Variational approaches toward nonlinear time-series analysis

Linear autoregressive models (AR) have broad applications toward spectral analyses and digital filtering of signals, as well as identification, prediction, and control of dynamical systems. However, the symmetric Gaussian properties of linear AR presume independence between the first and second order moments of system outputs. Using nonlinear AR to quantify interactions between the mean and autocovariance of a signal, analysts can discriminate asymmetric frequency components, estimate the relative contribution of system nonlinearities toward mean output, evaluate relative structural stability of a system, and predict critical regions where bifurcations in its dynamics might occur. Quadratic analyses are performed on simulated outputs from a nonlinear model and various experimental time series (Canadian lynx population, respiratory volume dynamics in rat, Cheyne-Stokes respiration in patients with congestive heart failure, intracranial pressures in patient with cerebral hematoma). These case studies demonstrate the utility of the techniques for evaluating the qualitative behaviors of dynamical systems in the presence of slowly varying inputs.