The use of finitely parametrized linear models such as ARMA models in
analysing time series data has been extensively studied and in recent years there has
been an increasing emphasis on the development of fast regression—based algorithms
for the problem of model identification. In this thesis we investigate the statistical
properties of pseudo—linear regression algorithms in the context of off-line and
online (real-time) identification. A review of these procedures is presented in Part I
in relation to the problem of identifying an appropriate ARMA model from observed
time series data. Thus, criteria introduced by Akaike and Rissanen are important
here to ensure a model of sufficient complexity is selected, based on the data.
In chapter 1 we survey published results pertaining to the statistical properties
of identification procedures in the off-line context and show there are important
differences as concerns the asymptotic performance of certain parameter estimation
algorithms. However, to effect the identification process in real-time recursive
estimation algorithms are required. Furthermore, these procedures need to be
adaptive to be applicable in practice. This is discussed in chapter 2. Technical
results and limit theorems required for the theoretical analysis conducted in Part II
are collated in chapter 3.
Chapters 4 and 5 of Part II are therefore devoted to the detailed investigation of
particular algorithms discussed in Part I. Chapter 4 deals with off-line parameter
estimation algorithms and in chapter 5, the important idea of a Description Length
Principle introduced by Rissanen, is examined in the context of the recursive
estimation of autoregressions. Empirical evidence from simulation experiments are
also reported in each chapter and in chapter 5, aspects of speech analysis are
incorporated in the simulation study. The simulation results bear out the theory and
the proofs of asymptotic results are given at the end of the chapter
