The Spectral Analysis of Time Series

Abstract

Preliminaries: Time Series and Spectra, Summary of Vector Space Geometry, Some Probability Notations and Properties. Models for Spectral Analysis - The Univariate Case: The Wiener Theory of Spectral Analysis, Stationary and Weakly Stationary Stochastic Processes, The Spectral Representation for Weakly Stationary Stochastic Processes - A Special Case, The General Spectral Representation for Weakly Stationary Processes, The Discrete and Continuous Components of the Process, Physical Realizations of the Different Kinds of Spectra, The Real Spectral Representation, Ergodicity and the Connection between the Wiener and Stationary Process Theories, Statistical Estimation of the Autocovariance and the Mean Ergodic Theorem. Sampling, Aliasing, and Discrete-Time Models: Sampling and the Aliasing Problem, The Spectral Model for Discrete-Time Series Linear Filters - General Properties with Applications to Continuous-Time Processes: Linear Filters, Combining Linear Filters, Inverting Linear Filters, Nonstationary Processes Generated by Time Varying Linear Filters. Multivariate Spectral Models and Their Applications: The Spectrum of a Multivariate Time Series-Wiener Theory, Multivariate Weakly Stationary Stochastic Processes, Linear Filters for Multivariate Time Series, The Bivariate Spectral Parameters, Their Interpretations and Uses. The Multivariate Spectral Parameters, Their Interpretations and Uses Digital Filters: General Properties of Digital Filters, The Effect of Finite Data Length, Digital Filters with Finitely Many Nonzero Weights, Digital Filters Obtained by Combining Simple Filters, Filters with Gapped Weights and Results Concerning the Filtering of Series with Polynomial Trends. Finite Parameter Models, Linear Prediction and Real-Time Filtering: Moving Averages, Autoregressive Processes, The Linear Prediction Problem, Mixed Autoregressive-Moving Average Processes and Recursive Prediction, Linear Filtering in Real Time. The Distribution Theory of Spectral Estimates with Applications to Statistical Inference: Distribution of the Finite Fourier Transform and the Periodogram. Distribution Theory for Univariate Spectral Estimators, Distribution Theory for Multivariate Spectral Estimators with Applications to Statistical Inference. Sampling Properties of Spectral Estimates, Experimental Design and Spectral Computations, Properties of Spectral Estimators and the Selection of Spectral Windows, Experimental Design, Methods for Computing Spectral Estimators, Data Processing Problems and Techniques.