taper, for a given spherical harmonic degree and two different angular orders.
SHRotateTapers Rotate orthogonal spherical-cap Slepian functions centered at the North pole to a different location.

Localization windows (arbitrary domain)

Function name Description
SHReturnTapersMap Calculate the eigenfunctions of the concentration problem for an arbitrary concentration region.
SHBiasKMask Calculate the multitaper (cross-)power spectrum expectation of a function localized by arbitrary windows derived from a mask.
SHMultiTaperMaskSE Perform a localized multitaper spectral analysis using arbitrary windows.
SHMultiTaperMaskCSE Perform a localized multitaper cross-spectral analysis using arbitrary windows.
ComputeDMap Compute the space-concentration kernel of a mask defined on the sphere.
Curve2Mask Given a set of latitude and longitude coordinates representing a closed curve, output a gridded mask.

Localization bias (general)

Function name Description
SHBias Calculate the (cross-)power spectrum expectation of a windowed function.

Slepian function expansions

Routine name Description
SlepianCoeffs Determine the expansion coefficients of a function for a given set of input Slepian functions.
SlepianCoeffsToSH Convert a function expressed in Slepian coefficients to spherical harmonic coefficients.
SHSCouplingMatrix Compute the spherical harmonic coupling matrix for a given set of Slepian functions.
SHSCouplingMatrixCap Compute the spherical harmonic coupling matrix for a given set of spherical-cap Slepian functions.
SHSlepianVar Calculate the theoretical variance of the power of a function expanded in spherical-cap Slepian functions.

Other routines

Function name Description
SphericalCapCoef Calculate the spherical harmonic coefficients of a spherical cap.

References

  • Grunbaum, F. A., L. Longhi, and M. Perlstadt, Differential operators commuting with finite convolution integral operators: some non-abelian examples, SIAM J. Appl. Math., 42, 941-955, doi:10.1137/0142067, 1982.
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