Timely detection of unusual and/or unexpected events in natural and man-made systems has deep scientific and practical relevance. We show that the recently proposed conceptually simple and easily calculated measure of permutation entropy can be effectively used to detect qualitative and quantitative dynamical changes. We illustrate our results on two model systems as well as on clinically characterized brain wave data from epileptic patients.

We used a high‐resolution nested climate modeling system to investigate the response of South Asian summer monsoon dynamics to anthropogenic increases in greenhouse gas concentrations. The simulated dynamical features of the summer monsoon compared well with reanalysis data and observations. Further, we found that enhanced greenhouse forcing resulted in overall suppression of summer precipitation, a delay in monsoon onset, and an increase in the occurrence of monsoon break periods. Weakening of the large‐scale monsoon flow and suppression of the dominant intraseasonal oscillatory modes were instrumental in the overall weakening of the South Asian summer monsoon. Such changes in monsoon dynamics could have substantial impacts by decreasing summer precipitation in key areas of South Asia.

Due to the ubiquity of time series with long-range correlation in many areas of science and engineering, analysis and modeling of such data is an important problem. While the field seems to be mature, three major issues have not been satisfactorily resolved. ͑i͒ Many methods have been proposed to assess long-range correlation in time series. Under what circumstances do they yield consistent results? ͑ii͒ The mathematical theory of long-range correlation concerns the behavior of the correlation of the time series for very large times. A measured time series is finite, however. How can we relate the fractal scaling break at a specific time scale to important parameters of the data? ͑iii͒ An important technique in assessing long-range correlation in a time series is to construct a random walk process from the data, under the assumption that the data are like a stationary noise process. Due to the difficulty in determining whether a time series is stationary or not, however, one cannot be 100% sure whether the data should be treated as a noise or a random walk process. Is there any penalty if the data are interpreted as a noise process while in fact they are a random walk process, and vice versa? In this paper, we seek to gain important insights into these issues by examining three model systems, the autoregressive process of order 1, on-off intermittency, and Lévy motions, and considering an important engineering problem, target detection within sea-clutter radar returns. We also provide a few rules of thumb to safeguard against misinterpretations of long-range correlation in a time series, and discuss relevance of this study to pattern recognition.

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Time series from complex systems with interacting nonlinear and stochastic subsystems and hierarchical regulations are often multiscaled. In devising measures characterizing such complex time series, it is most desirable to incorporate explicitly the concept of scale in the measures. While excellent scale-dependent measures such as ⑀ entropy and the finite size Lyapunov exponent ͑FSLE͒ have been proposed, simple algorithms have not been developed to reliably compute them from short noisy time series. To promote widespread application of these concepts, we propose an efficient algorithm to compute a variant of the FSLE, the scale-dependent Lyapunov exponent ͑SDLE͒. We show that with our algorithm, the SDLE can be accurately computed from short noisy time series and readily classify various types of motions, including truly low-dimensional chaos, noisy chaos, noise-induced chaos, random 1 / f ␣ and ␣-stable Levy processes, stochastic oscillations, and complex motions with chaotic behavior on small scales but diffusive behavior on large scales. To our knowledge, no other measures are able to accurately characterize all these different types of motions. Based on the distinctive behaviors of the SDLE for different types of motions, we propose a scheme to distinguish chaos from noise.

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BackgroundChaos and random fractal theories are among the most important for fully characterizing nonlinear dynamics of complicated multiscale biosignals. Chaos analysis requires that signals be relatively noise-free and stationary, while fractal analysis demands signals to be non-rhythmic and scale-free.Methodology/Principal FindingsTo facilitate joint chaos and fractal analysis of biosignals, we present an adaptive algorithm, which: (1) can readily remove nonstationarities from the signal, (2) can more effectively reduce noise in the signals than linear filters, wavelet denoising, and chaos-based noise reduction techniques; (3) can readily decompose a multiscale biosignal into a series of intrinsically bandlimited functions; and (4) offers a new formulation of fractal and multifractal analysis that is better than existing methods when a biosignal contains a strong oscillatory component.ConclusionsThe presented approach is a valuable, versatile tool for the analysis of various types of biological signals. Its effectiveness is demonstrated by offering new important insights into brainwave dynamics and the very high accuracy in automatically detecting epileptic seizures from EEG signals.