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Ensemble Empirical Mode Decomposition

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Title: Ensemble Empirical Mode Decomposition


1
Ensemble Empirical Mode Decomposition
Time-frequency Analysis and Wavelet Transform
course Oral Presentation
  • 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
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

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

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

6
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
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 (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
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

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
(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
10
EMD
  • Algorithm demo

Sifting
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
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
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

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 IMFs?
  • Information-rich, or just noise?

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
16
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
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

18
Efficacies of EEMD
EMD EEMD
19
Efficacies of EEMD
EMD EEMD
20
Applications
  • Signal processing
  • Example ECG

Denoising/ Detrending
Feature enhancement
21
Applications
  • Time-frequency analysis
  • Hilbert Spectrum
  • Hilbert Marginal Spectrum

IMFs
22
Applications
  • Time-frequency analysis

Hilbert Marginal Spectrum t 12.75 to 13.25
Hilbert Spectrum ?t 0.25, ?f 0.05
23
Applications
  • Time-frequency analysis

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

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

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 IMFs

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/Introduction20to20
    HHT.ppt
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