Chapter 9 - Spectral Analysis

Abstract

Publisher Summary This chapter explores the issues that are central to estimating spectral density functions from time series observations. It describes three classes of rational spectral density functions. The first two classes are the moving average (MA) and the autoregressive (AR) spectral models. These two classes of rational functions have the basic modeling tools in contemporary spectral estimation theory. The more general rational model is referred to as an ARMA model, which has a frequency characterization that is the composite of an MA model and an AR model. The chapter presents a schematic representation of the types of spectral estimation procedures to be developed. The chapter examines the theoretical autocorrelation characteristic of MA, AR, and ARMA random processes. It also reviews the classical problem of detection and frequency identification of the sinusoids in white noise time series to demonstrate the relative effectiveness of MA, AR, and ARMA modeling. The chapter describes a philosophy directed toward the rational modeling of wide-sense stationary (WSS) time series. The method is explicitly based on the Yule–Walker equations, which characterize the autocorrelation sequence associated with the rational time series being modeled. The chapter presents a procedure for extracting the prerequisite model order values that uses a singular value decomposition of an extended correlation matrix. An important by-product of this procedure is an adaptation of the ARMA modeling procedure, which improves spectral estimation. The chapter also describes an application of singular value decomposition (SVD) to ARMA modeling.