In signal processing, the multitaper method is a technique developed by David J. Thomson to estimate the power spectrum SX of a stationary ergodic finite-variance random process X, given a finite contiguous realization of X as data.
Contents |
The method
Consider a p-dimensional zero mean stationary stochastic process
.
Here T denotes the matrix transposition. In Neurophysiology for example, p refers to the total number of channels and hence 
can represent simultaneous measurement of electrical activity of those p channels. Let the sampling interval between observations be Δt, so that the Nyquist frequency is fN = 1 / (2Δt).
The multitaper spectral estimator utilizes several different data tapers which are orthogonal to each other. The multitaper cross-spectral estimator between channel l and m is the average of K direct cross-spectral estimators between the same pair of channels (l and m) and hence takes the form
Here, 
(for 
) is the k th direct cross spectral estimator between channel l and m and is given by
where
The Slepian sequences
The sequence {ht,k} is the data taper for the k th direct cross-spectral estimator and is chosen as follows:
We choose a set of K orthogonal data tapers such that each one provides a good protection against leakage. These are given by the Slepian sequences, after David Slepian (also known in literature as discrete prolate spheroidal sequences or DPSS for short) with parameter W and orders k = 0 to K − 1. The maximum order K is chosen to be less than the Shannon number 2NWΔt. The quantity 2W defines the resolution bandwidth for the Spectral concentration problem and 
. When l = m, we get the multitaper estimator for the auto-spectrum of the l th channel.
Advantages of multitaper method
In conventional nonparametric spectral analysis techniques, to reduce variance, we break up the data into overlapping segments (as in Welch's overlapped segment method), estimate the cross-spectrum or power spectrum for each segment and then average over the segments. Such methods have severe bias problems for short data. In the multitaper method, for reducing variance we average over different tapers using the full data. Since the data length is not shortened, the bias is smaller.
Applications of multitaper method
This technique is currently used in the spectral analysis toolkit of Chronux. An extensive treatment about the application of this method to analyze multi-trial, multi-channel data generated in Neuroscience experiments, Biomedical Engineering and others can be found here. Not limited to time series, the multitaper method can be reformulated for spectral estimation on the sphere using spherical harmonics for applications in geophysics and cosmology among others.
References
Percival, D. B., and A. T. Walden. Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques. Cambridge: Cambridge University Press, 1993.
Dahlen, F. A., and F. J. Simons. Spectral estimation on a sphere in geophysics and cosmology. Geophysical Journal International, 2008, doi:10.1111/j.1365-246X.2008.03854.x
F. J. Simons, M. A. Wieczorek and F. A. Dahlen. Spatiospectral concentration on a sphere. SIAM Review, 2006, doi:10.1137/S0036144504445765
Slepian, D. "Prolate spheroidal wave functions, Fourier analysis, and uncertainty - V: The discrete case." Bell System Technical Journal, Volume 57 (1978), 1371–430.
Thomson, D. J. "Spectrum estimation and harmonic analysis." In Proceedings of the IEEE, Volume 70 (1982), 1055–1096.
Wieczorek, M. A. and F. J. Simons. Minimum-Variance Multitaper Spectral Estimation on the Sphere. Journal of Fourier Analysis and Applications, 2007, 10.1007/s00041-006-6904-1
Partha Mitra and Hemant Bokil. Observed Brain Dynamics, Oxford University Press, USA 2007, Link for the book
Partha Mitra and B. Pesaran, "Analysis of Dynamic Brain Imaging Data." The Biophysical Journal, Volume 76 (1999), 691-708, arxiv.org/abs/q-bio/0309028
See also
External links
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
