An Introduction to Bispectral Analysis and Bilinear Time Series Models

ISBN-13: 9780387960395 / Angielski / Miękka / 1984 / 280 str.

T. Subba Rao; T. S. Rao; M. M. Gabr

Opis

The theory of time series models has been well developed over the last thirt, y years. Both the frequenc.y domain and time domain approaches have been widely used in the analysis of linear time series models. However. many physical phenomena cannot be adequately represented by linear models; hence the necessity of nonlinear models and higher order spectra. Recently a number of nonlinear models have been proposed. In this monograph we restrict attention to one particular nonlinear model. known as the "bilinear model." The most interesting feature of such a model is that its second order covariance analysis is ve similar to that for a linear model. This demonstrates the importance of higher order covariance analysis for nonlinear models. For bilinear models it is also possible to obtain analytic expressions for covariances. spectra. etc. which are often difficult to obtain for other proposed nonlinear models. Estimation of bispectrum and its use in the construction of tests for linearit, y and symmetry are also discussed. All the methods are illustrated with simulated and real data. The first author would like to acknowledge the benefit he received in the preparation of this monograph from delivering a series of lectures on the topic of bilinear models at the University of Bielefeld. Ecole Normale Superieure. University of Paris (South) and the Mathematisch Cen trum. Ams terdam.

Spis treści

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.