Spectral Estimation Using Multitaper Whittle Methods with a Lasso Penalty.

Area of Science:

• Time Series Analysis
• Spectral Estimation
• Statistical Signal Processing

Background:

• Traditional spectral estimation methods like periodograms are inconsistent.
• Advanced techniques such as lag window and multitaper estimators can still yield noisy results.
• Existing least squares approaches often rely on restrictive Gaussianity assumptions.

Purpose of the Study:

• To develop a semiparametric spectral estimation method for stationary time series.
• To overcome limitations of existing methods, including noise and Gaussianity assumptions.
• To achieve sparsity in spectral estimation for diverse basis functions.

Main Methods:

• Proposed an L1 penalized quasi-likelihood Whittle framework utilizing multitaper spectral estimates.
• Developed an Alternating Direction Method of Multipliers (ADMM) algorithm for efficient optimization.
• Introduced universal threshold and Generalized Information Criterion (GIC) strategies for parameter tuning.

Main Results:

• The new framework circumvents the Gaussianity assumption, enabling broader applicability.
• Achieved sparsity, leading to more refined spectral estimates.
• Demonstrated superior performance of tuning strategies compared to cross-validation.
• Established a fast convergence rate for the proposed spectral estimator.

Conclusions:

• The proposed L1 penalized Whittle framework provides a robust and efficient approach to spectral estimation.
• The method effectively handles noisy data and avoids restrictive statistical assumptions.
• Successfully applied to simulated data and real-world electroencephalogram (EEG) data analysis.