Bilinear Stochastic Models and Related Problems of Nonlinear Time Series Analysis: A Frequency Domain Approach

ISBN-13: 9780387988726 / Angielski / Miękka / 1999 / 270 str.

Gyorgy Terdik; Gyvrgy Terdik; Gyc6rgy Terdik

Opis

"Ninety percent of inspiration is perspiration. " 31] The Wiener approach to nonlinear stochastic systems 146] permits the representation of single-valued systems with memory for which a small per turbation of the input produces a small perturbation of the output. The Wiener functional series representation contains many transfer functions to describe entirely the input-output connections. Although, theoretically, these representations are elegant, in practice it is not feasible to estimate all the finite-order transfer functions (or the kernels) from a finite sam ple. One of the most important classes of stochastic systems, especially from a statistical point of view, is the case when all the transfer functions are determined by finitely many parameters. Therefore, one has to seek a finite-parameter nonlinear model which can adequately represent non linearity in a series. Among the special classes of nonlinear models that have been studied are the bilinear processes, which have found applica tions both in econometrics and control theory; see, for example, Granger and Andersen 43] and Ruberti, et al. 4]. These bilinear processes are de fined to be linear in both input and output only, when either the input or output are fixed. The bilinear model was introduced by Granger and Andersen 43] and Subba Rao 118], 119]. Terdik 126] gave the solution of xii a lower triangular bilinear model in terms of multiple Wiener-It(') integrals and gave a sufficient condition for the second order stationarity. An impor tant."

Spis treści

1 Foundations.- 1.1 Expectation of Nonlinear Functions of Gaussian Variables.- 1.2 Hermite Polynomials.- 1.2.1 Hermite polynomials of one Variable.- 1.2.2 Hermite polynomials of several variables.- 1.3 Cumulants.- 1.3.1 Definition of Cumulants.- 1.3.2 Basic Properties.- 1.4 Diagrams, and Moments and Cumulants for Gaussian Systems.- 1.4.1 Diagrams.- 1.4.2 Moments of Gaussian systems.- 1.4.3 Cumulants for Hermite polynomials.- 1.4.4 Products for Hermite polynomials.- 1.5 Stationary processes and spectra.- 1.5.1 Stochastic spectral representation.- 1.5.2 Complex Gaussian system.- 1.5.3 Spectra.- 2 The Multiple Wiener-Itô Integral.- 2.1 Functions of Spaces $$ \overline {L_{\Phi }^} $$ and $$ \widetilde{{L_{\Phi }^}} $$.- 2.2 The multiple Wiener-Itô Integral of second order.- 2.2.1 Definition I.- 2.2.2 Definition II.- 2.2.3 Definition III.- 2.3 The multiple Wiener-Itô integral of order n.- 2.3.1 Properties.- 2.3.2 Diagram Formula.- 2.3.3 Fock space.- 2.3.4 Stratonovich integral in frequency domain and the Hu-Meyer formula.- 2.4 Chaotic representation of stationary processes.- 2.4.1 Subordinated functionals of Gaussian processes.- 2.4.2 Spectra for processes with Hermite degree-2.- 2.4.3 The process F (Xt).- 3 Stationary Bilinear Models.- 3.1 Definition of bilinear models.- 3.2 Identification of a bilinear model with scalar states.- 3.2.1 Multiple spectral representation and stationarity.- 3.2.2 Spectra.- 3.2.3 The necessary and sufficient condition for the existence of 2nth order moment, scalar case.- 3.3 Identification of bilinear processes, general case.- 3.3.1 State space form of lower triangular bilinear models.- 3.3.2 Vector valued bilinear model with scalar input.- 3.3.3 Spectra.- 3.3.4 Necessary and sufficient condition for the existence of 2nth order moments of the state process.- 3.4 Identification of multiple-bilinear models.- 3.4.1 Chaotic representation and stationarity.- 3.4.2 Spectra.- 3.5 State space realization.- 3.5.1 The bilinear realization problem.- 3.5.2 Realization of the Hermite degree-N homogeneous polynomial model.- 3.5.3 Minimal realizations.- 3.6 Some bilinear models of interest.- 3.6.1 Simple bilinear model.- 3.6.2 Hermite degree-2 bilinear model.- 3.7 Identification of GARCH(1,1) Model.- 3.7.1 Spectrum of the State Process.- 3.7.2 Spectrum of the square of the observations.- 3.7.3 Bispectrum of the state process.- 3.7.4 Bispectrum of the process Yt.- 3.7.5 Simulation.- 4 Non-Gaussian Estimation.- 4.1 Estimating a parameter for non-Gaussian data.- 4.2 Consistency and asymptotic variance of the estimate.- 4.3 Asymptotic normality of the estimate.- 4.4 Asymptotic variance in the case of linear processes.- 4.4.1 A worked example and simulations.- 5 Linearity Test.- 5.1 Quadratic predictor.- 5.1.1 Quadratic predictor for a simple bilinear model.- 5.2 The test statistics.- 5.3 Comments on computing the test statistics.- 5.4 Simulations and real data.- 5.4.1 Homogeneous bilinear realizable time series with Hermite degree-2.- 5.4.2 Results of simulations.- 6 Some Applications.- 6.1 Testing linearity.- 6.1.1 Geomagnetic Indices.- 6.1.2 Results of testing weak linearity for simulated data at WUECON.- 6.1.3 GARCH model fitting.- 6.2 Bilinear fitting.- 6.2.1 Parameter estimation for bilinear processes.- 6.2.2 Bilinear fitting for real data.- Appendix A Moments.- Appendix B Proofs for the Chapter Stationary Bilinear Models.- Appendix C Proofs for Section 3.6.1.- Appendix D Cumulants and Fourier Transforms for GARCH(1,1).- Appendix E Proofs for the Chapter Non-Gaussian Estimation.- E.0.1 Proof for Section 4.4.- Appendix F Proof for the Chapter Linearity Test.- References.