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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 had relied in practice on Slepian tapers; this paper introduces minimum-bias and sinusoidal alternatives. It reports that, for fixed j, sinusoidal tapers converge to minimum-bias tapers at rate 1/N.

  • Problem

    In practice, Slepian tapers were the only family used, motivating the introduction of alternative orthonormal taper families.

    Source: §2 · p. 4

  • Method

    The paper introduces minimum-bias tapers that minimize local frequency bias and a sinusoidal orthonormal taper family.

    Source: §1 · p. 2

  • Results

    At rate 1/N, sinusoidal tapers converge to minimum-bias tapers as N increases.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The tapers require no auxiliary bandwidth parameter, and estimate bandwidth can be adjusted locally by adding or deleting tapers.

    Source: §8 · p. 15

    The analysis assumes the spectral density is twice continuously differentiable.

    Source: §2 · p. 3

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 excerpt is fragmentary and only identifies N as the number of points.

1 Introduction

The paper proposes minimum-bias and sinusoidal taper families for multitaper spectral analysis, then develops their error properties and applications. It also connects kernel smoothing to multitaper estimates and reports advantages when variable bandwidth is needed.

Multitaper framework

  • Multitaper spectral analysis averages K estimates from orthonormal tapers, and Slepian tapers [17] are the family routinely used in practice.The rank-K quadratic estimator can be represented as a weighted sum of K orthonormal rank-one estimators; Thomson originally proposed this class as multiple taper spectral analysis.Source: §1 · p. 2

Minimum-bias tapers

  • Minimum-bias tapers minimize local frequency bias subject to orthonormality, and the paper identifies weighted minimum-bias multitapering as the estimator minimizing expected squared local error.For continuous time, minimum-bias tapers have simple analytic expressions, with Papoulis’ optimal taper first; in discrete time, they satisfy a self-adjoint eigenvalue problem and may be computed numerically.Source: §1 · pp. 2–3

Sinusoidal tapers

  • Discrete sinusoidal tapers form an orthonormal family and converge to minimum-bias tapers at rate 1/N as N tends to infinity.The paper compares their local bias and band-limited concentration with those of Slepian tapers.Source: §1 · p. 2

Local error analysis

  • At frequencies where spectral density changes rapidly, fewer tapers should be used; the paper also determines the optimal taper count through local error analysis.The paper presents a local error analysis for the weighted minimum-bias multitaper estimate.Source: §1 · p. 3

Real-data application

  • The paper reports that its tapers outperform Slepian tapers whenever variable bandwidth is needed.It applies the spectral estimation techniques to real data.Source: §1 · p. 3

Kernel smoothing and adaptation

  • Kernel smoother spectral estimates are multitaper estimates, and smoothing the logarithm of a multitaper estimate significantly reduces variance versus smoothing the logarithm of a single-taper estimate.The paper also describes a data-adaptive method for estimating the spectrum.Source: §1 · p. 3

Kernel smoother connection

  • The leading principal components of kernel smoother spectral estimates resemble the minimum-bias tapers.Source: §1 · p. 3

2 Quadratic Estimators of the Power Spectrum

The section frames multitaper analysis as a low-rank, eigenvector-based form of quadratic spectral estimation and motivates choosing orthonormal taper families by their frequency-localization properties.

2 Quadratic Estimators of the Power Spectrum

  • Every quadratic, modulation-invariant power-spectrum estimator has a representation through a symmetric N×N matrix Q whose rank-K eigenvectors yield a multitaper representation.Source: §2 · pp. 3–4
  • The spectral inverse problem is to estimate S(f) from measurements of a stationary process, assuming S(f) is twice continuously differentiable.Source: §2 · p. 3
  • A multitaper estimate combines K orthogonal tapered periodograms and acts as a low-rank, “principal components” approximation; typically K≪N.Source: §2 · p. 4
  • In practice, analysts specify orthonormal tapers with desirable properties rather than starting from Q, motivating alternatives to the previously used Slepian family.Source: §2 · p. 4
  • Frequency localization is assessed through spectral concentration and local bias; local bias is proportional to the leading-order bias term as N→∞.Source: §2 · p. 4
  • Slepian tapers uniquely maximize spectral concentration among orthonormal families for a chosen bandwidth w; the first 2Nw windows concentrate within [−w,w], and later windows outside.Source: §2 · pp. 4–5

3 Minimum Bias Tapers

The section defines tapers that minimize local bias: the continuous-time solution is the first sine function, while discrete-time minimum-bias tapers are eigenvectors that can be approximated by sinusoidal tapers. The sinusoidal tapers yield a low-bias multitaper estimate through cancellation of frequency sidelobes.

3.1 Continuous Time Case

  • The continuous-time taper √2 sin(πt) minimizes the leading-order local bias, and the successive minimum-bias tapers are √2 sin(πkt).These successive tapers minimize the local-bias criterion in the subspace orthogonal to the preceding tapers; their frequency-window magnitude decays as f^-2 at large frequencies.Source: §3.1 · pp. 5–6

3.2 Discrete Time Case

  • Discrete minimum-bias tapers are eigenvectors of a matrix, ordered by increasing eigenvalue, with each eigenvalue equal to that taper’s minimum local bias.The eigenvector characterization gives the minimum in the subspace orthogonal to earlier tapers.Source: §3.2 · p. 7
  • Sinusoidal tapers form an orthonormal basis, with local bias k^2/(4N^2)(1 + O(1/N)); their K-taper estimate has bias Σ(k^2/(4N^2)) + O(K^2/N^3).The estimate also has a representation as a weighted sum of squared differences between Fourier transforms evaluated at f ± j/(2N+2).Source: §3.2 · p. 7
  • The sinusoidal estimate has low bias because its paired frequency sidelobes cancel, leaving a smaller sidelobe than the periodogram.The cancellation is between the Fourier-transform terms at f + j/(2N+2) and f − j/(2N+2).Source: §3.2 · p. 7
  • Parabolic weights μ_j = C(1 − j^2/K^2) are preferred because they minimize expected squared error asymptotically and yield an estimate smooth in frequency.Uniformly weighted sinusoidal multitaper estimates can be computed in O(N ln N) + O(KN) operations, versus O(KN ln N) operations plus taper computation for a generic estimate.Source: §3.2 · pp. 7–8
  • For all k, the sinusoidal tapers converge to discrete minimum-bias tapers at rate k/[4(N+2)] in L2, with the same rate in L∞.For N = 200 and k = 1, their spectral energies are nearly identical for |f| < 0.25, while near Nyquist the sinusoidal taper’s energy is roughly three times larger.Source: §3.2 · p. 8

4 Comparison of Spectral Localizations

Sinusoidal tapers closely match minimum-bias tapers in local bias and band-limited spectral concentration, while Slepian tapers can have much larger local bias. The comparisons vary with taper index and frequency range, and the concentration conclusion depends on measuring energy inside the target band.

4 Comparison of Spectral Localizations

  • Sinusoidal tapers come within 0.2% of optimal local bias, while Slepian tapers can have local bias many times larger than minimum-bias tapers.The section's summary notes an exception for Slepian tapers near k ≈ 1.2Nw.Source: §4 · pp. 9–10
  • For Slepian tapers, local bias is roughly comparable to minimum-bias tapers when Nw < k < 2Nw, with the smallest ratio near k ≈ 1.2Nw.As |k − 1.2Nw| increases, the bias ratio rapidly departs from its optimal value; Slepian tapers have better broad-band protection for k ≪ 2Nw, while minimum-bias tapers have better protection near k ∼ 2Nw.Source: §4 · pp. 9–10
  • For N = 50 and w = .08, minimum-bias and sinusoidal tapers are within 1.7% of optimal spectral concentration except at k = 8, 9.This conclusion uses concentration inside |f| < w; the ratio of energy outside that band can be quite large.Source: §4 · p. 9
  • The minimum-bias taper has a more concentrated central peak than the Slepian taper, while Slepian performs better near f ∼ w and minimum-bias has lower bias for f ∼ O(1).The passage attributes Slepian's middle-frequency advantage to maximizing in-band energy and its slower 1/f decay to Gibbs phenomenon; minimum-bias windows decay as 1/f^2.Source: §4 · p. 9
  • For |f| < 0.3w and |f| > 0.13, the minimum-bias taper has smaller range bias, whereas Slepian is better in the middle frequency range.For larger frequencies, the multitaper spectral energy is very similar to the single-taper comparison on a logarithmic scale when K << N.Source: §4 · pp. 9–10

5 Local Error Analysis and Optimal Multitapering

The section analyzes local multitaper error for Gaussian processes away from the frequency endpoints, identifies minimum-bias tapers as locally loss-optimal, and derives an optimal taper count for uniformly weighted sinusoidal tapers.

Local Error Analysis and Optimal Multitapering

  • The local error analysis assumes a Gaussian process and excludes frequencies near f = 0 and f = 1/2.Theorem 5.1 concerns the expected square error of the multitaper spectral estimate under these conditions.Source: §5 · p. 10
  • When the weights satisfy Σ_k µ_k ≠ 1, the multitaper estimate has bias even for white noise.Source: §5 · p. 10
  • For nonnegative weights, minimum-bias tapers construct the multitaper estimate that minimizes local loss.Theorem 5.2 states this optimality result.Source: §5 · p. 10
  • The optimal sinusoidal taper count is proportional to N^4/5 and varies with S(f)/S′′(f).Fewer tapers should be used when the spectrum varies more rapidly; changing K does not require recomputing minimum-bias or sinusoidal tapers, unlike Slepian tapers, which should be recomputed when K changes.Source: §5 · p. 11

6 Smoothed Multitaper Estimates

Kernel smoothing cannot improve the spectrum estimate beyond pure multitapering with minimum-bias tapers, but smoothing bias-corrected log multitaper estimates is useful for estimating the log-spectrum. The approach uses sinusoidal tapers and bandwidths adapted to log-spectrum curvature.

6 Smoothed Multitaper Estimates

  • An earlier comparison of kernel smoothing and multitaper estimation found that smoothed multiple-taper estimation worked best.Source: §6 · p. 12
  • Smoothing the quadratic multitaper spectrum cannot outperform the pure multitaper method with minimum-bias tapers.Kernel smoothing replaces the original quadratic estimator with another quadratic estimator.Source: §6 · p. 12
  • Smoothing log tapered periodograms inherits bias from the long lower tail of log[χ^2_2] and variance inflation of π^2/6, while smoothing the spectrum before taking logs is more sensitive to nonlocal bias.Source: §6 · p. 12
  • The hybrid estimate uses sinusoidal multitapers with μ_k = 1/K, then smooths ln[Ŝ_MT(f)] − B_K/K; for white noise, its variance is ψ′(K) ~ 1/K + 1/(2K²).B_K = ψ(K) − ln K is the bias of ln[χ²_(2K)].Source: §6 · pp. 12–13
  • Optimizing the asymptotic error gives w ~ N^-1/5 and K ~ N^8/15; the two (1 + 1/(2K)) factors reflect variance enhancement from the logarithm and sinusoidal-taper variance calculation.The expected error depends weakly on K when 1 ≪ K ≪ Nw.Source: §6 · p. 13
  • The optimal smoothing halfwidth scales as |θ′′(f)|^-2/5 N^-1/5, so it should be reduced where the log-spectrum varies rapidly.Because θ′′(f) is unknown, a preliminary estimate can set a variable halfwidth; these multistage estimators have convergence rate N^-4/5 and relative convergence rate at least N^-2/9.Source: §6 · p. 13

7 Application

On turbulent-plasma data, the sinusoidal multitaper estimate was more accurate than the converged-spectrum comparison and required less CPU time than the Slepian estimate. Local taper counts adapted the estimate to spectral shape, resolving the 1 MHz peak while reducing high-frequency bumps.

7 Application

  • The application used microwave-scattering data on Tokamak Fusion Test Reactor plasma fluctuations, with a quasicoherent 1 MHz peak and spectral density spanning over five orders of magnitude.Source: §7 · p. 13
  • Locally varying sinusoidal taper counts resolved the 1 MHz peak and high-curvature regions better than a fixed-24 Slepian estimate, while suppressing high-frequency bumps.The Slepian estimate used w = 60 kHz; a variable-count Slepian estimate with w = 100 kHz broadened the spectrum even more. The sinusoidal estimate used as few as 12 tapers near the peak and up to 40 between 1300 and 2400 kHz.Source: §7 · p. 14
  • A parabolically weighted, same-data plug-in pre-estimate selected taper counts locally, using a longer segment to reduce fluctuations in the count estimate.The optimal sinusoidal taper count was roughly 24 from 200 to 800 kHz, fell to 12 near the peak, and reached 40 in the flatter 1300–2400 kHz range.Source: §7 · p. 14
  • Against a converged spectrum based on N = 45,000, the sinusoidal taper estimate was more accurate, and the Slepian estimate required more CPU time.Source: §7 · p. 14

8 Conclusion

The conclusion presents minimum bias tapers as minimizing local bias and sinusoidal tapers as a simple analytic, low-bias approximation. For both families, estimate bandwidth is controlled by the number of tapers used.

8 Conclusion

  • Sinusoidal tapers have low bias because sidelobes from the two shifted Fourier transforms cancel.Source: §8 · p. 14
  • Minimum bias tapers minimize local bias, and the asymptotic quadratic estimate minimizing expected local squared error uses them in a multitaper estimate.The conclusion also notes that these tapers provide broad-band bias protection.Source: §8 · pp. 14–15
  • Sinusoidal tapers have a simple analytic form and, for fixed j, approximate minimum bias tapers to order 1/N.Source: §8 · p. 15
  • The kth sinusoidal taper concentrates spectral energy in bands from (k−1)/(2(N+1)) to (k+1)/(2(N+1)) in absolute frequency.Source: §8 · p. 15
  • Neither taper family has an auxiliary bandwidth parameter: bandwidth depends on taper count, which can be adjusted adaptively, unlike Slepian tapers that must be recomputed for a different bandwidth.Adaptive addition or deletion of tapers can construct an estimate with the optimal convergence properties of kernel smoothers; the conclusion describes fixed-bandwidth Slepian estimation as inherently inefficient.Source: §8 · p. 15

9 Appendix: Multitaper decomposition of kernel estimates

The appendix shows that a kernel-smoothed periodogram can have an exact minimum-bias multitaper decomposition for a parabolic kernel. A smoothed, tapered example likewise yields eigenvectors close to sinusoids and an estimate effectively using four tapers.

9 Appendix: Multitaper decomposition of kernel estimates

  • For a parabolic kernel, the kernel-smoothed periodogram’s eigenvectors are exactly the discrete minimum-bias tapers.This gives an exact minimum-bias taper decomposition of the smoothed periodogram.Source: §9 · p. 15
  • With N = 200, Tukey’s split-cosine taper, and a square box kernel of halfwidth .01, the first four eigenvectors are very close to sinusoids.The example evaluates the eigenvectors of the corresponding smoothed tapered periodogram.Source: §9 · p. 15
  • The resulting estimate is virtually a K = 4 multitaper spectral estimate; eigenvalues decrease sharply after k > 4, so higher eigenvectors contribute very little.Table 4 reports the estimate’s multitaper characterization.Source: §9 · pp. 15–16

Table Captions:

The section captions identify tables and figures addressing taper convergence, bias and spectral concentration, spectral-energy comparisons, plasma spectral estimates, and smooth-tapered-periodogram eigenvectors.

Table Captions:

  • Tables 2 and 3 cover the normalized bias term and spectral concentration, respectively.Source: Table Captions · p. 18, Table 2 · p. 25
  • The spectral-energy figures cover minimum bias, sinusoidal, and Slepian tapers, including a comparison of minimum bias and sinusoidal tapers.The captions specify N = 200 and k = 1 for the comparison figures.Source: Figure 1, Figure 2, Figure 3
  • The plasma-fluctuation estimate figure contrasts sinusoidal and Slepian multitaper estimates, noting artificial broadening at 1 MHz and undersmoothing for f ≥ 1.5 MHz with fixed-bandwidth Slepian tapers.Source: Figure 4
  • Table 1 reports the convergence of sinusoidal tapers to minimum bias tapers.Its caption specifies N = 50.Source: Table Captions · p. 18, Table 1 · p. 25
  • Table 4 and the first-eigenvectors figure present eigenvectors of the smooth tapered periodogram estimator.Source: Table Captions · p. 18, Table 4 · p. 26, Figure 6
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