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

Hilbert-Huang Transform (HHT) 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

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

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

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

Introduction Fourier STFT Wavelet HHT Basis A priori Adaptive Frequency Convolution: global, uncertainty Convolution: regional, uncertainty Differentiation: local, certainty Presentation Energy-frequency Energy-time-frequency Nonlinear No Yes Nonstationary Feature Extraction Discrete: No Continuous: Yes Theoretical Base Theory complete Empirical 1 Revised from [5] Page 6

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 priori bases for decomposition 2 Photo: 中央大學數據分析中心 http://rcada.ncu.edu.tw/member1.htm Page 7

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

EMD Algorithm3 Envelope construction Cubic spline interpolation (2) Sifting Subtracting envelope mean from the signal repeatedly (3) Subtracting the IMF from the original signal Quick intro. (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

EMD Algorithm: demo Sifting Page 10

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

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

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

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

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

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 Statistical approach, which is common for EMD due to the lack of analytical properties (cf. Fourier). Noise region 7 Zhaohua Wu and Norden E. Huang, 2004 Page 16

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

Efficacies of EEMD EMD EEMD Page 18

Efficacies of EEMD EMD EEMD Page 19

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

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

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

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

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

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

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

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