Estimation of parameters and eigenmodes of multivariate autoregressive models

Dynamical characteristics of a complex system can often be inferred from analysis of a stochastic time series model fitted to observations of the system. Oscillations in geophysical systems, for example, are sometimes characterized by principal oscillation patterns, eigenmodes of estimated autoregressive (AR) models of first order. This paper describes the estimation of eigenmodes of AR models of arbitrary order. AR processes of any order can be decomposed into eigenmodes with characteristic oscillation periods, damping times, and excitations. Estimated eigenmodes and confidence intervals for the eigenmodes and their oscillation periods and damping times can be computed from estimated models parameters. As a computationally efficient method of estimating the parameters of AR models from high-dimensional data, a stepwise least squares algorithm is proposed. This algorithm computes models of successively decreasing order. Numerical simulations indicate that, with the least squares algorithm, the AR model coefficients and the eigenmodes derived from the coefficients and eigenmodes are rough approximations of the confidence intervals inferred from the simulaitons.