1. ENSEMBLE EMPIRICAL MODE DECOMPOSITION Noise Assisted Signal Analysis (nasa) Part II EEMD Zhaohua Wu and N. E. Huang: Ensemble Empirical Mode Decomposition: A Noise Assisted Data Analysis Method. Advances in Adaptive Data Analysis, 1, 1-41, 2009

2. The Ensemble Effects The true answer is not the one without perturbation of noise.

3. EXAMPLE : ORIGINAL DATA

4. “LOCAL” -> “LOCAL”

5. Ensemble EMD Solves the mode mixing problem utilizing the uniformly distributed reference frame based on the white noise

6. EXAMPLE : ORIGINAL DECOMP.

7. Procedures for EEMD Add a white noise series to the targeted data; Decompose the data with added white noise into IMFs; Repeat step 1 and step 2 again and again, but with different white noise series each time; and Obtain the (ensemble) means of corresponding IMFs of the decompositions as the final result.

8. Definition of Signal in EEMD The signal used in EEMD is given by :

9. Definition of IMF in EEMD The truth defined by EEMD is given by the number of the ensemble approaching infinite:

10. The Standard Deviation of EEMD With the truth defined, the discrepancy, Δ, should be in which E{ } is the expected value as given in Equation.

11. Effect of the White Noise The effects of the added white noise should decrease following the well established statistical rule:

12. Data of the Noise Effects: Dotted line = theoretical; solid line = high frequency components; dashed line = low frequency components.

13. Procedure for EEMD Illustration

14. EXAMPLE : E1 DECOMP.

15. EXAMPLE : E10 DECOMP.

16. EXAMPLE : E100 DECOMP.

17. EXAMPLE : Intermittence DECOMP.

18. EXAMPLE : Difference Main IMF.

19. EXAMPLE : Difference Intermittent Signal.

20. EXAMPLE : Difference Intermittent Signal Details.

21. EXAMPLE : Instantaneous Frequency from Main Signal.

22. Summary: Numerical Data From the intermittency Example, we see that the Ensemble EMD can generate IMFs with comparable quality as the ones through the Intermittence test. More ensemble in the average will improve confidence in the EMD results. The main advantage of Ensemble EMD is that we do not need to determine the ‘ Intermittence test criteria ’ subjectively, which could become impossible for complicated data.

23. Example I : Geophysical Data Surface Temperature Data from Two Difference Satellite Radiometer channels

24. EXAMPLE I: ORIGINAL DATA

25. EXAMPLE I: DECOMPOSITION (I)

26. EXAMPLE I: DECOMPOSITION (II)

27. EXAMPLE I: NOISY DATA (added noise std=0.1)

28. NOISY DATA DECOMPOSITION (I) (added noise std=0.1)

29. NOISY DATA DECOMPOSITION (II) (added noise std=0.1)

30. EXAMPLE I: NOISY DATA (added noise std=0.2)

31. NOISY DATA DECOMPOSITION (I) (added noise std=0.2)

32. NOISY DATA DECOMPOSITION (II) (added noise std=0.2)

33. NOISY DATA DECOMPOSITION (II) (RSS_T2)

35. NOISY DATA DECOMPOSITION (II) (UAH_T2)

37. EXAMPLE I: CORR. COEF.’s

38. Summary: Radiometer Data Data from the two different channels should reflect a similar overall structure, especially for medium and long wave length. Straightforward sifting will have severe mode mixing for medium scale IMFs. It is impossible to select the proper scales for the ‘ Intermittence test ’ to separate the modes. Ensemble EMD provided an automatic dyadic filter to separate the modes. Ensemble EMD especially effective when the data contain intermittent signal as in UAH case as shown by the correlation coefficients between RSS and UAH series.

39. Example II : Geophysical Data SOI and the Sea Surface Temperature at Nino 34

40. EXAMPLE II: ORIGINAL DATA

41. EXAMPLE II: DECOMPOSITION (I)

42. EXAMPLE II: DECOMPOSITION (II)

43. EXAMPLE II: DECOMPOSITION (III)

44. EXAMPLE II: NOISY DATA (added noise std=0.4)

45. EXAMPLE II: DECOMPOSITION (I) (added noise std=0.4)

46. EXAMPLE II: DECOMPOSITION (II) (added noise std=0.4)

47. EXAMPLE II: DECOMPOSITION (III) (added noise std=0.4)

48. CTI: DECOMPOSITION (I)

49. CTI: DECOMPOSITION (II)

50. CTI: DECOMPOSITION (III)

51. SOI: DECOMPOSITION (I)

52. SOI: DECOMPOSITION (II)

53. SOI: DECOMPOSITION (III)

54. EXAMPLE II: CORR. COEF.’s

55. Summary : CTI and SOI Straightforward sift produced low correlated IMFs from the two difference time series. Correlation test indicates that the Ensemble EMD can separate the time scales in the IMFs, and improve the correlation significantly.

56. Application of EEMD to Sound March 30, 2005

57. Data : Hello

58. Data : Hello c3y

59. Data : IMF100

60. Hilbert Spectrum : IMF100 (500, 12000; 7x7^3)

61. Hilbert Spectrum : c3y (500, 12000; 7x7^3)

62. Hilbert Spectrum : IMF100 (500, 12000; 7x7^3)

63. Hilbert Spectrum : c3y (500, 12000; 7x7^3)

64. Summary True IMFs can be derived from adding finite amplitude of noise, rather than the case with infinitesimal noises. Ensemble EMD indeed enables the signals of similar scale collated together. No need for a priori criteria for intermittency.

65. Summary Sum of IMFs may not be an IMF. As the components produced by EEMD are the averaged values of many IMFs, they might not be IMFs: some of the component might have multi-extrema. More stringent stoppage criteria and/or trials in the ensemble can improve the situation.

66. Data

67. Stoppage Criteria : S=1

68. Stoppage Criteria : S=10

69. Stoppage Criteria : S=1 Detail

70. Stoppage Criteria : S=10 Detail

71. OI vs. Noise Level

72. OI vs. Stoppage Criteria

73. Orthogonal Index of S-Number & Noise Level

75. Same as last slide, but with log2 x-axis

76. Observations Is orthogonality index a good measure of the EEMD results? What if we use the excessive of extrema instead?

77. Observations It looks like the fixed-sifting number method can give better orthogonal index for the same noise-level. Fixed sifting number method requires un- adaptive IMF’s number. This must be determined by user. Too much computations. For the same case, the fixed sifting number method requires extremely more computations.

78. Observations Not all EEM produced components are IMFs for two reasons: –Sum of IMFs may not be IMF. –Leakage in the decomposition. There is no problem in re-constructing the data. We do need rectification to compute the instantaneous frequency and the time- frequency spectral representations.

79. Fourier Spectra of IMFs

80. Latest Development It has been noted that the EEMD could implemented with the added noise used in pairs: once with plus sign, and once with minus sign. (Yeh, et al, 2010) This pair-wise noise addition could reduce the noise in the final re-constitution of the signal.

81. Conclusions Ensemble EMD enables the EMD method to be a truly dyadic filter bank. By adding finite noise, the Ensemble EMD eliminates mode mixing in most cases automatically. Ensemble EMD, utilized the scale separation principle of EMD, represents a major improvement of the EMD method. The true IMF components should be the results of the Ensemble EMD rather than from the raw data. Sum of IMF is not necessarily an IMF; therefore, the ensemble EMD results might not be IMFs.

82. EEMD Is equivalent to an artificial ensemble mean of EMDs.