Bayesian Non-parametric Models for Time Series Decomposition

Abstract

The standard approach to analyzing brain electrical activity is to examine the spectral density function (SDF) and identify frequency bands, defined apriori, that have the most substantial relative contributions to the overall variance of the signal. However, a limitation of this approach is that the precise frequency and bandwidth of oscillations are not uniform across cognitive demands. Thus, these bands should not be arbitrarily set in any analysis. To overcome this limitation, we propose three Bayesian Non-parametric models for time series decomposition which are data-driven approaches that identifies (i) the number of prominent spectral peaks, (ii) the frequency peak locations, and (iii) their corresponding bandwidths (or spread of power around the peaks). The standardized SDF is represented as a Dirichlet process mixture based on a kernel derived from second-order auto-regressive processes which completely characterize the location (peak) and scale (bandwidth) parameters. A Metropolis-Hastings within Gibbs algorithm is developed for sampling from the posterior distribution of the mixture parameters for each project. Simulation studies demonstrate the robustness and performance of the proposed methods. The methods developed were applied to analyze local field potential (LFP) activity from the hippocampus of laboratory rats across different conditions in a non-spatial sequence memory experiment to identify the most prominent frequency bands and examine the link between specific patterns of brain oscillatory activity and trial-specific cognitive demands. The second application study 61 EEG channels from two subjects performing a visual recognition task to discover frequency-specific oscillations present across brain zones. The third application extends the model to characterize the data coming from 10 alcoholics and 10 controls across three experimental conditions across 30 trials. The proposed models generate a framework to condense the oscillatory behavior of populations across different tasks isolating the target fundamental components allowing the practitioner different perspectives of analysis.