Modelling and Adaptive Tracking of Nonstationarities for Both Linear and Nonlinear Systems

This thesis focuses on the modelling and adaptive tracking problem of both linear and nonlinear time-varying processes. Approaches for the estimation of time-varying parameters can broadly be classified into two categories: the adaptive recursive algorithm methods and the basis function approximation methods. Adaptive algorithms such as block least mean squares (LMS), recursive least squares (RLS) and Kalman filtering, are applied to estimate the time-varying parameters and are capable of tracking the transient variation providing that the variation is slow and smooth. For the basis function method, time-varying
parameters are expanded as a finite sequence of predetermined basis functions; the problem of time-varying estimation can then be reduced to a time invariant parameter estimation problem. The basis function expansion approaches are able to track process parameter changes even those with jumps, provided that appropriate basis functions are used. In this thesis, an attractive approach is to expand the time-varying parameters using wavelets as basis functions. Wavelets provide powerful tools for signal processing, with excellent approximation properties and
are well suited for approximating general nonstationary signals.

In this work, the application of data-based modelling techniques provides a powerful tool for electrophysiological data modelling and analysis, where a wavelet based modelling approach was applied to model the dynamics of nonstationary signals and capture its transient variations. The work in this thesis contains two parts. The first part deals with the estimation of time-varying linear models in both the time and frequency domains. The performance of tracking and capturing the transient changes of nonstationary systems by using time-varying system
identification and modelling in the time-frequency domain has been verified to be an effective approach which outperforms other existing traditional methods such as
sliding-window recursive least squares and Kalman filter algorithms. This technique has been used to investigate and interpret the properties of EEG oscillations of epileptic patients.

The second part deals with the estimation of nonlinear time-varying models, where a novel common model structure selection (CMSS) algorithm has been adapted and
extended to identify a robust time-varying common-structured (TVCS) model as a solution to time-varying nonlinear systems identification problems using an online
sliding-window approach. The main advantages of the proposed TVCS method are:
1) it produces a less biased or preferably unbiased robust model with better generalisation properties;
2) it enables rapid tracking of transient variations of
varying parameters and is more suitable for the estimation of parameters of inherently nonstationary processes. Results on time-varying nonlinear Granger causality analysis have also been investigated to detect and track nonlinear dynamical Granger causalities features.

A main contribution of this thesis is that linear and nonlinear models time-varying modelling techniques have been developed and applied to analyse and interpret
multi-frequency signals in both the time and frequency domains. Novel wavelet adaptive tracking algorithms were developed to track both linear and nonlinear system behaviours and the algorithms provide a new tool that can help clinicians interpret EEG signals. In addition, the newly developed methods are generally also applicable to other neuroscience signals. For example, possible applications of our proposed technique could be applied to describe and analyse the coding of speech signals into nerve-action potentials by the inner ear.