On empirical mode decomposition and its algorithms
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On Empirical Mode Decomposition and its Algorithms. G. Rilling, P. Flandrin (Cnrs - Éns Lyon, France) P. Gonçalvès (Inria Rhône-Alpes, France). outline. Empirical Mode Decomposition (EMD) basics examples algorithmic issues elements of performance evaluation perspectives. basic idea.

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On Empirical Mode Decomposition and its Algorithms

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On empirical mode decomposition and its algorithms

On Empirical Mode Decomposition and its Algorithms

G. Rilling, P. Flandrin (Cnrs - Éns Lyon, France)

P. Gonçalvès (Inria Rhône-Alpes, France)

IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing, Grado (I), June 9-11, 2003


Outline

outline

  • Empirical Mode Decomposition (EMD) basics

  • examples

  • algorithmic issues

  • elements of performance evaluation

  • perspectives


Basic idea

basic idea

  • « multimodal signal = fast oscillations on the top of slower oscillations »

  • Empirical Mode Decomposition (Huang)

    • identify locally the fastest oscillation

    • substract to the signal and iterate on the residual

    • data-driven method, locally adaptive and multiscale


Huang s algorithm

Huang’s algorithm

  • compute lower and upper envelopes from interpolations between extrema

    • substract mean envelope from signal

    • iterate until mean envelope = 0 and #{extrema} = #{zero-crossings} ± 1

  • substract the obtained Intrinsic Mode Function (IMF) from signal and iterate on residual


How emd works

how EMD works


Emd and am fm signals

EMD and AM-FM signals

  • quasi-monochromatic harmonic oscillations

  • self-adaptive time-variant filtering

  • example : 2 sinus FM + 1 Gaussian wave packet


On empirical mode decomposition and its algorithms

EMD


Time frequency signature

time-frequency signature


Time frequency signature1

time-frequency signature


Time frequency signature2

time-frequency signature


Time frequency signature3

time-frequency signature


Nonlinear oscillations

nonlinear oscillations

  • IMF ≠ Fourier mode and, in nonlinear situations, 1 IMF = many Fourier modes

  • example : 1 HF triangle + 1 MF tone + 1 LF triangle


On empirical mode decomposition and its algorithms

EMD


Issues

issues

  • algorithm ?

    • intuitive but ad-hoc procedure, not unique

    • several user-controlled tunings

  • performance ?

    • difficult evaluation since no analytical definition

    • numerical simulations


Algorithmic issues

algorithmic issues

  • interpolation

    • type ? cubic splines

    • border effects ? mirror symmetry

  • stopping criteria

    • mean zero ? 2 thresholds

    • variation 1 : « local EMD »

  • computational burden

    • about log2 N IMF ’s for N data points

    • variation 2 : « on-line EMD »


Performance evaluation

performance evaluation

  • extensive numerical simulations

  • deterministic framework

    • importance of sampling

    • ability to resolve multicomponent signals

  • a complement to stochastic studies

    • noisy signals + fractional Gaussian noise

    • PF et al., IEEE Sig. Proc. Lett., to appear


Emd of fractional gaussian noise

EMD of fractional Gaussian noise


1 emd and tone sampling

1. EMD and (tone) sampling


Case 1 oversampling

case 1 — oversampling


Equal height maxima

equal height maxima


Constant upper envelope

constant upper envelope


Equal height minima

equal height minima


Constant lower envelope

constant lower envelope


Zero mean tone imf

zero mean : tone = IMF


Case 2 moderate sampling

case 2 — moderate sampling


Fluctuating maxima

fluctuating maxima


Modulated upper envelope

modulated upper envelope


Fluctuating minima

fluctuating minima


Modulated lower envelope

modulated lower envelope


Non zero mean tone imf

non zero mean : tone ≠ IMF


Experiment 1

experiment 1

  • 256 points tone, with 0 ≤ f ≤1/2

  • error = normalized L2 distance comparing tone vs. IMF #1

  • minimum when 1/f even multiple of the sampling period


Experiment 2

experiment 2

  • 256 points tones, with 0 ≤ f1 ≤1/2 and f2 ≤ f1

  • error = weighted normalized L2 distance comparing tone #1 vs. IMF #1, and tone #2 vs. max{IMF #k, k ≥ 2}


Experiment 3

experiment 3

  • intertwining of amplitude ratio, sampling rate and frequency spacing

  • dominant effect when f1 ≤ 1/4 : constant-Q (wavelet-like) « confusion band »


Concluding remarks

concluding remarks

  • EMD is an appealing data-driven and multiscale technique

  • spontaneous dyadic filterbank structure in « stationary » situations, stochastic (fGn) or not (tones)

  • EMD defined as output of an algorithm: theoretical framework beyond numerical simulations?


P reprints matlab codes and demos www ens lyon fr flandrin emd html

(p)reprints, Matlab codes and demoswww.ens-lyon.fr/~flandrin/emd.html


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