E nsemble E mpirical M ode D ecomposition

EEnsemble nsemble EEmpirical mpirical MMode ode DDecompositionecomposition

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

Time-frequency Analysis and Wavelet Transform courseOral Presentation

Page 2

IntroductionIntroduction

Hilbert-Huang Transform (HHT)

Empirical ModeDecomposition

(EMD)

Hilbert Spectrum

(HS)

Ensemble EmpiricalMode Decomposition

(EEMD)

1998, [1]

2009, [4]

Studies on its properties: decomposing white noise

2003 – 2004, [2], [3]

Page 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

Page 4

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

Page 5

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

Page 6

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

Presentation Energy-frequency

Energy-time-frequency



Nonlinear No No No Yes

Nonstationary No Yes Yes Yes

Feature Extraction No Yes

Discrete: NoContinuous:

YesYes

Theoretical Base

Theory complete

Theory complete

Theory complete Empirical

Page 7

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

Page 8

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

Page 9

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

Page 10

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

IMF1

0 20 40 60 80 100 120-5

0

5

IMF2

0 20 40 60 80 100 120-2

0

2

resi

due

EMDEMD Algorithm: demo

Page 11

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

Page 12

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


Page 13

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

Page 14

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?

Page 15

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

Scalingstraight line in the ─ plotEln Tln

Page 16

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


Page 17

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

Page 18

Efficacies of EEMDEfficacies of EEMDEMD EEMD

Page 19

Efficacies of EEMDEfficacies of EEMDEMD EEMD

Page 20

ApplicationsApplications

0 200 400 600 800 1000 1200-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

originalIMF1+IMF2+IMF3

Feature enhancement0 200 400 600 800 1000 1200

-2.5

-2

-1.5

-1

-0.5

0

0.5

1


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


Denoising/Detrending

Signal processing- Example: ECG

Page 21

ApplicationsApplications Time-frequency analysis- Hilbert Spectrum

- Hilbert Marginal Spectrum

iii dttjtatH expRe,

T

dttHh0

,

tcjtcdttjtatjta iiiiii ˆexpexp

IMF’s

Page 22

ApplicationsApplications Time-frequency analysis

Hilbert Marginal Spectrumt = 12.75 to 13.25

104exp24

2exp115

10expRe

222 tttjtuttjtx

0 5 10 15 20 25 30 35 400

20

40

60

80

100

120Hilbert Marginal Spectrum

frequency (rad/s)

ampl

itude

time (sec)

frequ

ency

(rad

/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

Page 23

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)

frequ

ency

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)

frequ

ency

(rad

/s)

Hilbert Spectrum

0 5 10 150

5

10

15

20

25

30

35

5

10

15

20

25

30

35

Page 24

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

Page 25

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

Page 26

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

Page 27

[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

Documents

E nsemble E mpirical M ode D ecomposition