An Introduction to Bispectral Analysis and Bilinear Time Series Models.

Abstract

1 Introduction to Stationary time Series and Spectral Analysis.- 1.1 Some basic Definitions.- 1.2 Spectral Densities and Spectral Representations.- 1.3 Higher Order Spectra (Polyspectra).- 1.4 Bispectral Density Functions.- 1.5 Standard Linear Models - their spectra and bispectra.- 1.6 State Space Representation of Linear Time Series Models.- 1.7 Bispectra and Linear Processes.- 1.8 Invertibility of Time Series Models.- 2 The Estimation of Spectral and Bispectral Density Functions.- 2.1 Introduction.- 2.2 Estimation of the Spectral Density Function.- 2.3 Estimation of the Bispectral Density Function.- 2.4 Optimum Bispectral Window.- 2.5 Comparison of Bispectral Lag Windows.- 2.6 Bispectral Density Function of BL(1,0,1,1) Model.- 3 Practical Bispectral Analysis.- 3.1 The Choice of Truncation Point (M).- 3.2 Comparison of Parametric and Non-Parameteric Bispectral Estimates.- 3.3 Bispectral Analysis of some Time Series Data.- 3.4 Some Nonlinear Phenomena.- 4 Tests for Linearity and Gaussianity of Stationary time Series.- 4.1 General Introduction.- 4.2 Spectrum and Bispectrum of Linear Processes.- 4.3 Test for Symmetry and Linearity.- 4.4 Test for Linearity.- 4.5 Choice of Parameters.- 4.6 Numerical Illustrations.- 4.7 Applications to Real Time Series.- 5 Bilinear time Series Models.- 5.1 Non-Linear Representations in terms of independent random variables.- 5.2 Bilinear Time Series Models.- 5.3 Volterra Series Expansion of YBL(p) Models.- 5.4 Expressions for Covariances and Conditions for Stationarity.- 5.5 Invertibility of the VBL(p) Model.- 5.6 Conditions for Stationarity of the Diagonal Bilinear Model, DBL(?).- 5.7 Conditions for Stationarity of the Lower Triangular Bilinear Model, LTBL (?,?).- 5.8 Estimation of the Parameters of Bilinear Models.- 5.9 Determination of the Order of Bilinear Models.- 5.10 Numerical Illustrations.- 5.11 Sampling Properties of Parameter Estimations for the BL(1,0,1,1) Model.- 6 Estimation and Prediction for Subset Bilinear time Series Models with Applications.- 6.1 Introduction.- 6.2 An Algorithm for Fitting Subset Bilinear Models.- 6.3 Estimation of the Parameters of SBL(k?,m).- 6.4 Residuals.- 6.5 Fitting Subset Bilinear Models to Time Series Data.- 7 Markovian Representation and Existence Theorems for Bilinear time Series Models.- 7.1 Markovian Representations.- 7.2 Existence of the Bilinear Model BL(p,0,p,1).- Appendix A On the Kronecker Matrix Product.- Appendix B Linear Least Squares Solutions by Householder Transformations.- Appendix C Fitting the Best AR Model.- Appendix D Time Series Data.- Listing of Programs.- Program 1.- Program 2.- Program 3.- Program 4.- References.- Author Index.