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Norden E. Huang Research Center for Adaptive Data Analysis

An Introduction to HHT: Instantaneous Frequency, Trend, Degree of Nonlinearity and Non- stationarity. Norden E. Huang Research Center for Adaptive Data Analysis Center for Dynamical Biomarkers and Translational Medicine NCU, Zhongli , Taiwan, China.

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Norden E. Huang Research Center for Adaptive Data Analysis

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  1. An Introduction to HHT: Instantaneous Frequency, Trend, Degree of Nonlinearity and Non-stationarity Norden E. Huang Research Center for Adaptive Data Analysis Center for Dynamical Biomarkers and Translational Medicine NCU, Zhongli, Taiwan, China

  2. OutlineRather than the implementation details, I will talk about the physics of the method. • What is frequency? • How to quantify the degree of nonlinearity? • How to define and determine trend?

  3. What is frequency? It seems to be trivial. But frequency is an important parameter for us to understand many physical phenomena.

  4. Definition of Frequency Given the period of a wave as T ; the frequency is defined as

  5. Instantaneous Frequency

  6. Other Definitions of Frequency : For any data from linear Processes

  7. Definition of Power Spectral Density Since a signal with nonzero average power is not square integrable, the Fourier transforms do not exist in this case. Fortunately, the Wiener-Khinchin Theroem provides a simple alternative. The PSD is the Fourier transform of the auto-correlation function, R(τ), of the signal if the signal is treated as a wide-sense stationary random process:

  8. Fourier Spectrum

  9. Problem with Fourier Frequency • Limited to linear stationary cases: same spectrum for white noise and delta function. • Fourier is essentially a mean over the whole domain; therefore, information on temporal (or spatial) variations is all lost. • Phase information lost in Fourier Power spectrum: many surrogate signals having the same spectrum.

  10. Surrogate Signal:Non-uniqueness signal vs. Power Spectrum I. Hello

  11. The original data : Hello

  12. The surrogate data : Hello

  13. The Fourier Spectra : Hello

  14. The Importance of Phase

  15. To utilize the phase to define Instantaneous Frequency

  16. Prevailing Views onInstantaneous Frequency The term, Instantaneous Frequency, should be banished forever from the dictionary of the communication engineer. J. Shekel, 1953 The uncertainty principle makes the concept of an Instantaneous Frequency impossible. K. Gröchennig, 2001

  17. The Idea and the need of Instantaneous Frequency According to the classic wave theory, the wave conservation law is based on a gradually changing φ(x,t) such that Therefore, both wave number and frequency must have instantaneous values and differentiable.

  18. Hilbert Transform : Definition

  19. The Traditional View of the Hilbert Transform for Data Analysis

  20. Traditional Viewa la Hahn (1995) : Data LOD

  21. Traditional Viewa la Hahn (1995) : Hilbert

  22. Traditional Approacha la Hahn (1995) : Phase Angle

  23. Traditional Approacha la Hahn (1995) : Phase Angle Details

  24. Traditional Approacha la Hahn (1995) : Frequency

  25. The Real World Mathematics are well and good but nature keeps dragging us around by the nose. Albert Einstein

  26. Why the traditional approach does not work?

  27. Hilbert Transform a cos + b: Data

  28. Hilbert Transform a cos  + b : Phase Diagram

  29. Hilbert Transform a cos  + b : Phase Angle Details

  30. Hilbert Transform a cos  + b : Frequency

  31. The Empirical Mode Decomposition Method and Hilbert Spectral AnalysisSifting(Other alternatives, e.g., Nonlinear Matching Pursuit)

  32. Empirical Mode Decomposition: Methodology : Test Data

  33. Empirical Mode Decomposition: Methodology : data and m1

  34. Empirical Mode Decomposition: Methodology : data & h1

  35. Empirical Mode Decomposition: Methodology : h1 & m2

  36. Empirical Mode Decomposition: Methodology : h3 & m4

  37. Empirical Mode Decomposition: Methodology : h4 & m5

  38. Empirical Mode DecompositionSifting : to get one IMF component

  39. The Stoppage Criteria The Cauchy type criterion: when SD is small than a pre-set value, where Or, simply pre-determine the number of iterations.

  40. Empirical Mode Decomposition: Methodology : IMF c1

  41. Definition of the Intrinsic Mode Function (IMF): a necessary condition only!

  42. Empirical Mode Decomposition: Methodology : data, r1 and m1

  43. Empirical Mode DecompositionSifting : to get all the IMF components

  44. Definition of Instantaneous Frequency

  45. An Example of Sifting & Time-Frequency Analysis

  46. Length Of Day Data

  47. LOD : IMF

  48. Pair-wise % 0.0003 0.0001 0.0215 0.0117 0.0022 0.0031 0.0026 0.0083 0.0042 0.0369 0.0400 Overall % 0.0452 Orthogonality Check

  49. LOD : Data & c12

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