The Empirical Mode Decomposition Method Sifting

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The Empirical Mode Decomposition Method Sifting
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Goal of Data Analysis

• To define time scale or frequency.
• To define energy density.
• To define joint frequency-energy distribution as
  a function of time.
• To do this, we need a AM-FM decomposition of the
  signal. X(t) A(t) cos?(t) , where A(t)
  defines local energy and ?(t) defines the local
  frequency.

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Need for Decomposition

• Hilbert Transform (and all other IF computation
  methods) offers meaningful Instantaneous
  Frequency for IMFs.
• For complicate data, there should be more than
  one independent component at any given time.
• The decomposition should be adaptive in order to
  study data from nonstationary and nonlinear
  processes.
• Frequency space operations are difficult to track
  temporal changes.

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Why Hilbert Transform is not enough?

• Even though mathematicians told us that the
  Hilbert transform exists for all functions of
  Lp-class.

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Problems on Envelope

• A seemingly simple proposition but it is not so
  easy.

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Two examples
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Data set 1
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Data X1
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Data X1 Hilbert Transform
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Data X1 Envelopes
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Observations

• None of the two envelopes seem to make sense
• The Hilbert transformed amplitude oscillates too
  much.
• The line connecting the local maximum is almost
  the tracing of the data.
• It turns out that, though Hilbert transform
  exists, the simple Hilbert transform does not
  make sense physically.
• For envelopes, the necessary condition for
  Hilbert transformed amplitude to make sense is
  for IMF.

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Data X1 IMF
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Data x1 IMF1
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Data x2 IMF2
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Observations

• For each IMF, the envelope will make sense.
• For complicate data, we have to decompose it
  before attempting envelope construction.
• To be able to determine the envelope is
  equivalent to AM FM decomposition.

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Data set 2
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Data X2
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Data X2 Hilbert Transform
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Data X2 Envelopes
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Observations

• Even for this well behaved function, the
  amplitude from Hilbert transform does not serve
  as an envelope well. One of the reasons is that
  the function has two spectrum lines.
• Hilbert Transform represents higher frequency
  better.
• Complications for more complex functions are
  many.
• Here, the empirical envelope seems reasonable.

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Empirical Mode Decomposition

• Mathematically, there are infinite number of ways
  to decompose a functions into a complete set of
  components.
• The ones that give us more physical insight are
  more significant.
• In general, the fewer the number of representing
  components, the higher the information content
  Sparse representation.
• The adaptive method will represent the
  characteristics of the signal better.
• EMD is an adaptive method that can generate
  infinite many sets of IMF components to
  represent the original data.

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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 h2
m3
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Empirical Mode Decomposition Methodology h3
m4
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Empirical Mode Decomposition Methodology h2
h3
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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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Empirical Mode DecompositionSifting to get one
IMF component
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Empirical Mode Decomposition Methodology IMF
c1
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Definition of the Intrinsic Mode Function

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Empirical Mode DecompositionSifting to get all
the IMF components
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Empirical Mode DecompositionSifting to get all
the IMF components
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Empirical Mode DecompositionSifting to get all
the IMF components
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Empirical Mode DecompositionSifting to get all
the IMF components
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Observations

• All IMF components are the sums of spline
  functions.
• We selected cubic natural spline tomaintain the
  maximum smoothness.
• We will discuss spline function next time.

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Empirical Mode Decomposition Methodology data
r1
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Empirical Mode Decomposition Methodology
data, h1 r1
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Empirical Mode Decomposition Methodology IMFs
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Definition of Instantaneous Frequency
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Definitions of Frequency
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The Effects of Sifting

• The first effect of sifting is to eliminate the
  riding waves to make the number of extrema
  equals to that of zero-crossing.
• The second effect of sifting is to make the
  envelopes symmetric. The consequence is to make
  the amplitudes of the oscillations more even.

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Singularity points for Instantaneous Frequency
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Critical Parameters for EMD

• The maximum number of sifting allowed to extract
  an IMF, N.
• The criterion for accepting a sifting component
  as an IMF, the Stoppage criterion S.
• Therefore, the nomenclature for the IMF are
• CE(N, S) for extrema sifting
• CC(N, S) for curvature sifting

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The Stoppage Criteria S and SD
A. The S number S is defined as the
consecutive number of siftings, in which the
numbers of zero-crossing and extrema are the
same for these S siftings. B. If the mean is
smaller than a pre-assigned value. C. Fixed
sifting (iterating) time. D. SD is small than a
pre-set value, where
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Curvature Sifting

• Hidden Scales

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Empirical Mode Decomposition Methodology Test
Data
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Hidden Scales
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Hidden Scales
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Observations

• If we decide to use curvature, we have to be
  careful for what we ask for.
• For example, the Duffing pendulum would produce
  more than one components.
• Therefore, curvature sifting is used sparsely.
  It is useful in the first couple of components to
  get rid of noises.

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Intermittence Test

• To alleviate the Mode Mixing

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Sifting with Intermittence Test

• To avoid mode mixing, we have to institute a
  special criterion to separate oscillation of
  different time scales into different IMF
  components.
• The criteria is to select time scale so that
  oscillations with time scale shorter than this
  pre-selected criterion is not included in the IMF.

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Intermittence Sifting Data
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Intermittence Sifting IMF
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Intermittence Sifting Hilbert Spectra
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Intermittence Sifting Hilbert Spectra (Low)
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Intermittence Sifting Marginal Spectra
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Intermittence Sifting Marginal spectra (Low)
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Intermittence Sifting Marginal spectra (High)
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Critical Parameters for Sifting

• Because of the inclusion of intermittence test
  there will be one set of intermittence criteria.
• Therefore, the Nomenclature for IMF here are
• CEI(N,S n1, n2, )
• CCI(N, S n1, n2, )
• with n1, n2 as the intermittence test criteria.

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The mathematical Requirements for Basis

• The traditional Views

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IMF as Adaptive Basis

• According to the established mathematical
  paradigm, we should check the following
  properties of the basis
• Convergence
• completeness
• orthogonality
• Uniqueness

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Convergence
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Convergence Problem

• Given an arbitrary number, e, there always exists
  a large finite number N, such that Nth envelope
  mean, mN , satisfies mN e

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Convergence Problem

• Given an arbitrary number, e, there always exists
  a large finite number N, such that N- th sifting
  satisfies

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Convergence

• There is another convergence problem we have
  only finite number of components.
• Complete proof for convergence is underway.
• We can prove the convergence under simplified
  condition of linear segment fitting for sifting.
• Empirically, we found all cases converge in
  finite steps. The finite component, n, is less
  than or equal to log2M, with M as the total
  number of data points.

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Convergence

• The necessary condition for convergence is that
  the mean line should have less extrema than the
  original data.
• This might not be true if we use the middle
  points and a single spline the procedure might
  not converge.

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Completeness
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Completeness

• Completeness is given by the algebraic equation
• Therefore, the sum of IMF can be as close to the
  original data as required.
• Completeness is given.

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Orthogonality
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Orthogonality

• Definition Two vectors x and y are orthogonal if
  their inner product is zero.
• x y (x1 y1 x2 y2 x3 y3 ) 0.

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The need for an orthogonality check

• Orthogonal is required for

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Orthogonality

• Orthogonality is a requirement for any linear
  decomposition.
• For a nonlinear decomposition, as EMD, the
  orthogonality should not be a requirement, for
  nonlinear waves of different scale could share
  the same harmonics.
• Fortunately, the EMD is basically a Reynolds type
  decomposition , U ltUgt u, orthogonality is
  always approximately satisfied to the degree of
  nonlinearity.
• Orthogonality Index should be checked for each
  cases as a goodness of decomposition
  confirmation.

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Orthogonality Index
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Length Of Day Data
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LOD IMF
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Orthogonality Check

• Pair-wise
• 0.0003
• 0.0001
• 0.0215
• 0.0117
• 0.0022
• 0.0031
• 0.0026
• 0.0083
• 0.0042
• 0.0369
• 0.0400

• Overall
• 0.0452

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Uniqueness
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Uniqueness

• EMD, with different critical parameters, can
  generate infinite sets of IMFs.
• The result is unique only with respect to the
  critical parameters and sifting method selected
  therefore, all results should be properly named
  according to the nomenclature scheme proposed
  above.
• The present sifting is based on cubic spline.
  Different spline fitting in the sifting procedure
  will generate different results.
• The ensemble of IMF sets offers a Confidence
  Limit as function of time and frequency.

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Some Tricks in Sifting
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Some Tricks in Sifting

• Sometimes straightforward application of sifting
  will not generate good results.
• Invoking intermittence criteria is an alternative
  to get physically meaningful IMF components.
• By adding low level noise can improve the
  sifting.
• By using curvature may also help.

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An Example

• Adding Noise of small amplitude only,
• A prelude to the true Ensemble EMD

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Data 2 Coincided Waves
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IMF from Data of 2 Coincided Waves
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Data 2 Coincided Waves NoiseThe Amplitude of
the noise is 1/1000
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IMF form Data 2 Coincided Waves Noise
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IMF c1 and Component2 2 Coincided Waves
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IMF c2c3 and Component1 2 Coincided Waves
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A Flow Chart
Ensemble EMD
sifting
Data
IMF
OI
With Intermittence
IF
Hilbert Spectrum
Marginal Spectrum