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Minimum bias multiple taper spectral estimation

Kurt S. Riedel, Alexander Sidorenko

arXiv:1803.04078v2stat.MEeess.ASeess.SPmath.STphysics.data-an

TL;DR

Multitaper spectral analysis previously relied mainly on Slepian tapers, motivating alternatives with controllable local bandwidth. This paper proposes minimum bias and sinusoidal tapers and analyzes their spectral properties. The sinusoidal tapers converge to the minimum bias tapers at rate 1/N, while bandwidth is controlled by the number of tapers used.

  • Problem

    Previously, practical multitaper analysis used Slepian tapers, motivating the introduction of other orthonormal taper families.

  • Method

    The paper proposes minimum bias tapers that minimize local frequency bias and analytic sinusoidal tapers that converge to them.

  • Results

    At rate 1/N, the sinusoidal tapers converge to the minimum bias tapers, whose estimate’s bandwidth is determined solely by the number of tapers used.

  • Takeaways & Limitations

    Adaptively adding or deleting minimum bias or sinusoidal tapers can produce locally adjustable multitaper estimates with kernel-smoother convergence properties.

  • Takeaways & Limitations

    The analysis assumes that the spectral density is twice continuously differentiable.

Abstract

from arXiv · show

Two families of orthonormal tapers are proposed for multi-taper spectral analysis: minimum bias tapers, and sinusoidal tapers $\{ \bf{v}^{(k)}\}$, where $v_n^{(k)}=\sqrt{\frac{2}{N+1}}\sin\frac{πkn}{N+1}$, and $N$ is the number of points. The resulting sinusoidal multitaper spectral estimate is $\hat{S}(f)=\frac{1}{2K(N+1)} \sum_{j=1}^K |y(f+\frac{j}{2N+2}) -y(f-\frac{j}{2N+2})|^2$, where $y(f)$ is the Fourier transform of the stationary time series, $S(f)$ is the spectral density, and $K$ is the number of tapers. For fixed $j$, the sinusoidal tapers converge to the minimum bias tapers like $1/N$. Since the sinusoidal tapers have analytic expressions, no numerical eigenvalue decomposition is necessary. Both the minimum bias and sinusoidal tapers have no additional parameter for the spectral bandwidth. The bandwidth of the $j$th taper is simply $\frac{1}{N}$ centered about the frequencies $\frac{\pm j}{2N+2}$. Thus the bandwidth of the multitaper spectral estimate can be adjusted locally by simply adding or deleting tapers. The band limited spectral concentration, $\int_{-w}^w |V(f)|^2 df$, of both the minimum bias and sinusoidal tapers is very close to the optimal concentration achieved by the Slepian tapers. In contrast, the Slepian tapers can have the local bias, $\int_{-1/2}^{1/2} f^2 |V(f)|^2 df$, much larger than of the minimum bias tapers and the sinusoidal tapers.

EDICS: SP 3.1.1

The supplied passages contain only a fragment mentioning N and the number of points, plus an acknowledgment of comments and research funding.

  • A fragment states that N is the number of points.The passage begins mid-sentence and provides no further methodological context.
  • The authors thank D. J. Thomson and the referees for useful comments.
  • The research was funded by the U.S. Department of Energy.

1 Introduction

The paper introduces minimum bias and sinusoidal orthonormal tapers as alternatives to routinely used Slepian tapers for multitaper spectral analysis. The sinusoidal tapers converge to the minimum bias family, while minimum-bias multitaper estimates minimize expected squared local error.

  • 1 Introduction: Multitaper spectral analysis constructs a rank-K estimate by averaging K spectral-density estimates formed from orthonormal tapers.Slepian tapers are identified as the family routinely used in practice.
  • 1 Introduction: The paper proposes two new orthonormal taper families: minimum bias tapers and sinusoidal tapers.The minimum bias tapers are designed to minimize local frequency bias.
  • 1 Introduction: Minimum bias tapers minimize the local frequency-bias measure ∫ f^2|V(f)|^2df subject to orthonormality constraints.For continuous time they have simple analytic expressions; for discrete time they satisfy a self-adjoint eigenvalue problem and may be computed numerically.
  • 1 Introduction: At rate 1/N, sinusoidal tapers converge to the minimum bias tapers as N →∞.The sinusoidal tapers form an orthonormal family.
  • 1 Introduction: The minimum-bias multitaper estimate minimizes expected squared local error, and fewer tapers should be used where the spectral density changes rapidly.The paper also determines the optimal number of tapers through local error analysis.

2 Quadratic Estimators of the Power Spectrum

Quadratic, modulation-invariant power-spectrum estimators can be represented through symmetric matrices and their orthogonal eigenvectors. Multiple-taper estimation provides a low-rank approximation while motivating taper families selected for desirable spectral localization.

  • Quadratic estimator formulation: Every quadratic, modulation-invariant power-spectrum estimator has a representation involving a symmetric order-N matrix Q.
  • Multitaper representation: The rank K of Q determines an orthogonal eigenvector system whose associated tapered periodograms form the multitaper estimator.
  • Multitaper representation: Multiple-taper spectral estimation is a low-rank, principal-components approximation of a general quadratic estimator, with K normally much less than N.
  • Taper-family selection: Practice specifies orthonormal taper families with desirable properties rather than beginning from a given quadratic estimator; previously, Slepian tapers were used.
  • Frequency localization: Taper spectral windows are Fourier transforms of the tapers and are generally chosen to localize spectral density near zero frequency.
  • Frequency localization: Slepian tapers uniquely maximize concentration within [-w, w] for an orthonormal family, but depend on the free bandwidth parameter w.

3 Minimum Bias Tapers

Minimum bias tapers minimize the leading-order local bias, yielding sine tapers in continuous time and eigenvector-defined tapers in discrete time. Discrete sinusoidal tapers provide an orthonormal, analytically tractable approximation with low-bias multitaper estimates enabled by sidelobe cancellation.

  • Continuous-time minimum bias tapers: The continuous-time minimum-bias tapers are ν^(k)(t) = √2 sin(πkt), uniquely satisfying the stated orthogonality and minimum-bias requirements.Their Fourier-transform magnitude decays as f^-2 at large frequencies.
  • Discrete minimum bias tapers: The discrete minimum-bias tapers are eigenvectors of matrix A, ordered by increasing eigenvalues corresponding to their local-bias integrals.This construction follows from minimizing local bias within successively orthogonal subspaces.
  • Multitaper estimate: The uniformly weighted K-taper sinusoidal estimate has local bias K^2/(12N^2) + O(K^2…), while its frequency-domain representation uses paired Fourier evaluations around f.The paired terms arise from the sinusoidal multitaper representation in Equation (9).
  • Multitaper estimate: Paired Fourier sidelobes cancel, making the sinusoidal estimate’s sidelobe much smaller than the periodogram’s and explaining its low bias.The cancellation compares y(f + j/[2N+2]) with y(f − j/[2N+2]).
  • Weighting: Parabolic weighting, μ_j = C(1 − j^2/K^2), is preferred because it minimizes expected squared error asymptotically and produces a frequency-smooth estimate.The weights decrease smoothly to zero as taper index increases.

4 Comparison of Spectral Localizations

Sinusoidal tapers nearly match minimum-bias tapers in local bias and spectral concentration, while Slepian tapers can have substantially larger local bias except near k ≈ 1.2Nw. The comparison also shows that Slepian tapers offer better broad-band protection for k ≪ 2Nw, whereas minimum-bias tapers prevail near k ∼ 2Nw.

  • Local bias: The sinusoidal tapers come within 0.2% of the optimal local bias, whereas Slepian tapers can have many times larger local bias.This comparison uses N = 50.
  • Local bias: The kth Slepian taper has roughly the same local bias as the minimum-bias taper when Nw < k < 2Nw, with the smallest ratio near k ≈ 1.2Nw.As |k − 1.2Nw| increases, the local bias rapidly departs from the optimal value.
  • Spectral concentration: Both minimum-bias and sinusoidal tapers are within 1.7% of optimal spectral concentration except for k = 8, 9 when N = 50 and w = .08.Here, 2Nw = 8.
  • Spectral concentration: The concentration ratio can be close to one even when the ratio of spectral energy outside |f| < w is quite large.The conclusions therefore depend on using band-limited concentration as the frequency-concentration measure.
  • Overall comparison: Sinusoidal tapers perform nearly as well as minimum-bias tapers, while Slepian tapers provide better broad-band protection for k ≪ 2Nw and minimum-bias tapers are better near k ∼ 2Nw.The latter advantage is attributed to the Gibbs phenomena experienced by Slepian tapers.

5 Local Error Analysis and Optimal Multitapering

The section develops a local error analysis for multitaper spectral estimation and identifies minimum bias tapers as optimal for the local loss. For sinusoidal tapers, the optimal taper count scales as N^4/5 and decreases when the spectrum varies more rapidly.

  • Local bias: When the taper weights do not sum to one, the multitaper estimate has bias even for white noise.The condition is explicitly identified as Σ_k μ_k ≠ 1.
  • Optimal tapers: Minimum bias tapers minimize the local loss among multitaper constructions with nonnegative weights.This result is stated for the multitaper estimate under the local-loss criterion.
  • Sinusoidal tapers: The uniformly weighted sinusoidal multitaper estimate has an asymptotic local loss whose minimizer determines the optimal number of tapers.The asymptotic local-loss result is given in Theorem 5.3 and its minimizing taper count in Corollary 5.4.
  • Optimal taper count: N^4/5 is the scaling of the optimal taper count, which varies with the ratio of S(f) to S′′(f).Fewer tapers are recommended when the spectrum varies more rapidly.
  • Practical implications: Changing the number of sinusoidal or minimum bias tapers requires no taper recomputation, unlike Slepian tapers whose efficiency depends on K ∼ 2Nw.The section contrasts this flexibility with the Slepian requirement to change the bandwidth parameter when K changes.

6 Smoothed Multitaper Estimates

Kernel smoothing cannot outperform pure minimum-bias multitaper estimation for the spectrum itself, but combining multitapering with smoothing improves log-spectral estimation. The proposed bias-corrected sinusoidal-taper approach has diminishing logarithmic variance inflation and yields optimized smoothing and taper scalings.

  • Spectrum estimation: Kernel smoothing transforms the quadratic multitaper estimator into another quadratic estimator that cannot outperform pure multitapering with minimum-bias tapers.The smoothed estimator is represented by a modified quadratic-estimator matrix.
  • Log-spectrum estimation: The proposed hybrid computes a sinusoidal multitaper estimate, then smooths its bias-corrected logarithm using BK = ψ(K) − ln K.The correction removes the bias of ln[χ²_2K] before smoothing.
  • Log-spectrum estimation: For white noise, the log-spectral estimate has variance ψ′(K) ∼ 1/K + 1/(2K²), so logarithmic variance inflation rapidly vanishes.The variance enhancement from the logarithm therefore tends to zero as K increases.
  • Asymptotic optimization: Under 1 ≪ K ≪ Nw with uniformly weighted sinusoidal tapers, optimizing the asymptotic error gives w ∼ N^-1/5 and K ∼ N^8/15.The resulting smoothing halfwidth is much larger than K/N.
  • Adaptive smoothing: With K = N^8/15, the locally optimal halfwidth scales as wopt ∼ |θ′′(f)|^-2/5 N^-1/5, decreasing where the log-spectrum varies rapidly.Two-stage estimators first estimate θ′′(f) and then use a variable-halfwidth kernel; these schemes converge at N^-4/5 with relative convergence at least N^-2/9.

7 Application

An application to microwave scattering data from turbulent plasma fluctuations evaluates locally adaptive multitaper spectral estimates across a spectrum spanning over five orders of magnitude. Against a converged N = 45,000 reference, the sinusoidal taper estimate is more accurate and requires less CPU time than the Slepian estimate.

  • Application: The microwave scattering spectrum contains a quasicoherent 1 MHz peak and varies by over five orders of magnitude.The data measure turbulent plasma fluctuations in the Tokamak Fusion Test Reactor at Princeton.
  • Application: Fewer tapers are used near the peak, while parabolic taper weighting smooths the spectral estimate.The local taper count is determined with a multiple-stage plug-in method using a pre-estimate on the same data.
  • Application: Against a converged N = 45,000 spectrum, the sinusoidal taper estimate is more accurate and requires less CPU time than the Slepian multitaper estimate.The comparison uses a converged spectral estimate based on N = 45,000.

8 Conclusion

The section concludes that minimum bias and sinusoidal tapers provide low-bias multitaper spectral estimation, with sinusoidal tapers offering a simple analytic construction. Their bandwidth is controlled solely by the number of tapers, enabling adaptive local adjustment.

  • Conclusion: Sinusoidal tapers have low bias because sidelobes from paired frequency-shifted transforms cancel.This cancellation is built into the sinusoidal multitaper construction.
  • Conclusion: Minimum bias tapers minimize local bias, while the resulting multitaper estimate minimizes expected local squared error asymptotically.The conclusion also describes good broad-band bias protection.
  • Conclusion: Sinusoidal tapers have a simple analytic form, avoiding the need for numerical eigenvalue decomposition.Their analytic construction is presented as a practical advantage over numerically derived tapers.
  • Conclusion: The minimum bias and sinusoidal tapers have no auxiliary bandwidth parameter; spectral-estimate bandwidth depends solely on the number of used tapers.Adding or deleting tapers adaptively can produce an estimate with the convergence properties of kernel smoothers.

9 Appendix: Multitaper decomposition of kernel estimates

The appendix shows that kernel-smoothed spectral estimates can be represented by multitapers closely resembling the minimum bias and sinusoidal tapers. For the parabolic kernel, this correspondence is exact, while a tapered, locally smoothed example is virtually a K = 4 multitaper estimate.

  • Kernel–multitaper correspondence: Kernel smoother estimators have an equivalent multitaper representation, with equivalent tapers that strongly resemble minimum bias and sinusoidal tapers.In one special case, the smoothed periodogram decomposes exactly into minimum bias tapers.
  • Parabolic kernel: The eigenvectors of the parabolic-kernel smoothed periodogram are exactly the discrete minimum bias tapers.The relevant matrices share the same eigenvectors because the kernel-derived matrix is a linear combination of the identity and the matrix from Lemma 3.2.
  • Tapered smoothed example: For N = 200 with a Tukey split-cosine taper and square-box smoothing of halfwidth .01, the first 4 eigenvectors are very close to sinusoids.The corresponding matrix’s eigenvectors were computed for this tapered, locally smoothed periodogram.
  • Tapered smoothed example: After k > 4, the eigenvalues decrease sharply, so higher eigenvectors contribute very little to the overall estimate.The resulting spectral estimate is virtually a K = 4 multiple taper spectral estimate.

Table Captions:

The section’s tables and figures document taper convergence, normalized bias, spectral concentration, and eigenvectors of smooth tapered periodogram estimators. Additional figures compare spectral energy and estimated spectral densities for sinusoidal, minimum bias, and Slepian tapers.

  • Spectral energy: Figures 1–3 show spectral energy for minimum bias, sinusoidal, and Slepian tapers, including N = 200, k = 1, w = .01 and K = 3.The captions specify the taper comparisons and parameter settings for the displayed spectral-energy results.
  • Estimated spectral density: Figure 4 compares sinusoidal and Slepian multitaper spectral-density estimates for plasma fluctuations with w = 60 kHz.The caption states that the dashed line is the sinusoidal estimate and the solid line is the Slepian estimate.
  • Taper convergence: Table 1 reports convergence of the sinusoidal tapers to the minimum bias tapers.The convergence is identified in both the table caption and the corresponding paragraph caption.
  • Spectral concentration: Table 3 reports spectral concentration for the tapers.The caption names spectral concentration without providing numerical values.
  • Smooth tapered periodogram: Tables 4 and 4 reports eigenvectors of the smooth tapered periodogram estimator.The supplied captions identify eigenvectors as the subject of the tabulated results.
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