A Plea for Adaptive Data Analysis: An Introduction to HHT for Nonlinear and Nonstationary Data

1. A Plea for Adaptive Data Analysis:An Introduction to HHT for Nonlinear and Nonstationary Data Norden E. Huang Research Center for Adaptive Data Analysis National Central University Nanjing October 2009

2. Data Processing and Data Analysis • Processing [proces < L. Processus < pp of Procedere = Proceed: pro- forward + cedere, to go] : A particular method of doing something. • Data Processing >>>> Mathematically meaningful parameters • Analysis[Gr. ana, up, throughout + lysis, a loosing] : A separating of any whole into its parts, especially with an examination of the parts to find out their nature, proportion, function, interrelationship etc. • Data Analysis >>>> Physical understandings

3. Scientific Activities Collecting and analyzing data, synthesizing and theorizing the analyzed results are the core of scientific activities. Therefore, data analysis is a key link in this continuous loop.

4. Data Analysis There are, unfortunately, tensions between sciences and mathematics. Data analysis is too important to be left to the mathematicians. Why?!

5. Mathematicians Absolute proofs Logic consistency Mathematical rigor Scientists/Engineers Agreement with observations Physical meaning Working Approximations Different Paradigms Mathematics vs. Science/Engineering

6. Motivations for alternatives: Problems for Traditional Methods • Physical processes are mostly nonstationary • Physical Processes are mostly nonlinear • Data from observations are invariably too short • Physical processes are mostly non-repeatable.  Ensemble mean impossible, and temporal mean might not be meaningful for lack of stationarity and ergodicity. Traditional methods are inadequate.

7. Hilbert Transform : Definition

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

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

10. Traditional Viewa la Hahn (1995) : Hilbert

11. The Empirical Mode Decomposition Method and Hilbert Spectral AnalysisSifting

12. Empirical Mode Decomposition: Methodology : Test Data

13. Empirical Mode Decomposition: Methodology : data and m1

14. Empirical Mode DecompositionSifting : to get one IMF component

15. 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.

16. Empirical Mode Decomposition: Methodology : IMF c1

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

18. Definition of Instantaneous Frequency

19. 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. But how to find φ(x, t)?

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

21. Comparison between FFT and HHT

22. Comparisons: Fourier, Hilbert & Wavelet

23. Speech AnalysisHello : Data

24. Four comparsions D

25. An Example of Sifting

26. Length Of Day Data

27. LOD : IMF

28. 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

29. LOD : Data & c12

30. LOD : Data & Sum c11-12

31. LOD : Data & sum c10-12

32. LOD : Data & c9 - 12

33. LOD : Data & c8 - 12

34. LOD : Detailed Data and Sum c8-c12

35. LOD : Data & c7 - 12

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

37. LOD : Difference Data – sum all IMFs

38. Traditional Viewa la Hahn (1995) : Hilbert

39. Mean Annual Cycle & Envelope: 9 CEI Cases

40. 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

41. Hilbert’s View on Nonlinear Data

42. Duffing Type WaveData:x = cos(wt+0.3 sin2wt)

43. Duffing Type WavePerturbation Expansion

44. Duffing Type WaveWavelet Spectrum

45. Duffing Type WaveHilbert Spectrum

46. Duffing Type WaveMarginal Spectra

47. Ensemble EMDNoise Assisted Signal Analysis (nasa) Utilizing the uniformly distributed reference frame based on the white noise to eliminate the mode mixing Enable EMD to apply to function with spiky or flat portion The true result of EMD is the ensemble of infinite trials. Wu and Huang, Adv. Adapt. Data Ana., 2009

48. New Multi-dimensional EEMD • Extrema defined easily • Computationally inexpensive, relatively • Ensemble approach removed the Mode Mixing • Edge effects easier to fix in each 1D slice • Results are 2-directional Wu, Huang and Chen, AADA, 2009

49. What This Means • EMD separates scales in physical space; it generates an extremely sparse representation for any given data. • Added noises help to make the decomposition more robust with uniform scale separations. • Instantaneous Frequency offers a total different view for nonlinear data: instantaneous frequency needs no harmonics and is unlimited by uncertainty principle. • Adaptive basis is indispensable for nonstationary and nonlinear data analysis • EMD establishes a new paradigm of data analysis

50. Comparisons