On the Application of the SDLE to the Analysis of Complex Time Series
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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.
KeywordsHeart Rate Variability Lyapunov Exponent Detrended Fluctuation Analysis Chaos Theory Hurst Parameter
This work is supported in part by NSF grants CMMI-1031958 and 0826119.
- 1.Fairley P (2004) The unruly power grid. IEEE Spectrum 41:22–27Google Scholar
- 2.Bassingthwaighte JB, Liebovitch LS, West BJ (1994) Fractal physiology. Oxford University Press, New YorkGoogle Scholar
- 7.Gao JB, Hu J, Tung WW, Zheng Y (2011) Multiscale analysis of economic time series by scale-dependent Lyapunov exponent. Quantitative Finance DOI:10.1080/14697688.2011.580774Google Scholar
- 8.Mit-bih normal sinus rhythm database & bidmc congestive heart failure database available at http://www.physionet.org/physiobank/database/#ecg.
- 18.Kaneko K, Tsuda I (2000) Complex systems: chaos and beyond. Springer, BerlinGoogle Scholar
- 40.Hu J, Gao JB, Tung WW (2009) Characterizing heart rate variability by scale-dependent lyapunov exponent. Chaos (special issue on Controversial Topics in Nonlinear Science: Is the Normal Heart Rate Chaotic? it is one of the most downloaded papers in that issue) 19:028506Google Scholar
- 43.Takens F (1981) Detecting strange attractors in turbulence. In: Rand DA, Young LS (eds.) Dynamical systems and turbulence, lecture notes in mathematics, vol 898. Springer, Berlin, p 366Google Scholar
- 49.Frisch U (1995) Turbulence—the legacy of A.N. Kolmogorov. Cambridge University PressGoogle Scholar
- 56.Hon EH, Lee ST (1965) Electronic evaluations of the fetal heart rate patterns preceding fetal death: further observations. Am J Obstet Gynecol 87:814–826Google Scholar
- 57.Task Force of the European Society of Cardiology & the North American Society of Pacing & Electrophysiology (1996) Heart rate variability: Standards of measurement, physiological interpretation,and clinical use. Circulation 93:1043–1065Google Scholar
- 61.Pincus SM, Viscarello RR (1992) Approximate entropy: a regularity statistic for fetal heart rate analysis. Obst Gynecol 79:249–255Google Scholar
- 62.Hu J, Gao JB, Wang XS (2009) Multifractal analysis of sunspot time series: the effects of the 11-year cycle and fourier truncation. J Stat Mech. DOI: 10.1088/1742-5468/2009/02/P02066Google Scholar