Robust spectrotemporal decomposition by iteratively reweighted least squares

Abstract

Classical nonparametric spectral analysis uses sliding windows to capture the dynamic nature of most real-world time series. This universally accepted approach fails to exploit the temporal continuity in the data and is not well-suited for signals with highly structured time-frequency representations. For a time series whose time-varying mean is the superposition of a small number of oscillatory components, we formulate nonparametric batch spectral analysis as a Bayesian estimation problem. We introduce prior distributions on the time-frequency plane that yield maximum a posteriori (MAP) spectral estimates that are continuous in time yet sparse in frequency. Our spectral decomposition procedure, termed spectrotemporal pursuit, can be efficiently computed using an iteratively reweighted least-squares algorithm and scales well with typical data lengths. We show that spectrotemporal pursuit works by applying to the time series a set of data-derived filters. Using a link between Gaussian mixture models, ℓ 1 minimization, and the expectation-maximization algorithm, we prove that spectrotemporal pursuit converges to the global MAP estimate. We illustrate our technique on simulated and real human EEG data as well as on human neural spiking activity recorded during loss of consciousness induced by the anesthetic propofol. For the EEG data, our technique yields significantly denoised spectral estimates that have significantly higher time and frequency resolution than multitaper spectral estimates. For the neural spiking data, we obtain a new spectral representation of neuronal firing rates. Spectrotemporal pursuit offers a robust spectral decomposition framework that is a principled alternative to existing methods for decomposing time series into a small number of smooth oscillatory components.A cross nearly all fields of science and engineering, dynamic behavior in time-series data, due to evolving temporal and/or spatial features, is a ubiquitous phenomenon. Common examples include speech (1), image, and video (2) signals; neural spike trains (3) and EEG (4) measurements; seismic and oceanographic recordings (5); and radar emissions (6). Because the temporal and spatial dynamics in these time series are often complex, nonparametric spectral techniques, rather than parametric, modelbased approaches (7), are the methods most widely applied in the analysis of these data. Nonparametric spectral techniques based on Fourier methods (8, 9), wavelets (10, 11), and data-dependent approaches, such as the empirical mode decomposition (EMD) (12, 13), use sliding windows to take account of the dynamic behavior. Although analysis with sliding windows is universally accepted, this approach has several drawbacks.First, the spectral estimates computed in a given window do not use the estimates computed in adjacent windows, hence the resulting spectral representations do not fully capture the degree of smoothness inherent in the underlying signal. Second, the uncertainty principle (14) imposes stringent limits on the spectral resolution...

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