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A Plea for Adaptive Data Analysis: Instantaneous Frequencies and Trends For Nonstationary Nonlinear Data

A Plea for Adaptive Data Analysis: Instantaneous Frequencies and Trends For Nonstationary Nonlinear Data. Norden E. Huang Research Center for Adaptive Data Analysis National Central University Zhongli, Taiwan, China Hot Topic Conference, 2011.

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A Plea for Adaptive Data Analysis: Instantaneous Frequencies and Trends For Nonstationary Nonlinear Data

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  1. A Plea for Adaptive Data Analysis:Instantaneous Frequencies and Trends For Nonstationary Nonlinear Data Norden E. Huang Research Center for Adaptive Data Analysis National Central University Zhongli, Taiwan, China Hot Topic Conference, 2011

  2. In search of frequency I found the trend and other information Instantaneous Frequencies and Trends For Nonstationary Nonlinear Data

  3. 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

  4. How to define frequency? It seems to be trivial. But frequency is an important parameter for us to understand many physical phenomena.

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

  6. Traditional Definition of Frequency • frequency = 1/period. • Definition too crude • Only work for simple sinusoidal waves • Does not apply to nonstationary processes • Does not work for nonlinear processes • Does not satisfy the need for wave equations

  7. 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.

  8. Instantaneous Frequency

  9. Hilbert Transform : Definition

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

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

  12. Traditional Viewa la Hahn (1995) : Hilbert

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

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

  15. Traditional Approacha la Hahn (1995) : Frequency

  16. Why the traditional approach does not work?

  17. Hilbert Transform a cos  + b : Data

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

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

  20. Hilbert Transform a cos  + b : Frequency

  21. The Empirical Mode Decomposition Method and Hilbert Spectral AnalysisSifting

  22. Empirical Mode Decomposition: Methodology : Test Data

  23. Empirical Mode Decomposition: Methodology : data and m1

  24. Empirical Mode Decomposition: Methodology : data & h1

  25. Empirical Mode Decomposition: Methodology : h1 & m2

  26. Empirical Mode Decomposition: Methodology : h3 & m4

  27. Empirical Mode Decomposition: Methodology : h4 & m5

  28. Empirical Mode DecompositionSifting : to get one IMF component

  29. 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.

  30. Effects of Sifting • To remove ridding waves • To reduce amplitude variations • The systematic study of the stoppage criteria leads to the conjecture connecting EMD and Fourier Expansion.

  31. Empirical Mode Decomposition: Methodology : IMF c1

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

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

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

  35. Definition of Instantaneous Frequency

  36. An Example of Sifting & Time-Frequency Analysis

  37. Length Of Day Data

  38. LOD : IMF

  39. 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

  40. LOD : Data & c12

  41. LOD : Data & Sum c11-12

  42. LOD : Data & sum c10-12

  43. LOD : Data & c9 - 12

  44. LOD : Data & c8 - 12

  45. LOD : Detailed Data and Sum c8-c12

  46. LOD : Data & c7 - 12

  47. LOD : Detail Data and Sum IMF c7-c12

  48. LOD : Difference Data – sum all IMFs

  49. Properties of EMD Basis The Adaptive Basis based on and derived from the data by the empirical method satisfy nearly all the traditional requirements for basis empirically and a posteriori: Complete Convergent Orthogonal Unique

  50. The combination of Hilbert Spectral Analysis and Empirical Mode Decomposition has been designated by NASA as HHT (HHT vs. FFT)

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