E nsemble E mpirical M ode D ecomposition

1. Time-frequency Analysis and Wavelet Transform course Oral Presentation Ensemble Empirical Mode Decomposition Instructor: Jian-Jiun Ding Speaker: Shang-Ching Lin 2010. Nov. 25

2. Introduction Hilbert-Huang Transform (HHT) Empirical Mode Decomposition (EMD) Hilbert Spectrum (HS) 1998, [1] Ensemble Empirical Mode Decomposition (EEMD) Studies on its properties: decomposing white noise 2009, [4] 2003 – 2004, [2], [3] Page 2

3. Introduction • Motivation • Traditional methods are not suitable for analyzing nonlinear AND nonstationary data series, which is often resulted from real-world physical processes. • “Though we can assume all we want, the reality cannot be bent by the assumptions.” (N. E. Huang) → A plea for adaptive data analysis Page 3

4. Introduction • Drawbacks of Fourier-based analysis • Decomposing signal into sinusoids • May not be a good representation of the signal • Assuming linearity, even stationarity • Short-time Fourier Transform: window function introduces finite mainlobe and sidelobes, being artifacts • Spectral resolution limited by uncertainty principle: can not be "local" enough Page 4

5. Introduction • Wavelet analysis • Using a priori basis • Efficacy sensitive to inter-subject, even intra-subject variations • Fails to catch signal characteristics if the waveforms do not match Page 5

6. Introduction 1 Revised from [5] Page 6

7. EMD • Empirical mode decomposition (EMD) • Proposed by Norden E. Huang et al., in 1998 • Decomposing the data into a set of intrinsic mode functions (IMF’s) • Verified to be highly orthogonal • Time-domain processing: can be very local  No uncertainty principle limitation • Not assuming linearity, stationarity, or any a prioribases for decomposition 2 Photo: 中央大學數據分析中心 http://rcada.ncu.edu.tw/member1.htm Page 7

8. EMD • Intrinsic Mode Functions (IMF) • Definition (1) | (# of extremas) – (# of zero crossings ) | ≤ 1 (2) Symmetric: the mean of envelopes of local maxima and minima is zero at ant point IMF = oscillatory mode embedded in the data ↔ sinusoids in Fourier analysis • Lower order ↔ faster oscillation • Can be viewed as AM-FM signal • Analytic signal Page 8

9. EMD • Envelope construction • Cubic spline interpolation • Algorithm3 (2) Sifting Subtracting envelope mean from the signal repeatedly (3) Subtracting the IMFfrom the original signal (4) Repeat (1)~(3) Until the number of extrema of the residue ≤ 1 3 Revised from Ruqiang Yan et al., “A Tour of the Hilbert-Huang Transform: An Empirical Tool for Signal Analysis” Page 9

10. EMD • Algorithm: demo Sifting Page 10

11. EMD • Problem • End effects • Not stable • i.e. sensitive to noise • Mode mixing4 • When processing intermittent signals • Solution: Ensemble EMD 4 Zhaohua Wu and Norden E. Huang, 2009 Page 11

12. EEMD • Ensemble Empirical Mode Decomposition (EEMD) • Proposed by Norden E. Huang et al., in 2009 • Inspired by the study on white noise using EMD • EMD: equivalently a dyadic filter bank5 5 Zhaohua Wu and Norden E. Huang, 2004 Page 12

13. EEMD • Algorithm • Adding noise to the original data to form a “trial” i.e. (2) Performing EMD on each (3) For each IMF, take the ensemble mean among the trials as the final answer Page 13

14. EEMD • A noise-assisted data analysis • Noise: act as the reference scale • Perturbing the data in the solution space • To be cancelled out ideally by averaging • What can we say about the content of the IMF’s? • Information-rich, or just noise? Page 14

15. Properties of EMD • Information content test • ─ relationship6 • Same area under the plot  • After some manipulations…  Energy Mean period Energy Period straight line in the ─ plot Scaling Energy Mean period 6 Zhaohua Wu and Norden E. Huang, 2004 Page 15

16. Properties of EMD • Information content test • ─ relationship ↔ information content • Distribution of each IMF: approx. normal7 • Energy is argued to be χ2 distributed • Degree of freedom = energy in the IMF  Energy spread line (in terms of percentiles) can be derived, and the confidence level of an IMF being noise can be deduced Signals with information Noise region 7 Zhaohua Wu and Norden E. Huang, 2004 Page 16

17. Efficacies of EEMD • Analysis of real-world data • Climate data • El Niño-Southern Oscillation (ENSO) phenomenon: The Southern Oscillation Index (SOI) and the Cold Tongue Index (CTI) are negatively related • Great improvement from EMD to EEMD Page 17

18. Efficacies of EEMD EMD EEMD Page 18

19. Efficacies of EEMD EMD EEMD Page 19

20. Applications • Signal processing • Example: ECG Denoising/ Detrending Feature enhancement Page 20

21. Applications • Time-frequency analysis • Hilbert Spectrum • Hilbert Marginal Spectrum IMF’s Page 21

22. Applications • Time-frequency analysis Hilbert Marginal Spectrum t = 12.75 to 13.25 Hilbert Spectrum Δt = 0.25, Δf = 0.05 Page 22

23. Applications • Time-frequency analysis HHT (using EEMD) Cohen (Cone-shape) Gabor Transform Gabor-Wigner WDF Page 23

24. Discussion • Pros • NOT assuming linearity nor stationarity • Fully adaptive • No requirement for a priori knowledge about the signal • Time-domain operation • Reconstruction extremely easy • EEMD: the results are not IMF’s in a strict sense • NOT convolution/ inner product/ integration based • Generally EMD is fast, but EEMD is not Page 24

25. Discussion • Pros • Capable of de-trending • In time-frequency analysis • Resolution not limited by the uncertainty principle • In Filtering • Fourier filters • Harmonics also filtered → distortion of the fundamental signal • EEMD • Confidence level of an IMF being noise can be deduced • Similar to the filtering using Discrete Wavelet Transform Page 25

26. Discussion • Cons • Lack of theoretical background and good mathematical (analytical) properties • Usually appealing to statistical approaches • Found useful in many applications without being proven mathematically, as the wavelet transform in the late 1980s • Challenge • Interpretation of the contents of the IMF’s Page 26

27. Reference • [1] N. E. Huang et al., “The Empirical Mode Decomposition Method and the Hilbert Spectrum for Non-stationary Time Series Analysis,” Proc. Roy. Soc. London, 454A, pp. 903-995, 1998 • [2] Patrick Flandrin, Gabriel Rilling and Paulo Gonçalvès, “Empirical Mode Decomposition as a Filter Bank,” IEEE Signal Processing Letters, Volume 10, No. 20, pp.1-4, 2003 • [3] Z. Wu and N. E. Huang, “A Study of the Characteristics of White Noise Using the Empirical Mode Decomposition,” Proc. R. Soc. Lond., Volume 460, pp.1597-1611, 2004 • [4] Z. Wu and N. E. Huang, “Ensemble Empirical Mode Decomposition: A Noise-Assisted Data Analysis Method,” Advances in Adaptive Data Analysis, Volume 1, No. 1, pp. 1-41, 2009 • [5] N. E. Huang, “Introduction to Hilbert-Huang Transform and Some Recent Developments,” The Hilbert-Huang Transform in Engineering, pp.1-23, 2005 • [6] R. Yan and R. X. Gao, “A Tour of the Hilbert-Huang Transform: An Empirical Tool for Signal Analysis,” Instrumentation & Measurement Magazine, IEEE, Volume 10, Issue 5, pp. 40-45, October 2007 • [7] Norden E. Huang, “An Introduction to Hilbert-Huang Transform: A Plea for Adaptive Data Analysis”(Internet resource; Powerpoint file) http://wrcada.ncu.edu.tw/Introduction%20to%20HHT.ppt Page 27