A Plea for Adaptive Data Analysis An Introduction to HHT

1. A Plea for Adaptive Data AnalysisAn Introduction to HHT Norden E. Huang Research Center for Adaptive Data Analysis National Central University

2. Ever since the advance of computer, there is an explosion of data. The situation has changed from a thirsty for data to that of drinking from a fire hydrant.We are drowning in data, but thirsty for knowledge!

3. Henri Poincaré Science is built up of facts*, as a house is built of stones; but an accumulation of facts is no more a science than a heap of stones is a house. * Here facts are indeed data.

4. Scientific Activities Collecting, analyzing, synthesizing, and theorizing are the core of scientific activities. Theory without data to prove is just hypothesis. Therefore, data analysis is a key link in this continuous loop.

5. Data Analysis Data analysis is too important to be left to the mathematicians. Why?!

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

7. Mathematicians Idealized Spaces Perfect world in which everything is known Inconsistency in the different spaces and the real world Scientists/Engineers Real Space Real world in which knowledge is incomplete and limited Constancy in the real world within allowable approximation Different Paradigms IIMathematics vs. Science/Engineering

8. Rigor vs. Reality As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Albert Einstein

9. 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

10. Traditional Data Analysis All traditional ‘data analysis’ methods are either developed by or established according to mathematician’s rigorous rules. They are really ‘data processing’ methods. In pursue of mathematic rigor and certainty, however, we are forced to idealize, but also deviate from, the reality.

11. Traditional Data Analysis As a result, we are forced to live in a pseudo-real world, in which all processes are Linear and Stationary

12. 削足適履 Trimming the foot to fit the shoe.

13. Available ‘Data Analysis’ Methodsfor Nonstationary (but Linear) time series • Spectrogram • Wavelet Analysis • Wigner-Ville Distributions • Empirical Orthogonal Functions aka Singular Spectral Analysis • Moving means • Successive differentiations

14. Available ‘Data Analysis’ Methodsfor Nonlinear (but Stationary and Deterministic) time series • Phase space method • Delay reconstruction and embedding • Poincaré surface of section • Self-similarity, attractor geometry & fractals • Nonlinear Prediction • Lyapunov Exponents for stability

15. Typical Apologia • Assuming the process is stationary …. • Assuming the process is locally stationary …. • As the nonlinearity is weak, we can use perturbation approach …. Though we can assume all we want, but the reality cannot be bent by the assumptions.

16. 掩耳盜鈴 Stealing the bell with muffed ears

17. 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.

18. The Job of a Scientist The job of a scientist is to listen carefully to nature, not to tell nature how to behave. Richard Feynman To listen is to use adaptive method and let the data sing, and not to force the data to fit preconceived modes.

19. Characteristics of Data from Nonlinear Processes

20. Duffing Pendulum x

21. Duffing Equation : Data

22. Hilbert Transform : Definition

23. Hilbert Transform Fit

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

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

26. Traditional Viewa la Hahn (1995) : Hilbert

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

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

29. Traditional Approacha la Hahn (1995) : Frequency

30. Why the traditional approach does not work?

31. Hilbert Transform a cos  + b : Data

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

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

34. Hilbert Transform a cos  + b : Frequency

35. The Empirical Mode Decomposition Method and Hilbert Spectral AnalysisSifting

36. Empirical Mode Decomposition: Methodology : Test Data

37. Empirical Mode Decomposition: Methodology : data and m1

38. Empirical Mode Decomposition: Methodology : data & h1

39. Empirical Mode Decomposition: Methodology : h1 & m2

40. Empirical Mode Decomposition: Methodology : h3 & m4

41. Empirical Mode Decomposition: Methodology : h4 & m5

42. Empirical Mode DecompositionSifting : to get one IMF component

43. The Stoppage Criteria The Cauchy type criterion: when SD is small than a pre-set value, where

44. Empirical Mode Decomposition: Methodology : IMF c1

45. Definition of the Intrinsic Mode Function (IMF)

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

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

48. Definition of Instantaneous Frequency

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

50. Instantaneous Frequency