Time Series Decomposition into Oscillation Components and Phase Estimation

Abstract

Many time series are naturally considered as a superposition of several oscillation components. For example, electroencephalogram (EEG) time series include oscillation components such as alpha, beta, and gamma. We propose a method for decomposing time series into such oscillation components using state-space models. Based on the concept of random frequency modulation, gaussian linear state-space models for oscillation components are developed. In this model, the frequency of an oscillator fluctuates by noise. Time series decomposition is accomplished by this model like the Bayesian seasonal adjustment method. Since the model parameters are estimated from data by the empirical Bayes' method, the amplitudes and the frequencies of oscillation components are determined in a data-driven manner. Also, the appropriate number of oscillation components is determined with the Akaike information criterion (AIC). In this way, the proposed method provides a natural decomposition of the given time series into oscillation components. In neuroscience, the phase of neural time series plays an important role in neural information processing. The proposed method can be used to estimate the phase of each oscillation component and has several advantages over a conventional method based on the Hilbert transform. Thus, the proposed method enables an investigation of the phase dynamics of time series. Numerical results show that the proposed method succeeds in extracting intermittent oscillations like ripples and detecting the phase reset phenomena. We apply the proposed method to real data from various fields such as astronomy, ecology, tidology, and neuroscience.

State Space Oscillator Models for Neural Data Analysis

“…The linear systems approach we have described may also provide more specificity in detecting oscillations, since the temporal structure or impulse response implied by the AR components is specific to oscillators. Matsuda and Komaki [10] have described a very similar oscillatory model, albeit composed of separate sinusoidal and cosinusoidal components intended to facilitate phase estimation. Their work has focused on phase estimation, rather than identification and separation of oscillations.…”

Estimation of Time-Varying Spectral Peaks and Decomposition of EEG Spectrograms

“…present in windowed methods [9], [40], allows for smoother estimates of time-varying properties. Our expanded statespace model even includes discrete On/Off-switching of spectral peaks, a feature not captured by other state-space approaches [43]- [47]. The functional forms of the peaks used in our method result in estimates that more closely match the shapes of the spectral peaks observed in the EEG than sinusoidal or AR(MA) model estimates [40], [42]- [46].…”

“…Tarvainen et al [41] estimate a state-space autoregressive-moving average (ARMA) model for nonstationary EEG. Dubois et al [42] and Matsuda and Komaki [43] use state-space models of time-varying parameterized oscillations, estimated via unscented particle filter (UPF) and Kalman smoother (KS), respectively. Yet other approaches have been proposed to estimate instantaneous amplitudes and frequencies of signals comprising variable sinusoids [44]- [46].…”

“…While this method enables tracking of variable peaks, it encountered several limitations and difficulties. First, the method, like the state-space approaches of [43]- [46], does not handle the phenomena of peaks quickly turning off (disappearing) and on (reappearing), as with sleep spindles in EEG of non-REM sleep. More importantly, the high dimensionality of the observations, i.e., the instantaneous spectral estimates, quickly leads to degenerate particle weightings.…”

A State Space Modeling Approach to Real-Time Phase Estimation

“…Brain rhythms are often non-sinusoidal (Cole & Voytek, 2017) and broadband (Buzsaki, 2004; Roopun et al, 2008), making development of an accurate bandpass filter difficult. While the contemporary modeling approach using filters accurately estimates the phase across a range of contexts, several limitations exist that limit the phase estimate accuracy (Matsuda & Komaki, 2017; Siegle & Wilson, 2014): (i) By depending on bandpass filters, existing real-time phase estimators are susceptible to non-sinusoidal distortions of the waveform, and inappropriate filter choices may miss the center frequency or total bandwidth of the rhythm. (ii) Phase resets, moments when the phase slips as a result of stimulus presentation or spontaneous dynamics, cannot be tracked using filters, which control the maximum possible instantaneous frequency.…”