1. ENSEMBLE EMPIRICAL MODE DECOMPOSITION Noise Assisted Signal Analysis (nasa) Part I Preliminary
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
The title of my talk is “The impact of ENSO on NAO variability”. This work was done with Drs. Kirtman and Schneider at COLA.

2. Theoretical Foundations
Intermittency test, though ameliorates the mode mixing, destroys the adaptive nature of EMD.
The EMD study of white noise guarantees a uniformed frame of scales.
The cancellation of white noise with sufficient number of ensemble.

3. Theoretical Background I
Intermittency

4. 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 longer than this pre-selected criterion is not included in the IMF.

5. Observations
Intermittency test ameliorates the mode mixing considerably.
Intermittency test requires a set of subjective criteria.
EMD with intermittency is no longer totally adaptive.
For complicated data, the subjective criteria are hard, or impossible, to determine.

6. Effects of EMD (Sifting)
To separate data into components of similar scale.
To eliminate ridding waves.
To make the results symmetric with respect to the x-axis and the amplitude more even.
Note: The first two are necessary for valid IMF, the last effect actually cause the IMF to lost its intrinsic properties.

7. Theoretical Background II
A Study of White Noise

8. Wu, Zhaohua and N. E. Huang, 2004:
A Study of the Characteristics of White Noise Using the Empirical Mode Decomposition Method, Proceedings of the Royal Society of London , A 460,

9. Methodology
Based on observations from Monte Carlo numerical experiments on 1 million white noise data points.
All IMF generated by 10 siftings.
Fourier spectra based on 200 realizations of 4,000 data points sections.
Probability density based on 50,000 data points data sections.

10. IMF Period Statistics IMF 1 2 3 4 5 6 7 8 9 number of peaks
IMF 1 2 3 4 5 6 7 8 9
number of peaks
347042
168176
83456
41632
20877
10471
5290
2658
1348
Mean period
2.881
5.946
11.98
24.02
47.90
95.50
189.0
376.2
741.8
period in year
0.240
0.496
0.998
2.000
3.992
7.958
15.75
31.35
61.75

11. Fourier Spectra of IMFs

12. Empirical Observations : I Mean Energy

13. Empirical Observations : II Normalized spectral area is constant

14. Empirical Observations : III Normalized spectral area is constant

15. Empirical Observations : IV Computation of mean period

16. Empirical Observations : III The product of the mean energy and period is constant

17. Monte Carlo Result : IMF Energy vs. Period

18. Empirical Observation: Histograms IMFs By Central Limit theory IMF should be normally distributed.

19. Fundamental Theorem of Probability
If we know the density function of a random variable, x, then we can express the density function of any random variable, y, for a given y=g(x). The procedure is as follows:

20. Fundamental Theorem of Probability
If we know the density function of a random variable, x, is normal, then x-square should be

21. Chi and Chi-Square Statistics

22. DEGREE OF FREEDOM
Random samples of length N contains N degree of freedom
Each Fourier component contains one degree of freedom
For EMD, the shares of DOF is proportional to its share of energy; therefore, the degree of freedom for each IMF is given as

23. Chi-Squared Energy Density Distributions

24. Histograms : IMF Energy Density
By Central Limit theory, IMF should be normally distributed; therefore, its energy should be Chi-squared distributed.

25. Chi-Squared Energy Density Distributions
By Central Limit theory, IMF should be normally distributed; therefore, its energy should be Chi-squared distributed.

26. Formula of Confidence Limit for IMF Distributions I

27. Formula of Confidence Limit for IMF Distributions II

28. Formula of Confidence Limit for IMF Distributions III

29. Formula of Confidence Limit for IMF Distributions IV
For a Gaussian distribution, it is often to relate α to the standard deviation, σ , i.e., α confidence level corresponds to k σ, where k varies with α. For example, having values , , -0.0, 0.675, and for the first, 25th, 50th, 75th and 99th percentiles (with α being 0.01, 0.25, 0.5, 0.75, 0.99), respectively.

30. Formula of Confidence Limit for IMF Distributions V

31. Formula of Confidence Limit for IMF Distributions VI

32. Confidence Limit for IMF Distributions

33. Data and IMFs SOI

34. Statistical Significance for SOI IMFs
IMF 4, 5, 6 and 7 are 99% statistical significance signals.
1 mon
1 yr
10 yr
100 yr

35. Summary Not all IMF have the same statistical significance.
Based on the white noise study, we have established a method to determine the statistical significant components.
References:
Wu, Zhaohua and N. E. Huang, 2003: A Study of the Characteristics of White Noise Using the Empirical Mode Decomposition Method, Proceedings of the Royal Society of London A460,
Flandrin, P., G. Rilling, and P. Gonçalvès, 2003: Empirical Mode Decomposition as a Filterbank, IEEE Signal Proc Lett. 11 (2):

36. Observations The white noise signal consists of signal of all scales.
EMD separates the scale dyadically.
The white noise provide a uniformly distributed frame of scales through EMD.

37. Different Approaches but reach the same end.
Flandrin, P., G. Rilling and P. Goncalves, 2004: Empirical Mode Decomposition as a filter bank. IEEE Signal Process. Lett., 11, Flandrin, P., P. Goncalves and G. Rilling, 2005: EMD equivalent filter banks, from interpretation to applications. Introduction to Hilbert-Huang Transform and its Applications, Ed. N. E. Huang and S. S. P. Shen, p World Scientific, New Jersey,

38. Fractional Gaussian Noise aka Fractional Brownian Motion

39. Examples

40. Flandrin’s results

41. Flandrin’s results

42. Flandrin’s results

43. Flandrin’s results

44. Flandrin’s results

45. Flandrin’s results : Delta Function

46. Flandrin’s results : Delta Function

47. Theoretical Background III
Effects of adding White Noise

48. Some Preliminary
Robert John Gledhill, 2003: Methods for Investigating Conformational Change in Biomolecular Simulations, University of Southampton, Department of Chemistry, Ph D Thesis.
He investigated the effect of added noise as a tool for checking the stability of EMD.

49. Some Preliminary
His basic assumption is that the correct result is the one without noise:

50. Test results Top Whole data perturbed; bottom only 10% perturbed.

51. Test results

52. Observations
They made the critical assumption that the unperturbed signal gives the correct results.
When the amplitude of the added perturbing noise is small, the discrepancy is small.
When the amplitude of the added perturbing noise is large, the discrepancy becomes bi-modal.