[Paper Review] Large Spectral Density Matrix Estimation by Thresholding

Yiming Sun, Yige Li+2 more|arXiv (Cornell University)|Dec 3, 2018

Functional Brain Connectivity StudiesNeuroscience27 references19 citations

TL;DR

This paper proposes a thresholding-based method for estimating high-dimensional spectral density matrices of multivariate time series using averaged periodograms, achieving consistent estimation under the high-dimensional regime log p/n → 0 when the true spectrum is approximately sparse. The approach enables automatic edge selection in functional connectivity networks and introduces a novel concentration inequality for averaged periodograms, offering improved theoretical guarantees over shrinkage-based methods.

ABSTRACT

Spectral density matrix estimation of multivariate time series is a classical problem in time series and signal processing. In modern neuroscience, spectral density based metrics are commonly used for analyzing functional connectivity among brain regions. In this paper, we develop a non-asymptotic theory for regularized estimation of high-dimensional spectral density matrices of Gaussian and linear processes using thresholded versions of averaged periodograms. Our theoretical analysis ensures that consistent estimation of spectral density matrix of a $p$-dimensional time series using $n$ samples is possible under high-dimensional regime $\log p / n ightarrow 0$ as long as the true spectral density is approximately sparse. A key technical component of our analysis is a new concentration inequality of average periodogram around its expectation, which is of independent interest. Our estimation consistency results complement existing results for shrinkage based estimators of multivariate spectral density, which require no assumption on sparsity but only ensure consistent estimation in a regime $p^2/n ightarrow 0$. In addition, our proposed thresholding based estimators perform consistent and automatic edge selection when learning coherence networks among the components of a multivariate time series. We demonstrate the advantage of our estimators using simulation studies and a real data application on functional connectivity analysis with fMRI data.

Motivation & Objective

• To develop a non-asymptotic theory for regularized estimation of high-dimensional spectral density matrices under the regime log p/n → 0.
• To enable consistent estimation of spectral density matrices when the true spectrum is approximately sparse, even with small sample sizes (n ≪ p²).
• To provide a method that performs automatic edge selection in coherence networks among time series components, enhancing interpretability.
• To establish theoretical consistency for thresholded averaged periodogram estimators, complementing existing shrinkage-based approaches with weaker dependence on p²/n → 0.
• To demonstrate the method’s superiority through simulations and real fMRI data analysis in neuroscience.

Proposed method

• The method uses thresholded versions of averaged periodograms as estimators of the spectral density matrix for multivariate Gaussian and linear processes.
• It applies hard, lasso, and adaptive lasso thresholding to the averaged periodogram to promote sparsity and reduce estimation error.
• A new concentration inequality is derived for the average periodogram around its expectation, which is central to the theoretical analysis and of independent interest.
• Theoretical analysis establishes consistency under the high-dimensional regime log p/n → 0, provided the true spectral density is approximately sparse.
• The method enables automatic edge selection by shrinking weak coherence values to zero, producing sparse, interpretable functional connectivity networks.
• Theoretical bounds are derived that explicitly link estimation error to the degree of approximate sparsity in the true spectral density matrix.

Experimental results

Research questions

• RQ1 Can consistent estimation of high-dimensional spectral density matrices be achieved when log p/n → 0, even under approximate sparsity?
• RQ2 How does thresholding-based estimation compare to shrinkage-based methods in terms of theoretical consistency and sample size requirements?
• RQ3 Can thresholding of averaged periodograms lead to automatic and meaningful edge selection in functional connectivity networks?
• RQ4 What is the role of the new concentration inequality for averaged periodograms in enabling non-asymptotic theoretical guarantees?
• RQ5 Does the proposed method outperform existing estimators in finite-sample settings with small n and large p, particularly in fMRI data analysis?

Key findings

• The proposed thresholding estimators achieve consistent estimation of spectral density matrices under the high-dimensional regime log p/n → 0, provided the true spectrum is approximately sparse.
• The method outperforms shrinkage-based estimators in terms of sample size efficiency, as shrinkage methods require the stronger condition p²/n → 0.
• Simulation results show that adaptive lasso thresholding achieves the highest F1 scores (e.g., 93.64% at p=96, n=600), indicating superior precision and recall in edge detection.
• In fMRI data analysis with p=86 brain regions and n=200 samples, the thresholding approach successfully identified coherent brain networks with improved interpretability.
• The heat maps of coherence matrices estimated via adaptive lasso thresholding (Figure 3) show clearer, sparser, and more structured patterns compared to diagonal shrinkage (Figure 4).
• Theoretical analysis confirms that the estimation error is bounded and converges to zero as long as the true spectral density is approximately sparse, even in high-dimensional settings.