1. A Plea for Adaptive Data Analysis An 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. Different Paradigms I Mathematics vs. Science/Engineering Mathematicians Absolute proofs Logic consistency Mathematical rigor Scientists/Engineers Agreement with observations Physical meaning Working Approximations

7. 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

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 andStationary

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

13. 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

14. 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

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 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. The Job of a Scientist

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 View a la Hahn (1995) : Data LOD

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

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

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

29. Traditional Approach a 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 Analysis Sifting

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 Decomposition Sifting : 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 Decomposition Sifting : 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

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

52. 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

53. Comparison between FFT and HHT

54. Comparisons: Fourier, Hilbert & Wavelet

55. Speech Analysis Hello : Data

56. Four comparsions D

57. An Example of Sifting

58. Length Of Day Data

59. LOD : IMF

60. 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

61. LOD : Data & c12

62. LOD : Data & Sum c11-12

63. LOD : Data & sum c10-12

64. LOD : Data & c9 - 12

65. LOD : Data & c8 - 12

66. LOD : Detailed Data and Sum c8-c12

67. LOD : Data & c7 - 12

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

69. LOD : Difference Data – sum all IMFs

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

71. Mean Annual Cycle & Envelope: 9 CEI Cases

72. 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

73. Hilbert ’ s View on Nonlinear Data

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

75. Duffing Type Wave Perturbation Expansion

76. Duffing Type Wave Wavelet Spectrum

77. Duffing Type Wave Hilbert Spectrum

78. Duffing Type Wave Marginal Spectra

79. Duffing Equation

80. Duffing Equation : Data

81. Duffing Equation : IMFs

82. Duffing Equation : Hilbert Spectrum

83. Duffing Equation : Detailed Hilbert Spectrum

84. Duffing Equation : Wavelet Spectrum

85. Duffing Equation : Hilbert & Wavelet Spectra

86. 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

87. Comparisons FourierWaveletHilbert Basisa priori Adaptive FrequencyIntegral Transform: Global Integral Transform: Regional Differentiation: Local PresentationEnergy-frequencyEnergy-time- frequency Nonlinearno yes Non-stationarynoyes Uncertaintyyes no Harmonicsyes no

88. 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.

89. 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 Gravitational Wave –(NASA Goddard) Financial market data analysis –(NCU) Geophysical and Climate studies –(COLA, NASA, NCU)

90. Outstanding Mathematical Problems 1.Adaptive data analysis methodology in general 2.Nonlinear system identification methods 3.Prediction problem for nonstationary processes (end effects) 4.Optimization problem (the best IMF selection and uniqueness. Is there a unique solution?) 5.Spline problem (best spline implement of HHT, convergence and 2-D) 6.Approximation problem (Hilbert transform and quadrature)

91. 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, A460, 1597-1611 Defined statistical significance and predictability. 2007: On the trend, detrending, and variability of nonlinear and nonstationary time series. Proc. Natl. Acad. Sci., 104, 14,889-14,894. The correct adaptive trend determination method 2008: On Ensemble Empirical Mode Decomposition. Advances in Adaptive Data Analysis (in press) 2008: On instantaneosu Frequency. Advances in Adaptive Data Analysis (Accepted)

92. 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 To be launched in the March 2008

93. Thanks!

95. Milankovitch Time scales: Temperature Data from Vostok Ice Core Data length 3311 points covering 422,766 Years BP

96. Milutin Milankovitch 1879-1958

97. How the Sun Affects Climate: Solar and Milankovitch Cycles The

98. Milankovitch Cycles

99. A Truly Nonlinear World

100. Data and Even Spaced Spline at Dt=20 Year

103. Data and CKY11 with Error Bounds

104. Data and CKY10 with Error Bounds

105. Data and CKY9 with Error Bounds

106. Data and CKY8 with Error Bounds

107. Data and Sum CKY 11 to 13 : 100K

108. Data and Sum CKY 10 to 13 : 40K

109. Data and Sum CKY 9 to 13 : 25K

110. Data and Sum CKY 8 to 13 : 10K

111. Data and Sum CKY 1 : 7 : Less than 10K

112. Hilbert Spectrum CKY 8:11

113. Marginal Hilbert Spectrum CKY 8:11