Minimum bias multiple taper spectral estimation (1803.04078v2)

Abstract: Two families of orthonormal tapers are proposed for multi-taper spectral analysis: minimum bias tapers, and sinusoidal tapers ${ \bf{v}^{(k)}}$, where $v_n^{(k)}=\sqrt{\frac{2}{N+1}}\sin\frac{\pi kn}{N+1}$, 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.

• The paper introduces MB and sinusoidal tapers that minimize local bias in spectral estimates compared to traditional Slepian tapers.
• It derives analytical expressions for sinusoidal tapers, enabling fast convergence and eliminating the need for numerical eigen decomposition.
• The methodology allows adaptive bandwidth adjustments, enhancing spectral concentration and accuracy in nonstationary signal analysis.

Analytical Approaches in Spectral Estimation: A Rigorous Examination

In the paper of multitaper spectral analysis, the paper by Riedel and Sidorenko introduces two innovative families of orthonormal tapers: minimum bias (MB) tapers and sinusoidal tapers. These tapers are analyzed for their efficacy in reducing bias in spectral estimates, offering an alternative to the traditionally used Slepian tapers within the multitaper framework. The authors derive analytical expressions for the sinusoidal tapers, which are characterized by their simple forms and convergence properties, and provide an in-depth comparison of their performance against Slepian tapers.

Theoretical Insights and Methodology

The paper delineates the structural basis for employing MB and sinusoidal tapers in spectral analysis, focusing on minimizing local bias—$\int_{-1/2}^{1/2} f^2 |V(f)|^2 df$—in spectral estimates. This is achieved by proposing an orthonormal system of functions, which is derived as analytical sinusoidal sequences. The sinusoidal tapers are defined as $$v_n^{(k)}=\sqrt{\frac{2}{N+1}}\sin\frac{\pi kn}{N+1}$$ where $N$ represents the number of data points. Such tapers asymptotically converge to the MB tapers with the convergence rate of $1/N$, eliminating the need for numerical eigenvalue decomposition.

Both MB and sinusoidal tapers are shown to be devoid of additional bandwidth parameters, lending themselves to adaptive adjustments of bandwidth by simply adding or removing tapers. This adaptability is crucial in addressing the rapid changes in the spectral density, allowing for a more nuanced estimation without the requirement of recomputation that characterizes Slepian tapers.

Comparative Analysis and Results

In terms of theoretical contributions, the paper proposes a quadratic spectral estimator ($\widehat{S}(f)$), more adaptable than traditional models. This estimator is expressed as a weighted sum of K orthonormal rank-one spectral estimators, emphasizing minimal frequency bias and allowing for localized bandwidth adjustment. Notably, the paper provides a comparative analysis showing that both MB and sinusoidal tapers achieve similar spectral concentration, nearly equivalent to that of the Slepian tapers, while exhibiting significantly lower local bias in situations where the spectral band needs variable width.

Numerical results indicate that the sinusoidal tapers are within 0.2% of the optimal local bias provided by MB tapers, significantly outperforming Slepian tapers which may exhibit higher local bias depending on parameters such as bandwidth. These quantitative metrics establish a rigorous empirical foundation for the proposed methods.

Implications and Future Directions

The practical implications of this research are profound for spectral analysis in signal processing, particularly in fields requiring high-frequency resolution or analysis of nonstationary processes. The ability to change taper numbers dynamically without computational overhead allows for greater adaptability in handling complex data, which is advantageous in real-world applications. Future research may explore further optimization of taper weighting schemes and adaptive strategies to further fine-tune the bias-variance trade-off inherent in spectral estimations.

Moreover, the paper sets a stage for broadening the applicability of multiple tapering methods beyond stationary time series, venturing into non-stationary time series analysis, where variability and irregularity in frequency content pose greater challenges. Further integration with data-adaptive smoothing techniques may refine the real-time application potential of these methods.

In summary, Riedel and Sidorenko's work provides a substantial contribution to spectral estimation by introducing methodological innovations that streamline computational processes while enhancing accuracy. As this methodology matures, it could redefine standard practices in statistical signal processing and beyond.