MATH 3290 Mathematical Modeling

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MATH 3290 Mathematical Modeling
Tutorial on the Empirical Mode Decomposition
Method (EMD)
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First, review of the procedure of EMD
methodThe main idea of the EMD method is Sifting
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Empirical Mode Decomposition Methodology Test
Data
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Empirical Mode Decomposition Methodology data
and m1
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Empirical Mode Decomposition Methodology data
h1
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Empirical Mode Decomposition Methodology h1
m2
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Empirical Mode Decomposition Methodology h3
m4
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Empirical Mode Decomposition Methodology h4
m5
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Empirical Mode DecompositionSifting to get one
IMF component
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Two Stoppage Criteria SD
Standard Deviation is small than a pre-set
value, where
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Stoppage Criteria

• It is critical that we use the correct stoppage
  criterion.
• Over shifting, we can prove that the envelopes
  defined has to be a straight line.
• If the data is not monotonically increasing or
  decreasing, the straight lines would be
  horizontal lines.

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Empirical Mode Decomposition Methodology IMF
c1
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Definition of the Intrinsic Mode Function (IMF)

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Empirical Mode DecompositionSifting to get all
the IMF components
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Empirical Mode Decomposition data
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Empirical Mode Decomposition IMFs and residue
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Definition of Instantaneous Frequency
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Definition of Frequency
Given the period of a wave as T the frequency
is defined as
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Equivalence

• The definition of frequency is equivalent to
  defining velocity as
• Velocity Distance / Time

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Instantaneous Frequency
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The combination of Hilbert Spectral Analysis and
Empirical Mode Decomposition is designated as

• Hilbert-Huang Transform
• (HHT vs. FFT)

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Comparison between FFT and HHT
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The Idea behind EMD

• To be able to analyze data from the nonstationary
  and nonlinear processes and reveal their physical
  meaning, the method has to be Adaptive.
• Adaptive requires a posteriori (not a priori)
  basis. But the present established mathematical
  paradigm is based on a priori basis.
• Only a posteriori basis could fit the varieties
  of nonlinear and nonstationary data without
  resorting to the mathematically necessary (but
  physically nonsensical) harmonics.

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The Idea behind EMD

• The method has to be local.
• Locality requires differential operation to
  define properties of a function.
• Take frequency, for example. The traditional
  established mathematical paradigm is based on
  Integral transform. But integral transform
  suffers the limitation of the uncertainty
  principle.

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Global Temperature Anomaly

• Annual Data from 1856 to 2003

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Global Temperature Anomaly 1856 to 2003
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IMF Mean of 10 Sifts CC(1000, I)
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Data and Trend C6
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Rate of Change Overall Trends EMD and Linear
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What This Means

• Instantaneous Frequency offers a total different
  view for nonlinear data instantaneous frequency
  with no need for harmonics and unlimited by
  uncertainty.
• Adaptive basis is indispensable for nonstationary
  and nonlinear data analysis
• HHT establishes a new paradigm of data analysis

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Comparisons
Fourier Wavelet Hilbert
Basis a priori a priori Adaptive
Frequency Integral transform Global Integral transform Regional Differentiation Local
Presentation Energy-frequency Energy-time-frequency Energy-time-frequency
Nonlinear no no yes
Non-stationary no yes yes
Uncertainty yes yes no
Harmonics yes yes no
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Conclusion

• Adaptive method is a scientifically meaningful
  way to analyze data.
• It is a way to find out the underlying physical
  processes therefore, it is indispensable in
  scientific research.
• It is physical, direct, and simple.

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• History of EMD HHT
• 1998 The Empirical Mode Decomposition Method and
  the Hilbert Spectrum for Non-stationary Time
  Series Analysis, Proc. Roy. Soc. London, A454,
  903-995. The invention of the basic method of
  EMD, and Hilbert transform for determining the
  Instantaneous Frequency and energy.
• 1999 A New View of Nonlinear Water Waves The
  Hilbert Spectrum, Ann. Rev. Fluid Mech. 31,
  417-457.
• Introduction of the intermittence in
  decomposition.
• 2003 A confidence Limit for the Empirical mode
  decomposition and the Hilbert spectral analysis,
  Proc. of Roy. Soc. London, A459, 2317-2345.
• Establishment of a confidence limit without the
  ergodic assumption.
• 2004 A Study of the Characteristics of White
  Noise Using the Empirical Mode Decomposition
  Method, Proc. Roy. Soc. London, 460, 1597-1611.
• Defined statistical significance and
  predictability.

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• Recent Developments in HHT
• 2007 On the trend, detrending, and variability
  of nonlinear and nonstationary time series.
  Proc. Natl. Acad. Sci., 104, 14,889-14,894.
• The correct adaptive trend determination method
• 2009 On Ensemble Empirical Mode Decomposition.
  Advances in Adaptive Data Analysis. (Advances in
  Adaptive data Analysis, 1, 1-41)
• 2009 On instantaneous Frequency. Advances in
  Adaptive Data Analysis (Advances in Adaptive Data
  Analysis. Advances in Adaptive data Analysis, 1,
  177-229).
• 2009 Multi-Dimensional Ensemble Empirical Mode
  Decomposition. Advances in Adaptive Data Analysis
  (Advances in Adaptive Data Analysis. Advances in
  Adaptive data Analysis, 1, 339-372).
• 2010 The Time-Dependent Intrinsic Correlation
  based on the Empirical Mode Decomposition
  (Advances in Adaptive Data Analysis. Advances in
  Adaptive data Analysis, 2, 233-265).
• 2010 On Hilbert Spectral Analysis (to appear in
  AADA).

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Current Efforts and Applications

• Non-destructive Evaluation for Structural Health
  Monitoring
• (DOT, NSWC, DFRC/NASA, KSC/NASA Shuttle, THSR)
• Vibration, speech, and acoustic signal analyses
• (FBI, and DARPA)
• Earthquake Engineering
• (DOT)
• Bio-medical applications
• (Harvard, Johns Hopkins, UCSD, NIH, NTU, VHT, AS)
• Climate changes
• (NASA Goddard, NOAA, CCSP)
• Cosmological Gravity Wave
• (NASA Goddard)
• Financial market data analysis
• (NCU)
• Theoretical foundations
• (Princeton University and Caltech)

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Reference

• Huang, M. L. Wu, S. R. Long, S. S. Shen, W. D.
  Qu, P. Gloersen, and K. L. Fan (1998)The
  empirical mode decomposition and the Hilbert
  spectrum for nonlinear and non-stationary time
  series analysis. Proc. Roy. Soc. Lond., 454A,
  903-993.
• Flandrin, P., G. Rilling, and P. Gonçalves
  (2004) Empirical mode decomposition as a filter
  bank. IEEE Signal Proc Lett., 11, 112-114.
• Research Center for Adaptive Data Analysis,
  National Central University
• http//rcada.ncu.edu.tw/research1.htm