1. An Introduction to Hilbert-Huang Transform: A Plea for Adaptive Data Analysis Norden E. Huang Research Center for Adaptive Data Analysis National Central University

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. 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.

3. Data Analysis Why we do it? How did we do it? What should we do?

4. Why?

5. Why do we have to analyze data? Data are the only connects we have with the reality; data analysis is the only means we can find the truth and deepen our understanding of the problems.

6. Ever since the advance of computer and sensor technology, there is an explosion of very complicate data. The situation has changed from a thirsty for data to that of drinking from a fire hydrant.

7. 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 our data.

8. Data and Data Analysis Data Analysis is the key step in converting the ‘ facts ’ into the edifice of science. It infuses meanings to the cold numbers, and lets data telling their own stories and singing their own songs.

9. Science vs. Philosophy Data and Data Analysis are what separate science from philosophy: With data we are talking about sciences; Without data we can only discuss philosophy.

10. 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.

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

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

13. Different Paradigms II Mathematics vs. Science/Engineering 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

14. 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

15. How?

16. Data Processing vs. Analysis All traditional ‘data analysis’ methods are really for ‘data processing’. They are either developed by or established according to mathematician’s rigorous rules. Most of the methods consist of standard algorithms, which produce a set of simple parameters. They can only be qualified as ‘data processing’, not really ‘data analysis’. Data processing produces mathematical meaningful parameters; data analysis reveals physical characteristics of the underlying processes.

17. Data Processing vs. Analysis In pursue of mathematic rigor and certainty, however, we lost sight of physics and are forced to idealize, but also deviate from, the reality. As a result, we are forced to live in a pseudo-real world, in which all processes are Linear and Stationary

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

19. Available Data Analysis Methods for Nonstationary (but Linear) time series Spectrogram Wavelet Analysis Wigner-Ville Distributions Empirical Orthogonal Functions aka Singular Spectral Analysis Moving means Successive differentiations

20. Available Data Analysis Methods for Nonlinear (but Stationary and Deterministic) time series Phase space method Delay reconstruction and embedding Poincar é surface of section Self-similarity, attractor geometry & fractals Nonlinear Prediction Lyapunov Exponents for stability

21. 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.

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

23. 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. ÈEnsemble mean impossible, and temporal mean might not be meaningful for lack of stationarity and ergodicity. ÈTraditional methods are inadequate.

24. What?

25. 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 methods and let the data sing, and not to force the data to fit preconceived modes. The Job of a Scientist

26. How to define nonlinearity? Based on Linear Algebra: nonlinearity is defined based on input vs. output. But in reality, such an approach is not practical. The alternative is to define nonlinearity based on data characteristics.

27. Characteristics of Data from Nonlinear Processes

28. Duffing Pendulum x

29. Hilbert Transform : Definition

30. Hilbert Transform Fit

31. Conformation to reality rather then to Mathematics We do not have to apologize, we should use methods that can analyze data generated by nonlinear and nonstationary processes. That means we have to deal with the intrawave frequency modulations, intermittencies, and finite rate of irregular drifts. Any method satisfies this call will have to be adaptive.

32. The Traditional Approach of Hilbert Transform for Data Analysis

33. Traditional Approach a la Hahn (1995) : Data LOD

34. Traditional Approach a la Hahn (1995) : Hilbert

35. Traditional Approach a la Hahn (1995) : Phase Angle

36. Traditional Approach a la Hahn (1995) : Phase Angle Details

37. Traditional Approach a la Hahn (1995) : Frequency

38. Why the traditional approach does not work?

39. Hilbert Transform a cos  + b : Data

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

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

42. Hilbert Transform a cos  + b : Frequency

43. The Empirical Mode Decomposition Method and Hilbert Spectral Analysis Sifting

44. Empirical Mode Decomposition: Methodology : Test Data

45. Empirical Mode Decomposition: Methodology : data and m1

46. Empirical Mode Decomposition: Methodology : data & h1

47. Empirical Mode Decomposition: Methodology : h1 & m2

48. Empirical Mode Decomposition: Methodology : h3 & m4

49. Empirical Mode Decomposition: Methodology : h4 & m5

50. Empirical Mode Decomposition Sifting : to get one IMF component

51. Two Stoppage Criteria : S and SD A.The S number : S is defined as the consecutive number of siftings, in which the numbers of zero- crossing and extrema are the same for these S siftings. B. SD is small than a pre-set value, where

52. Empirical Mode Decomposition: Methodology : IMF c1

53. Definition of the Intrinsic Mode Function (IMF)

54. Empirical Mode Decomposition Sifting : to get all the IMF components

55. Empirical Mode Decomposition: Methodology : data & r1

56. Empirical Mode Decomposition: Methodology : data and m1

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

58. Empirical Mode Decomposition: Methodology : IMFs

59. Definition of Instantaneous Frequency

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

61. Equivalence : The definition of frequency is equivalent to defining velocity as Velocity = Distance / Time

62. Instantaneous Frequency

63. The combination of Hilbert Spectral Analysis and Empirical Mode Decomposition is designated as HHT (HHT vs. FFT)

64. Jean-Baptiste-Joseph Fourier 1807 “On the Propagation of Heat in Solid Bodies” 1812 Grand Prize of Paris Institute “Théorie analytique de la chaleur” ‘... the manner in which the author arrives at these equations is not exempt of difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigor. ’ 1817 Elected to Académie des Sciences 1822 Appointed as Secretary of Math Section paper published Fourier’s work is a great mathematical poem. Lord Kelvin

65. Comparison between FFT and HHT

66. Comparisons: Fourier, Hilbert & Wavelet

67. An Example of Sifting

68. Length Of Day Data

69. LOD : IMF

70. Orthogonality Check 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

71. LOD : Data & c12

72. LOD : Data & Sum c11-12

73. LOD : Data & sum c10-12

74. LOD : Data & c9 - 12

75. LOD : Data & c8 - 12

76. LOD : Detailed Data and Sum c8-c12

77. LOD : Data & c7 - 12

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

79. LOD : Difference Data – sum all IMFs

80. Traditional View a la Hahn (1995) : Hilbert

81. Mean Annual Cycle & Envelope: 9 CEI Cases

82. Mean Hilbert Spectrum : All CEs

83. Tidal Machine

84. 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 a posteriori: Complete Convergent Orthogonal Unique

85. Hilbert ’ s View on Nonlinear Data

86. Duffing Type Wave Data: x = cos(wt+0.3 sin2wt)

87. Duffing Type Wave Perturbation Expansion

88. Duffing Type Wave Wavelet Spectrum

89. Duffing Type Wave Hilbert Spectrum

90. Duffing Type Wave Marginal Spectra

91. Duffing Equation

92. Duffing Equation : Data

93. Duffing Equation : IMFs

94. Duffing Equation : Hilbert Spectrum

95. Duffing Equation : Detailed Hilbert Spectrum

96. Duffing Equation : Wavelet Spectrum

97. Duffing Equation : Hilbert & Wavelet Spectra

98. Speech Analysis Nonlinear and nonstationary data

99. Speech Analysis Hello : Data

100. Four comparsions D

101. Global Temperature Anomaly Annual Data from 1856 to 2003

102. Global Temperature Anomaly 1856 to 2003

103. IMF Mean of 10 Sifts : CC(1000, I)

104. Statistical Significance Test

105. Data and Trend C6

106. Rate of Change Overall Trends : EMD and Linear

107. What This Means Instantaneous Frequency offers a total different view for nonlinear data: instantaneous frequency with no need for harmonics and unlimited by uncertainty. Adaptive basis is indispensable for nonstationary and nonlinear data analysis HHT establishes a new paradigm of data analysis

108. Comparisons FourierWaveletHilbert Basisa priori Adaptive FrequencyConvolution: Global Convolution: Regional Differentiation: Local PresentationEnergy-frequencyEnergy-time- frequency Nonlinearno yes Non-stationarynoyes Uncertaintyyes no Harmonicsyes no

109. Conclusion Adaptive method is the only scientifically meaningful way to analyze data. It is the only way to find out the underlying physical processes; therefore, it is indispensable in scientific research. It is physical, direct, and simple.

110. History of HHT 1998: The Empirical Mode Decomposition Method and the Hilbert Spectrum for Non-stationary Time Series Analysis, Proc. Roy. Soc. London, A454, 903-995. The invention of the basic method of EMD, and Hilbert transform for determining the Instantaneous Frequency and energy. 1999: A New View of Nonlinear Water Waves – The Hilbert Spectrum, Ann. Rev. Fluid Mech. 31, 417-457. Introduction of the intermittence in decomposition. 2003: A confidence Limit for the Empirical mode decomposition and the Hilbert spectral analysis, Proc. of Roy. Soc. London, A459, 2317-2345. Establishment of a confidence limit without the ergodic assumption. 2004: A Study of the Characteristics of White Noise Using the Empirical Mode Decomposition Method, Proc. Roy. Soc. London, (in press) Defined statistical significance and predictability. 2004: On the Instantaneous Frequency, Proc. Roy. Soc. London, (Under review) Removal of the limitations posted by Bedrosian and Nuttall theorems for instantaneous Frequency computations.

111. Current Applications Non-destructive Evaluation for Structural Health Monitoring –(DOT, NSWC, and DFRC/NASA, KSC/NASA Shuttle) Vibration, speech, and acoustic signal analyses –(FBI, MIT, and DARPA) Earthquake Engineering –(DOT) Bio-medical applications –(Harvard, UCSD, Johns Hopkins) Global Primary Productivity Evolution map from LandSat data –(NASA Goddard, NOAA) Cosmological Gravity Wave –(NASA Goddard) Financial market data analysis –(NCU)

112. Advances in Adaptive data Analysis: Theory and Applications A new journal to be published by the World Scientific Under the joint Co-Editor-in-Chief Norden E. Huang, RCADA NCU Thomas Yizhao Hou, CALTECH in the January 2008

113. Oliver Heaviside 1850 - 1925 Why should I refuse a good dinner simply because I don't understand the digestive processes involved.