1. 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

2. 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]
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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
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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
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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
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6. 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]
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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 priori bases for decomposition
2 Photo: 中央大學數據分析中心
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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
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9. 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”
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10. EMD
Algorithm: demo
Sifting
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11. 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
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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
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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
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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?
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15. 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
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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
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
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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
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18. Efficacies of EEMD
EMD EEMD
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19. Efficacies of EEMD
EMD EEMD
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20. Applications Signal processing Example: ECG Denoising/ Detrending
Feature enhancement
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21. Applications Time-frequency analysis Hilbert Spectrum
Hilbert Marginal Spectrum
IMF’s
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22. Applications Time-frequency analysis Hilbert Marginal Spectrum
Hilbert Spectrum
Δt = 0.25, Δf = 0.05
Hilbert Marginal Spectrum
t = to 13.25
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23. Applications Time-frequency analysis HHT (using EEMD) Gabor Transform
Cohen (Cone-shape)
WDF
Gabor-Wigner
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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
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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
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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
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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 , 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 , 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 , October 2007
[7] Norden E. Huang, “An Introduction to Hilbert-Huang Transform: A Plea for Adaptive Data Analysis”(Internet resource; Powerpoint file)
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