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Empirical Mode Decompositionand Hilbert-Huang Transform
Emine Can 2010
Empirical Mode Decomposition and Hilbert-Huang Transform
2
Data Analysis
too short total data span
non-stationar
y
nonlinear process
Data Processing Methodso Spectrogram o Wavelet Analysiso Wigner-Ville Distribution (Heisenberg wavelet)o Evolutionary spectrum o Empirical Orthogonal Function Expansion (EOF)o Other methods
Fourier Spectral Analysis Energy-frequency distributions =Spectrum≈Fourier Transform of the dataRestrictions: * the system must be linear * the data must be strictly periodic or stationary
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Modifications of Fourier SA
Empirical Mode Decomposition and Hilbert-Huang Transform
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Hilbert Transform
Instantaneous Frequency
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Intrinsic Mode FunctionsEmpirical Mode Decomposition
Complicated Data Set
(Energy-Frequency-Time)
Empirical Mode Decomposition and Hilbert-Huang Transform
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A method that any complicated data set can be decomposed intoa finite and often small number of `intrinsic mode functions' thatadmit well-behaved Hilbert Transforms.
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Emperical Mode Decomposition (EMD)
Intrinsic Mode Functions(IMF)
IMF is a function that satisfies two conditions: 1- In the whole data set, the number of extrema and the number of zero crossings must either equal or differ at most by one 2- At any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero
Empirical Mode Decomposition and Hilbert-Huang Transform
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The empirical mode decomposition method: the sifting process
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Empirical Mode Decomposition and Hilbert-Huang Transform
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10 20 30 40 50 60 70 80 90 100 110 120
-2
-1
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1
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IMF 1; iteration 0
The sifting processComplicated Data Set x(t)
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-1
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IMF 1; iteration 0
The sifting process1. identify all upper extrema of x(t).
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IMF 1; iteration 0
The sifting process2. Interpolate the local maxima to form an upper envelope u(x).
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IMF 1; iteration 0
The sifting process3. identify all lower extrema of x(t).
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IMF 1; iteration 0
The sifting process4. Interpolate the local minima to form an lower envelope l(x).
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IMF 1; iteration 0
5. Calculate the mean envelope: m(t)=[u(x)+l(x)]/2.
The sifting process
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IMF 1; iteration 0
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residue
The sifting process6. Extract the mean from the signal: h(t)=x(t)-m(t)
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IMF 1; iteration 1
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residue
The sifting process7. Check whether h(t) satisfies the IMF condition. YES: h(t) is an IMF, stop sifting. NO: let x(t)=h(t), keep sifting.
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IMF 1; iteration 1
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residue
The sifting process
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IMF 1; iteration 1
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residue
The sifting process
10 20 30 40 50 60 70 80 90 100 110 120
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IMF 1; iteration 1
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residue
The sifting process
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IMF 1; iteration 1
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residue
The sifting process
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-1.5
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IMF 1; iteration 1
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residue
The sifting process
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
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IMF 1; iteration 1
10 20 30 40 50 60 70 80 90 100 110 120
-1.5
-1
-0.5
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residue
The sifting process
Empirical Mode Decomposition and Hilbert-Huang Transform
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Empirical Mode Decomposition and Hilbert-Huang Transform
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The signal is composed of 1. a “high frequency” triangular
waveform whose amplitude is slowly (linearly) growing.
2. a “middle frequency” sine wave whose amplitude is quickly (linearly) decaying
3. a “low frequency” triangular waveform
Empirical Mode Decomposition and Hilbert-Huang Transform
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A criterion for the sifting process to stop: Standard deviation, SD, computed from the two consecutive sifting results is in limited size.
:residue after the kth iteration of the 1st IMF
A typical value for SD can be set between 0.2 and 0.3.
The sifting processStop criterion
Empirical Mode Decomposition and Hilbert-Huang Transform
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Hilbert Transform
*
Instantaneous Frequency:
Analytic Signal:
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Advantages*Adaptive,highly efficient,applicable to nonlinear and non-stationary processes.
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Traditional methods EMD
- Not appropriate for nonlinear & nonstationary signals.
- Predefined basis and/or system model.
- Distorted information extracted.
- Full theoretical basis.
- Adequate for both nonlinear & nonstationary.
- Adaptive – data driven basis.
- Preserves physical meaning.
- Sharper spectrum
- Lack of theoretical analysis.
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Applications of EMD
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• nonlinear wave evolution,• climate cycles,• earthquake engineering,• submarine design,• structural damage detection,• satellite data analysis,• turbulence flow,• blood pressure variations and heart arrhythmia,• non-destructive testing,• structural health monitoring,• signal enhancement,• economic data analysis,• investigation of brain rythms• Denoising• …
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References• “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series
analysis” Huang et al., The Royal Society, 4 November 1996.• Rilling Gabriel, Flandrin Patrick , Gon¸calv`es Paulo, “On Empirical Mode Decomposition and Its
Algorithms”• Stephen McLaughlin and Yannis Kopsinis.ppt “Empirical Mode Decomposition: A novel algorithm for
analyzing multicomponent signals” Institute of Digital Communications (IDCOM)• “Hilbert-Huang Transform(HHT).ppt” Yu-Hao Chen, ID:R98943021, 2010/05/07
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