A Confidence Limit for Hilbert Spectrum

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A Confidence Limit for Hilbert Spectrum

• Through stoppage criteria

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Need a confidence limit

• As we have presented here, EMD could generate
  infinite many sets of IMFs. In this case, which
  one of the infinite many sets really represents
  the true physics? To answer this question, we
  need a confidence limit on our result.
• Traditional methods also had the similar problem
  of generating many answers. Take Fourier
  analysis for example we have to assume
  trigonometric series is basis. How about other
  basis? Why not slightly distorted sinusoidal
  wave as basis? .

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Confidence Limit for Fourier Spectrum

• The Confidence limit for Fourier Spectral
  analysis is based on ergodic assumption.
• It is derived by dividing the data into M
  sections, and substituting temporal (or spatial)
  average as ensemble average.
• This approach is valid for linear and stationary
  processes, and the sub-sections have to be
  statistically independent.
• By dividing the data into subsections, the
  resolution will suffer.

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Statistical Independence
The probability function of x and y jointly, f(x,
y), is equal to f(x) times f(y) if x and y are
statistically independent. For any number of
variables, x1, x2, ..., xn, if the joint
probability is the product of the several
probability functions, then the variables are all
statistically independent. Independent variables
are noncorrelated, but not necessarily
conversely. James James Mathematics
Dictionary
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LOD Data
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Confidence Limit for Fourier Spectrum
Confidence Limit from 7 sections, each 2048
points.
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Are the sub-sections statistically independent?

• For narrow band signals, most likely they would
  not be independent.

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Confidence Limit for Hilbert Spectrum

• Any data can be decomposed into infinitely many
  different constituting component sets.
• EMD is a method to generate infinitely many
  different IMF representations based on different
  sifting parameters.
• Some of the IMFs are better than others based on
  various properties for example, Orthogonal
  Index.
• A Confidence Limit for Hilbert Spectral analysis
  can be based on an ensemble of valid IMF
  resulting from different sifting parameters S
  covering the parameter space fairly.
• It is valid for nonlinear and nonstationary
  processes.

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Different Kinds of Confidence Limit

• The basic idea is to generate various IMFs, treat
  the mean as the true answer, and obtain the
  confidence limit based on the STD from the
  various solutions.
• By stoppage criteria
• By Ensemble EMD
• By down sampling
• Different spline methods

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Empirical Mode DecompositionSifting to get one
IMF component
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None of the above methods depends on ergodic
assumption.

• We are truly achieving an ensemble mean.

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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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Critical Parameters for EMD

• The maximum number of sifting allowed to extract
  an IMF, N.
• Note N is originally set to guarantee
  convergence of sifting, but later found to be
  superfluous.
• 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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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.
• Note N is originally set to guarantee
  convergence of sifting, but later found to be
  superfluous.

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Effects of EMD (Sifting)

• To separate data into components of similar
  scale.
• To eliminate ridding waves.
• To make the results symmetric with respect to the
  x-axis and the amplitude more even.
• Note The first two are necessary for valid IMF,
  the last effect actually cause the IMF to lost
  its intrinsic properties.

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LOD Data
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IMF CE(100, 2)
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IMF CE(100, 10)
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Orthogonal Index as function of N and S Contour
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Orthogonality Index as function of N and S
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Confidence Limit without Intermittence Criteria

• Number of IMF for different siftings may not be
  the same therefore, average of IMF is, in
  general, not possible. However, we can take the
  mean of the Hilbert Spectra, for we can make all
  the spectra having the same frequency and time
  ranges.

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Hilbert Spectrum CE(100, 2)
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Mean Hilbert Spectrum All CEs
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STD Hilbert Spectrum All CEs
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Marginal Mean STD Hilbert Spectra All CEs
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Mean and STD of Marginal Hilbert Spectra
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Confidence Limit with Intermittence Criteria

• In general, the number of IMFs can be controlled
  to the same therefore, averages of IMFs and
  Hilbert Spectra are all possible.

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IMF CEI(100,2 4,-13,452,-10)
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IMF CEI(100,10 4,-13,452,-10)
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Envelopes of Selected Annual Cycle IMFs
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Orthogonal Indices for CEI cases
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IMF Mean CEI 9 cases
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IMF STD CEI 9 cases
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Mean Hilbert Spectrum All CEIs
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STD Hilbert Spectrum for All CEIs
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Marginal mean STD Hilbert Spectra All CEIs
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Mean Marginal Hilbert Spectrum Confidence Limit
All CEIs
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Mean Marginal Hilbert Spectrum Confidence Limit
All CEs
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Individual Annual Cycle IMFs 9 CEI Cases
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Details of Individual Annual Cycle CEIs
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Mean Annual Cycle Envelope 9 CEI Cases
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Individual Envelopes for Annual Cycle IMFs
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Mean Envelopes for Annual Cycle IMFs
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Optimal Sifting Parameters

• The Maximum sifting number should be set very
  high to guarantee that the stoppage criterion is
  always satisfied.
• The Stoppage criterion should be selected by
  considering the difference between the individual
  case with the mean to see if there is an optimal
  range where the difference is minimum.
• The difference can be computed from the Hilbert
  spectra or IMF components. It turn out that the
  IMF is a more sensitive way to determine the
  optimal sifting parameters.

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Computation of the Differences
where V(t) can be IMF or Hilbert Spectrum.
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IMF for CEI Cases Annual Cycle
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IMF for CEI Cases Half-monthly tidal Cycle
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Hilbert Spectrum Deviation Individual form the
mean CEI
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Hilbert Spectrum Deviation Individual form the
mean CE
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Another Example using Earthquake Data
Earthquake data has no fixed time scale
therefore, it is not possible to sift with
intermittence. The only way to compute the
confidence limit is use an ensemble of Hilbert
Spectra.
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Earthquake Data
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Mean Hilbert Spectrum
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Another Example using Earthquake Data
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Optimal Selection of Stoppage Criterion

• From the above tests, we can see that the Hilbert
  spectrum difference is less sensitive to the
  changes of stoppage criterion S than IMFs.
• From the IMF tests, we suggest that the S number
  should be set in the range of 3 to 10.
• This selection is in agreement with our past
  experiences however, additional quantitative
  tests should be conduct for other data types.

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Summary

• The Confidence limit presented here exists only
  with respect to the EMD method used.
• The Confidence limit presented here is only one
  of many possibilities. Instead of using OI as
  criterion, we can also use the STD of different
  trials to get a feeling of the stability of the
  analysis.
• Instead of Stoppage criteria, we can us different
  spline methods, down sampling and study their
  variations.
• Most interestingly, we could use Ensemble EMD, to
  be discussed next.

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Envelope of IMF c1