Matsuda Takeru, Komaki Fumiyasu
Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo 113-8656, Japan
Department of Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo, Tokyo 113-8656, and RIKEN Brain Science Institute, Wako, Saitama 351-0198, Japan
Neural Comput. 2017 Feb;29(2):332-367. doi: 10.1162/NECO_a_00916. Epub 2016 Nov 21.
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.
许多时间序列自然地被视为几个振荡成分的叠加。例如，脑电图（EEG）时间序列包括诸如阿尔法、贝塔和伽马等振荡成分。我们提出了一种使用状态空间模型将时间序列分解为这些振荡成分的方法。基于随机频率调制的概念，开发了用于振荡成分的高斯线性状态空间模型。在这个模型中，振荡器的频率受噪声波动影响。时间序列分解通过这个模型完成，类似于贝叶斯季节调整方法。由于模型参数通过经验贝叶斯方法从数据中估计，振荡成分的幅度和频率以数据驱动的方式确定。此外，振荡成分的适当数量通过赤池信息准则（AIC）确定。通过这种方式，所提出的方法将给定的时间序列自然地分解为振荡成分。在神经科学中，神经时间序列的相位在神经信息处理中起着重要作用。所提出的方法可用于估计每个振荡成分的相位，并且相对于基于希尔伯特变换的传统方法具有几个优点。因此，所提出的方法能够研究时间序列的相位动态。数值结果表明，所提出的方法成功地提取了诸如涟漪等间歇性振荡并检测到相位重置现象。我们将所提出的方法应用于来自天文学、生态学、潮汐学和神经科学等各个领域的实际数据。
