Novel recursive algorithms for multichannel time-varying autoregressive process with applications

Abstract

This thesis aims to propose novel adaptive methods to estimate the coefficients of time-varying autoregressive (TVAR) processes, especially for multivariate AR (MVAR) processes and study their application to spectral and connectivity analyses of electroencephalography (EEG) signals. The AR model has been widely used to model certain non-stationary processes in nature, such as biomedical signals, economics, etc. Moreover, most real world signals are time-varying in nature, making their modeling complicated. First, we propose a bias-compensated (BC) variable forgetting factor (VFF) recursive least square (RLS) algorithm with extended instrumental variable (EIV) for estimating TVAR models in multivariate setting with the presence of spatially colored measurement noise. By utilizing the EIV estimator, the noise covariance can be reliably estimated and hence the bias can be compensated from the estimator. A robust smoothing method and an adaptive model selection method i for respectively estimating the noise parameters and controlling the BC contribution are proposed for reducing the variance of the estimator. Simulation results show that the proposed VFF-EIV-BC method offers considerably better performance than conventional methods. It is also applied to estimate the spectrum and functional connectivity of real EEG data from motor imagery tasks, where the high-alpha and beta event-related desynchronization associated with the tasks can be clearly revealed. Secondly, a new adaptive fading Kalman filter (AF-KF) with variable number of measurements (AF-KF-VNM) is proposed for estimating the spectra of the EEG signals and identifying their functional connectivity. The proposed AF-KF-VNM algorithm uses a new adaptive fading method to adaptively update the model parameters of the KF for improved state estimation and utilizes multiple measurements for better adaptation to the nonstationary signal observations. Experimental results on simulated data for modeling the TV directed interactions in multivariate neural data show that the proposed AF-KF-VNM method yields better tracking performance than other approaches tested. The proposed algorithm is then integrated into a novel methodology for combined functional Magnetic Resonance Imaging (fMRI) activation maps and EEG spectrum analyses for quantifying the differences in spectrum contents and information flows between two conditions in a visual oddball paradigm. The results and findings show that the proposed methodology agrees well with the literature and is capable of revealing significant frequency components and information flow involved as well as their temporal variations. Finally, based on the single-channel local polynomial modeling (LPM), we propose a novel multi-channel LPM-based (MLPM) approach to time-varying brain connectivity analyses of EEG signals. The proposed MLPM algorithm uses the intersection of confidence intervals (ICI) method to adaptively select the local optimal bandwidth at each sample for better adaptation to nonstationary signals. ii Experimental results on simulated data show that the proposed MLPM method yields better tracking performance, temporal and frequency resolutions than other approaches tested. The proposed algorithm is also used to estimate information flows between the target and standard conditions in an auditory oddball paradigm. The frequency contents and functional connectivities of the EEG signals are characterized.