On the Application of the SDLE to the Analysis of Complex Time Series

Complex systems often generate highly nonstationary and multiscale signals because of nonlinear and stochastic interactions among their component systems and hierarchical regulations imposed by the operating environments. Rapid accumulation of such complex data in life sciences, systems biology, nano-sciences, information systems, and physical sciences has made it increasingly important to develop complexity measures that incorporate the concept of scale explicitly, so that different behaviors of signals on varying scales can be simultaneously characterized by the same scale-dependent measure. The scale-dependent Lyapunov exponent (SDLE) discussed here is such a measure and can be used as the basis for a unified theory of multiscale analysis of complex data. The SDLE can readily characterize deterministic low-dimensional chaos, noisy chaos, random 1 ∕ f α processes, random Levy processes, stochastic oscillations, and processes with multiple scaling behavior. It can also readily deal with many types of nonstationarity, detect intermittent chaos, and accurately detect epileptic seizures from EEG data and distinguish healthy subjects from patients with congestive heart failure from heart rate variability (HRV) data. More importantly, analyses of EEG and HRV data illustrate that commonly used complexity measures from information theory, chaos theory, and random fractal theory can be related to the values of the SDLE at specific scales, and useful information on the structured components of the data is also embodied by the SDLE.