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要旨:
Multiresolution wavelet analysis (MWA) is a powerful data processing tool that provides a characterization of complex signals over multiple time scales. Typically, the standard deviations of wavelet coefficients are computed depending on the resolution level and such quantities are used as measures for diagnosing different types of system behavior. To enhance the capabilities of this tool, we propose a combination of MWA with detrended fluctuation analysis (DFA) of detail wavelet coefficients. We find that such an MWA&DFA approach is capable of revealing the correlation features of wavelet coefficients in independent ranges of scales, which provide more information about the complex organization of datasets compared to variances or similar statistical measures of the standard MWA. Using this approach, we consider changes in the dynamics of coupled chaotic systems caused by transitions between different types of complex oscillations. We also demonstrate the potential of the MWA&DFA method for characterizing different physiological conditions by analyzing the electrical brain activity in mice.
Complex signals with multiple time scales are often analyzed by decomposing them into simpler components that can be easily characterized with clear numerical quantities. Fourier transform and Hilbert transform are classic examples of decompositions aimed at introducing simpler characteristics: the magnitudes of harmonic components, instantaneous amplitudes, or frequencies. In the case of time-varying dynamics with localized features of the datasets, an expansion in terms of wavelet functions is often provided, which can be very different and efficient under the fulfillment of some general constraints. The statistics of the multiresolution wavelet analysis (MWA) decomposition coefficients enable detecting short-term changes in system behavior and can be applied for diagnostics in many areas of science and technology. Unlike the generally used analysis of the standard deviation of detail wavelet coefficients vs resolution level, we propose an enhanced MWA dealing with the correlation analysis of these coefficients in order to identify information about their relationship. We show that such an analysis offers wider opportunities for studying complex systems and demonstrates several examples of its application, including chaotic oscillations in mathematical models of interacting systems and electroencephalograms (EEGs)
