Ensemble Empirical Mode Decomposition

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Ensemble Empirical Mode Decomposition
Time-frequency Analysis and Wavelet Transform
course Oral Presentation

• Instructor Jian-Jiun Ding
• Speaker Shang-Ching Lin
• 2010. Nov. 25

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

• Empirical mode decomposition (EMD)
• Proposed by Norden E. Huang et al., in 1998
• Decomposing the data into a set of intrinsic mode
  functions (IMFs)
• 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 ?????????? http//rcada.ncu.edu.tw/mem
ber1.htm
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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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EMD

• Algorithm3

• Envelope construction
• Cubic spline interpolation

(2) Sifting Subtracting envelope mean from
the signal repeatedly
(3) Subtracting the IMF from 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
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EMD

• Algorithm demo

Sifting
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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
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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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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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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 IMFs?
• Information-rich, or just noise?

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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
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Properties of EMD

• Information content test
• - relationship ? information
  content
• Distribution of each IMF approx. normal7
• Energy is argued to be ?2 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
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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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Efficacies of EEMD
EMD EEMD
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Efficacies of EEMD
EMD EEMD
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Applications

• Signal processing
• Example ECG

Denoising/ Detrending
Feature enhancement
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Applications

• Time-frequency analysis
• Hilbert Spectrum
• Hilbert Marginal Spectrum

IMFs
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Applications

• Time-frequency analysis

Hilbert Marginal Spectrum t 12.75 to 13.25
Hilbert Spectrum ?t 0.25, ?f 0.05
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Applications

• Time-frequency analysis

HHT (using EEMD)
Cohen (Cone-shape)
Gabor Transform
Gabor-Wigner
WDF
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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 IMFs in a strict sense
• NOT convolution/ inner product/ integration based
• Generally EMD is fast, but EEMD is not

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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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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 IMFs

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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/Introduction20to20
  HHT.ppt