Local Spectral Analysis via a Bayesian Mixture of Smoothing Splines

Rosen, Ori; Stoffer, David S.; Wood, Sally

doi:10.1198/jasa.2009.0118

Cited by 47 publications

(56 citation statements)

References 18 publications

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“…Spectrotemporal pursuit offers a principled alternative to existing methods, such as EMD, for decomposing a noisy time-series into a small number of oscillatory components (13); in spectrotemporal pursuit, this is easily handled using the regularization parameters α 1 and α 2 , respectively, in the cases of f 1 ð·Þ and f 2 ð·Þ, which we estimate from the time-series using cross-validation (30 35 propose an algorithm to estimate the nonparametric, nonstationary spectrum of a Dahlhaus locally stationary process. Lastly, dynamic models with time-varying sparsity have been proposed in several contexts (36,37).…”

Section: Discussionmentioning

confidence: 99%

Robust spectrotemporal decomposition by iteratively reweighted least squares

Ba¹,

Babadi²,

Purdon³

et al. 2014

Proc. Natl. Acad. Sci. U.S.A.

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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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Section: Discussionmentioning

confidence: 99%

Robust spectrotemporal decomposition by iteratively reweighted least squares

Ba¹,

Babadi²,

Purdon³

et al. 2014

Proc. Natl. Acad. Sci. U.S.A.

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“…Also, the exponential model has played an important role in the Bayesian estimation of the spectrum (Wahba, 1980;Carter and Kohn, 1997), for regularized spectral estimation, where smoothness priors are enforced by shrinking higher order cepstral coefficients toward zero, and has been recently considered in the estimation of time-varying spectra (Rosen, Stoffer and Wood, 2009, and Rosen, Wood and Stoffer, 2012). Among other uses of the EXP model we mention discrimination and clustering of time series, as in Fokianos and Savvides (2008).…”

Section: Introductionmentioning

confidence: 99%

The Exponential Model for the Spectrum of a Time Series: Extensions and Applications

Proietti

Luati

2013

SSRN Journal

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Abstract. The exponential model for the spectrum of a time series and its fractional extensions are based on the Fourier series expansion of the logarithm of the spectral density. The coefficients of the expansion form the cepstrum of the time series. After deriving the cepstrum of important classes of time series processes, also featuring long memory, we discuss likelihood inferences based on the periodogram, for which the estimation of the cepstrum yields a generalized linear model for exponential data with logarithmic link, focusing on the issue of separating the contribution of the long memory component to the log-spectrum.We then propose two extensions. The first deals with replacing the logarithmic link with a more general Box-Cox link, which encompasses also the identity and the inverse links: this enables nesting alternative spectral estimation methods (autoregressive, exponential, etc.) under the same likelihood-based framework.Secondly, we propose a gradient boosting algorithm for the estimation of the log-spectrum and illustrate its potential for distilling the long memory component of the log-spectrum.

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“…We express h h h as a linear combination of basis functions, h h h = X X Xβ β β, where the columns of the design matrix X X X are the Demmler-Reinsch basis functions evaluated at the Fourier frequencies, and β β β is vector of unknown coefficients. We follow Wood et al (2002) and Rosen et al (2009) and retain only the basis functions corresponding to the J = 10 largest eigenvalues, resulting in significant computational saving. For linear smoothing splines, the jth column of X X X, j = 1, .…”

Section: Priorsmentioning

confidence: 99%

“…These results indicate that if the true process is stationary, AdaptSPEC does not overfit by dividing the time series into more than one segment. We simulate data from two processes used in Rosen et al (2009), given by…”

Section: Stationary Ar(3) Processmentioning

confidence: 99%

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AdaptSPEC: Adaptive Spectral Estimation for Nonstationary Time Series

Rosen

Wood

Stoffer

2012

Journal of the American Statistical Association

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We propose a method for analyzing possibly nonstationary time series by adaptively dividing the time series into an unknown but finite number of segments and estimating the corresponding local spectra by smoothing splines. The model is formulated in a Bayesian framework, and the estimation relies on reversible jump Markov chain Monte Carlo (RJMCMC) methods. For a given segmentation of the time series, the likelihood function is approximated via a product of local Whittle likelihoods. Thus, no parametric assumption is made about the process underlying the time series. The number and lengths of the segments are assumed unknown and may change from one MCMC iteration to another. The frequentist properties of the method are investigated by simulation, and applications to EEG and the El Niño Southern Oscillation phenomenon are described in detail.

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Local Spectral Analysis via a Bayesian Mixture of Smoothing Splines

Cited by 47 publications

References 18 publications

Robust spectrotemporal decomposition by iteratively reweighted least squares

Robust spectrotemporal decomposition by iteratively reweighted least squares

The Exponential Model for the Spectrum of a Time Series: Extensions and Applications

AdaptSPEC: Adaptive Spectral Estimation for Nonstationary Time Series

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