1. The Empirical Mode Decomposition Method Sifting

2. Goal of Data Analysis To define time scale or frequency. To define energy density. To define joint frequency-energy distribution as a function of time. To do this, we need a AM-FM decomposition of the signal: X(t) = A(t) cosθ(t), where A(t) defines local energy and θ(t) defines the local frequency. This is a Generalized Fourier Expansion.

3. Generalized Fourier Expansion

4. Need for Decomposition Hilbert Transform (and all other IF computation methods) offers meaningful Instantaneous Frequency for IMFs. For complicate data, there should be more than one independent component at any given time. The decomposition should be adaptive in order to study data from nonstationary and nonlinear processes. Frequency space operations are difficult to track temporal changes.

5. Why Hilbert Transform is not enough? Even though mathematicians told us that the Hilbert transform exists for all functions of Lp-class.

6. Problems on ‘Envelope’ A seemingly simple proposition but it is not so easy.

7. Two examples

8. Data set 1

9. Data X1

10. Data X1 Hilbert Transform

11. Data X1 Envelopes

12. Observations None of the two “envelopes” seem to make sense in term of Generalized Fourier Expansion (GFE): The Hilbert transformed amplitude oscillates too much. The line connecting the local maximum is almost the tracing of the data. It turns out that, though Hilbert transform exists, the simple Hilbert transform does not make sense physically. For “envelopes” to make sense in terms of GFE, the necessary condition for Hilbert transformed amplitude to make sense is for IMF.

13. Data X1 IMF

14. Data x1 IMF1

15. Data x2 IMF2

16. Observations For each IMF, the envelope in GFE will make sense. For complicate data, we have to decompose it before attempting envelope construction. To be able to determine the envelope is equivalent to AM & FM decomposition.

17. Data set 2

18. Data X2

19. Data X2 Hilbert Transform

20. Data X2 Envelopes

21. Observations Even for this well behaved function, the amplitude from Hilbert transform does not serve as an “envelope” well. One of the reasons is that the function has two spectrum lines. Hilbert Transform represents higher frequency better. Complications for more complex functions are many. Here, the empirical envelope seems reasonable.

22. Empirical Mode Decomposition Mathematically, there are infinite number of ways to decompose a functions into a complete set of components. The ones that give us more physical insight are more significant. In general, the fewer the number of representing components, the higher the information content: Sparse representation. The adaptive method will represent the characteristics of the signal better. EMD is an adaptive method that can generate infinite many sets of IMF components to represent the original data.

23. Empirical Mode Decomposition: Methodology : Test Data

24. Empirical Mode Decomposition: Methodology : data and m1

25. Empirical Mode Decomposition: Methodology : data & h1

26. Empirical Mode Decomposition: Methodology : h1 & m2

27. Empirical Mode Decomposition: Methodology : h2 & m3

28. Empirical Mode Decomposition: Methodology : h3 & m4

29. Empirical Mode Decomposition: Methodology : h2 & h3

30. Empirical Mode Decomposition: Methodology : h4 & m5

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

33. Empirical Mode Decomposition: Methodology : IMF c1

34. Definition of the Intrinsic Mode Function

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

39. Observations All IMF components are the sums of spline functions. We selected cubic natural spline to maintain the maximum smoothness. We will discuss spline function next time.

40. Empirical Mode Decomposition: Methodology : data & r1

41. Empirical Mode Decomposition: Methodology : data, h1 & r1

42. Empirical Mode Decomposition: Methodology : IMFs

43. Definition of Instantaneous Frequency

44. Definitions of Frequency

45. The Effects of Sifting The first effect of sifting is to eliminate the riding waves : to make the number of extrema equals to that of zero-crossing. The second effect of sifting is to make the envelopes symmetric. The consequence is to make the amplitudes of the oscillations more even.

46. Singularity points for Instantaneous Frequency

47. Critical Parameters for EMD The maximum number of sifting allowed to extract an IMF, N. The criterion for accepting a sifting component as an IMF, the Stoppage criterion S. Therefore, the nomenclature for the IMF are CE(N, S) : for extrema sifting CC(N, S) : for curvature sifting

48. The 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. If the mean is smaller than a pre-assigned value. C. Fixed sifting (iterating) time. D. SD is small than a pre-set value, where

49. Curvature Sifting Hidden Scales

50. Empirical Mode Decomposition: Methodology : Test Data

51. Hidden Scales

53. Observations If we decide to use curvature, we have to be careful for what we ask for. For example, the Duffing pendulum would produce more than one components. Therefore, curvature sifting is used sparsely. It is useful in the first couple of components to get rid of noises.

54. Intermittence Test To alleviate the Mode Mixing

55. Sifting with Intermittence Test To avoid mode mixing, we have to institute a special criterion to separate oscillation of different time scales into different IMF components. The criteria is to select time scale so that oscillations with time scale shorter than this pre- selected criterion is not included in the IMF.

56. Intermittence Sifting : Data

57. Intermittence Sifting : IMF

58. Intermittence Sifting : Hilbert Spectra

59. Intermittence Sifting : Hilbert Spectra (Low)

60. Intermittence Sifting : Marginal Spectra

61. Intermittence Sifting : Marginal spectra (Low)

62. Intermittence Sifting : Marginal spectra (High)

63. Critical Parameters for Sifting Because of the inclusion of intermittence test there will be one set of intermittence criteria. Therefore, the Nomenclature for IMF here are CEI(N,S: n1, n2, …) CCI(N, S: n1, n2, …) with n1, n2 as the intermittence test criteria.

64. The mathematical Requirements for Basis The traditional Views

65. IMF as Adaptive Basis According to the established mathematical paradigm, we should check the following properties of the basis: Convergence completeness orthogonality Uniqueness

66. Convergence

67. Convergence Problem Given an arbitrary number, ε, there always exists a large finite number N, such that Nth envelope mean, m N, satisfies | m N | ≤ε:

68. Convergence Problem Given an arbitrary number, ε, there always exists a large finite number N, such that N- th sifting satisfies

69. Convergence There is another convergence problem: we have only finite number of components. Complete proof for convergence is underway. We can prove the convergence under simplified condition of linear segment fitting for sifting. Empirically, we found all cases converge in finite steps. The finite component, n, is less than or equal to log 2 M, with M as the total number of data points.

70. Convergence The necessary condition for convergence is that the mean line should have less extrema than the original data. This might not be true if we use the middle points and a single spline; the procedure might not converge.

71. Completeness

72. Completeness is given by the algebraic equation Therefore, the sum of IMF can be as close to the original data as required. Completeness is given.

73. Orthogonality

74. Definition: Two vectors x and y are orthogonal if their inner product is zero. x ∙y = (x 1 y 1 + x 2 y 2 + x 3 y 3 + … ) = 0.

75. The need for an orthogonality check Orthogonal is required for:

76. Orthogonality Orthogonality is a requirement for any linear decomposition. For a nonlinear decomposition, as EMD, the orthogonality should not be a requirement, for nonlinear waves of different scale could share the same harmonics. Fortunately, the EMD is basically a Reynolds type decomposition, U = + u’, orthogonality is always approximately satisfied to the degree of nonlinearity. Orthogonality Index should be checked for each cases as a goodness of decomposition confirmation.

77. Orthogonality

78. Orthogonality Index

79. Length Of Day Data

80. LOD : IMF

81. 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

82. Uniqueness

83. EMD, with different critical parameters, can generate infinite sets of IMFs. The result is unique only with respect to the critical parameters and sifting method selected; therefore, all results should be properly named according to the nomenclature scheme proposed above. The present sifting is based on cubic spline. Different spline fitting in the sifting procedure will generate different results. The ensemble of IMF sets offers a Confidence Limit as function of time and frequency.

84. Some Tricks in Sifting

85. Sometimes straightforward application of sifting will not generate good results. Invoking intermittence criteria is an alternative to get physically meaningful IMF components. By adding low level noise can improve the sifting. By using curvature may also help.

86. An Example Adding Noise of small amplitude only, A prelude to the true Ensemble EMD

87. Data: 2 Coincided Waves

88. IMF from Data of 2 Coincided Waves

89. Data: 2 Coincided Waves + Noise The Amplitude of the noise is 1/1000

90. IMF form Data 2 Coincided Waves + Noise

91. IMF c1 and Component2 : 2 Coincided Waves

92. IMF c2+c3 and Component1 : 2 Coincided Waves

93. A Flow Chart Data IMF sifting With Intermittence Hilbert Spectrum IF Marginal Spectrum OI Ensemble EMD