A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the empirical mode decomposition method with which any complicated data set can be dec...

/pdf/the-empirical-mode-decomposition-and-the-hilbert-spectrum-qkmom366dy.pdf

The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis

We survey the newly developed Hilbert spectral analysis method and its applications to Stokes waves, nonlinear wave evolution processes, the spectral form of the random wave field, and turbulence. Our emphasis is on the inadequacy of presently available methods in nonlinear and nonstationary data analysis. Hilbert spectral analysis is here proposed as an alternative. This new method provides not only a more precise definition of particular events in time-frequency space than wavelet analysis, but also more physically meaningful interpretations of the underlying dynamic processes.

/pdf/a-new-view-of-nonlinear-water-waves-the-hilbert-spectrum-42def0zzxb.pdf

A new view of nonlinear water waves: the Hilbert spectrum

Almost all variables in biology are nonstationarily stochastic. For these variables, the conventional tools leave us a feeling that some valuable information is thrown away and that a complex phenomenon is presented imprecisely. Here, we apply recent advances initially made in the study of ocean waves to study the blood pressure waves in the lung. We note first that, in a long wave train, the handling of the local mean is of predominant importance. It is shown that a signal can be described by a sum of a series of intrinsic mode functions, each of which has zero local mean at all times. The process of deriving this series is called the “empirical mode decomposition method.” Conventionally, Fourier analysis represents the data by sine and cosine functions, but no instantaneous frequency can be defined. In the new way, the data are represented by intrinsic mode functions, to which Hilbert transform can be used. Titchmarsh [Titchmarsh, E. C. (1948) Introduction to the Theory of Fourier Integrals (Oxford Univ. Press, Oxford)] has shown that a signal and i times its Hilbert transform together define a complex variable. From that complex variable, the instantaneous frequency, instantaneous amplitude, Hilbert spectrum, and marginal Hilbert spectrum have been defined. In addition, the Gumbel extreme-value statistics are applied. We present all of these features of the blood pressure records here for the reader to see how they look. In the future, we have to learn how these features change with disease or interventions.

https://www.pnas.org/content/pnas/95/9/4816.full.pdf

Engineering analysis of biological variables: An example of blood pressure over 1 day

Almost all variables in biology are nonsta- tionarily stochastic. For these variables, the conventional tools leave us a feeling that some valuable information is thrown away and that a complex phenomenon is presented impre- cisely. Here, we apply recent advances initially made in the study of ocean waves to study the blood pressure waves in the lung. We note first that, in a long wave train, the handling of the local mean is of predominant importance. It is shown that a signal can be described by a sum of a series of intrinsic mode functions, each of which has zero local mean at all times. The process of deriving this series is called the ''empirical mode decomposition method.'' Conventionally, Fourier analysis represents the data by sine and cosine functions, but no instantaneous frequency can be defined. In the new way, the data are represented by intrinsic mode functions, to which Hilbert transform can be used. Titchmarsh (Titchmarsh, E. C. (1948) Introduction to the Theory of Fourier Integrals (Oxford Univ. Press, Oxford)) has shown that a signal and i times its Hilbert transform together define a complex variable. From that complex variable, the instantaneous frequency, instan- taneous amplitude, Hilbert spectrum, and marginal Hilbert spectrum have been defined. In addition, the Gumbel extreme- value statistics are applied. We present all of these features of the blood pressure records here for the reader to see how they look. In the future, we have to learn how these features change with disease or interventions.

Engineering analysis of biological variables: An example of blood pressure over 1 day (Fourier spectrumyHilbert spectrumyGumbel extreme-value statisticsypulmonary artery)

In order to identify coherent structure of turbulent shear flow, a new combination of familiar techniques of signal processing, called wavelet correlation analysis, is developed based on the wavelet transform. The wavelet correlation analysis provides the unique capability for decomposing the correlation of arbitrary signals over a two-dimensional time delay-period plane. By analyzing two superposition functions implicating several pure frequencies, the correlation of periodic oscillations at several frequencies can well be separated and observed clearly. Coherent structures in the intermediate region of a plane turbulent jet are investigated using the wavelet correlation method

S.R. Long

Papers

A 2-D complex wavelet analysis of an unsteady wind-generated surface wave field