Nonlinear Time Series Analysis in a Nutshell

Nonlinear time series analysis is a practical spinoff from complex dynamical systems theory and chaos theory. It allows one to characterize dynamical systems in which nonlinearities give rise to a complex temporal evolution. Importantly, this concept allows extracting information that cannot be resolved using classical linear techniques such as the power spectrum or spectral coherence. Applications of nonlinear time series analysis to signals measured from the brain contribute to our understanding of brain functions and malfunctions and thereby help to advance cognitive neuroscience and neurology. In this chapter, we show how a combination of a nonlinear prediction error and the Monte Carlo concept of surrogate time series can be used to attempt to distinguish between purely stochastic, purely deterministic, and deterministic dynamics superimposed with noise. The framework of nonlinear time series analysis comprises a wide variety of measures that allow one to extract different characteristic features of a dynamical system underlying some measured signal (Kantz and Schreiber 1997). These include the correlation dimension as an estimate of the number of independent degrees of freedom, the Lyapunov exponent as a measure for the divergence of similar system states in time, prediction errors as detectors for characteristic traits of deterministic dynamics, or different information theory measures. The aforementioned nonlinear time series measures are univariate, i.e., they are applied to single signals measured from individual dynamics. In contrast, bivariate measures are used to analyze pairs of signals measured simultaneously from two dynamics. Such bivariate time series analysis measures aim to distinguish whether the two dynamics are independent or interacting through some coupling. Some of these bivariate measures aim to extract not only the strength, but also the direction of these couplings. The Monte Carlo concept of surrogates allows one to test the results of the different nonlinear measures against well-specified null hypotheses. contents