ensemble empirical mode decomposition instructor: jian-jiun ding speaker: shang-ching lin 2010. nov....

EEnsemble nsemble EEmpirical mpirical MMode ode DDecompositionecomposition

Instructor: Jian-Jiun DingSpeaker: Shang-Ching Lin2010. Nov. 25

Time-frequency Analysis and Wavelet Transform courseOral Presentation

IntroductionIntroduction

Hilbert-Huang Transform (HHT)Hilbert-Huang Transform (HHT)

Empirical Mode

Decomposition

(EMD)

Empirical Mode

Decomposition

(EMD)

Hilbert Spectrum

(HS)

Hilbert Spectrum

(HS)

Ensemble Empirical

Mode Decomposition

(EEMD)

Ensemble Empirical

Mode Decomposition

(EEMD)

1998, [1]

2009, [4]

Studies on its properties: decomposing white noise

2003 – 2004, [2], [3]

IntroductionIntroduction 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

IntroductionIntroduction 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

IntroductionIntroduction 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

IntroductionIntroduction

1 Revised from [5]

Fourier STFT Wavelet HHT

Basis A priori A priori A priori Adaptive

FrequencyConvolution:

global, uncertainty

Convolution: regional,

uncertainty

Convolution: regional,

uncertainty

Differentiation: local, certainty

PresentationEnergy-

frequencyEnergy-time-



frequency

Nonlinear No No No Yes

Nonstationary No Yes Yes Yes

Feature Extraction

No YesDiscrete: NoContinuous:

YesYes

Theoretical Base

Theory complete

Theory complete

Theory complete

Empirical

EMDEMD 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

EMDEMD 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 tjtatxHTjtxtz exp

Algorithm3

3 Revised from Ruqiang Yan et al., “A Tour of the Hilbert-Huang Transform: An Empirical Tool for Signal Analysis”

(1) 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

EMDEMD

Sifting

0 20 40 60 80 100 120-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Original signal

0 20 40 60 80 100 120-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Original signal

0 20 40 60 80 100 120-1.5

-1

-0.5

0

0.5

1

1.5

2sifting for IMF1, pass 1

proto-IMFenvelopes

envelope mean

0 20 40 60 80 100 120-1

-0.5

0

0.5

1

1.5sifting for IMF1, pass 2

proto-IMFenvelopes

envelope mean

0 20 40 60 80 100 120-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8


proto-IMFenvelopes

envelope mean

0 20 40 60 80 100 120-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8


proto-IMFenvelopes

envelope mean

0 20 40 60 80 100 120-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1sifting for IMF1: pass 5 - done!

proto-IMFenvelopes

envelope mean

0 20 40 60 80 100 120-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8IMF1

0 20 40 60 80 100 120-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5


proto-IMFenvelopes

envelope mean

0 20 40 60 80 100 120-2

-1.5

-1

-0.5

0

0.5

1

1.5

2


proto-IMFenvelopes

envelope mean

0 20 40 60 80 100 120-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2


proto-IMFenvelopes

envelope mean

0 20 40 60 80 100 120-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5sifting for IMF2: pass 4 - done!

proto-IMFenvelopes

envelope mean

0 20 40 60 80 100 120-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5IMF2

0 20 40 60 80 100 120-0.2

0

0.2

0.4

0.6

0.8

1

1.2Residue

0 20 40 60 80 100 120-1

0

1

orig

inal

Original signal and its EMD

0 20 40 60 80 100 120-1

0

1

IMF

1

0 20 40 60 80 100 120-5

0

5

IMF

2

0 20 40 60 80 100 120-2

0

2

resi

due

EMDEMD Algorithm: demo

EMDEMD 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

EEMDEEMD 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


EEMDEEMD Algorithm(1) 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

tntxtxi

txi

EEMDEEMD 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?

Properties of EMDProperties of EMD Information content test- ─ relationship6

- Same area under the plot

- After some manipulations…


Eln TlnEnergy Mean period

Energy Period

Energy Mean period

Scaling

straight line in the ─ plotEln Tln

Properties of EMDProperties 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

Eln Tln

Noise region

Signals with information


Efficacies of EEMDEfficacies 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

Efficacies of EEMDEfficacies of EEMD

EMD EEMD

ApplicationsApplications

0 200 400 600 800 1000 1200-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

original

IMF1+IMF2+IMF3

Feature enhancement0 200 400 600 800 1000 1200

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

originalIMF3+IMF4+IMF5

0 200 400 600 800 1000 1200 1400 1600 1800 2000-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

original

IMF3+IMF4+IMF5

Denoising/Detrending

Signal processing- Example: ECG

ApplicationsApplications Time-frequency analysis- Hilbert Spectrum

- Hilbert Marginal Spectrum

iii dttjtatH expRe,

T

dttHh0

,

tcjtcdttjtatjta iiiiii ˆexpexp

IMF’s

ApplicationsApplications Time-frequency analysis

Hilbert Marginal Spectrumt = 12.75 to 13.25

10

4exp24

2exp115

10expRe

222 tt

tjtut

tjtx

0 5 10 15 20 25 30 35 400

20

40

60

80

100

120Hilbert Marginal Spectrum

frequency (rad/s)

ampl

itude

time (sec)

freq

uenc

y (r

ad/s

)

Hilbert Spectrum

0 5 10 150

5

10

15

20

25

30

35

5

10

15

20

25

30

35

Hilbert SpectrumΔt = 0.25, Δf = 0.05

ApplicationsApplications Time-frequency analysis

-10 -5 0 5 10-4

-3

-2

-1

0

1

2

3

4

Gabor Transform

-10 -5 0 5 10-4

-2

0

2

4

WDF

time (sec)

freq

uenc

y

alpha = 1

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

1

2

3

4

5

Cohen (Cone-shape)

-10 -5 0 5 10-10

-5

0

5

10

-10 -5 0 5 10-10

-5

0

5

10(c) (d)

Gabor-Wigner

HHT (using EEMD)

time (sec)

freq

uenc

y (r

ad/s

)

Hilbert Spectrum

0 5 10 150

5

10

15

20

25

30

35

5

10

15

20

25

30

35

DiscussionDiscussion 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

DiscussionDiscussion 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

DiscussionDiscussion 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

[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

ReferenceReference

ensemble empirical mode decomposition instructor: jian-jiun ding speaker: shang-ching lin 2010. nov....

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