Complex systems often generate highly nonstationary and multiscale signals, due to nonlinear and stochastic interactions among their component systems and hierarchical regulations imposed by the operating environments. The further advances in the fields of life sciences, systems biology, nano-sciences, information systems, and physical sciences, have made it increasingly important to develop complexity measures that incorporate the concept of scale explicitly, so that different behaviors of the signals on varying scales can be simultaneously characterized by the same scale-dependent measure. Here, we propose such a measure, the scale-dependent Lyapunov exponent (SDLE), and develop a unified theory of multiscale analysis of complex data. We show that the SDLE can readily characterize low-dimensional chaos and random 1/fα processes, as well as 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, our 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.

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