Algorithm for Vector Autoregressive Model Parameter Estimation Using an Orthogonalization Procedure

We review the derivation of the fast orthogonal search algorithm, first proposed by Korenberg, with emphasis on its application to the problem of estimating coefficient matrices of vector autoregressive models. New aspects of the algorithm not previously considered are examined. One of these is the application of the algorithm to estimate coefficient matrices of a vector autoregressive process with time-varying coefficients when multiple realizations of the said process are available. Computer simulations were also performed to characterize the statistical properties of the estimates. The results show that even for shorter time series the algorithm works well and obtains good estimates of the time-varying parameters. Statistical characterization indicates that the standard deviation of the estimates decreases as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${1 \mathord{\left/ {\vphantom {1 {\sqrt N }}} \right. \kern-\nulldelimiterspace} {\sqrt N }}{\text{ }}\left( N \right.$$ \end{document} being the length of the time series), a typical behavior of least-squares estimators. Another key aspect of the approach, which has previously been considered, is its direct extension to the parameter estimation of vector nonlinear autoregressive models. Nonlinear terms can be added to the model and the same algorithm can be applied to effectively estimate their associated parameters. Using chaotic time series generated from the Lorenz equations, the algorithm produces a model that captures the nonlinear structure of the data and exhibits the same chaotic attractor as that of the original system. © 2002 Biomedical Engineering Society.