Algorithms in time series

Abstract

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 chapters Show more