A Plea for Adaptive Data Analysis: Instantaneous Frequencies and Trends For Nonstationary Nonlinear Data

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A Plea for Adaptive Data AnalysisInstantaneous
Frequencies and Trends For Nonstationary
Nonlinear Data

• Norden E. Huang
• Research Center for Adaptive Data Analysis
• National Central University
• Zhongli, Taiwan, China
• Hot Topic Conference, 2011

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In search of frequency I found the trend and
other information

• Instantaneous Frequencies and Trends
• For Nonstationary Nonlinear Data

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Prevailing Views onInstantaneous Frequency
The term, Instantaneous Frequency, should be
banished forever from the dictionary of the
communication engineer. J. Shekel, 1953 The
uncertainty principle makes the concept of an
Instantaneous Frequency impossible. K.
Gröchennig, 2001
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How to define frequency?

• It seems to be trivial.
• But frequency is an important parameter for us to
  understand many physical phenomena.

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Definition of Frequency
Given the period of a wave as T the frequency
is defined as
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Traditional Definition of Frequency

• frequency 1/period.
• Definition too crude
• Only work for simple sinusoidal waves
• Does not apply to nonstationary processes
• Does not work for nonlinear processes
• Does not satisfy the need for wave equations

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The Idea and the need of Instantaneous Frequency
According to the classic wave theory, the wave
conservation law is based on a gradually changing
f(x,t) such that
Therefore, both wave number and frequency must
have instantaneous values and differentiable.
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Instantaneous Frequency
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Hilbert Transform Definition
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The Traditional View of the Hilbert Transform
for Data Analysis
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Traditional Viewa la Hahn (1995) Data LOD
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Traditional Viewa la Hahn (1995) Hilbert
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Traditional Approacha la Hahn (1995) Phase
Angle
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Traditional Approacha la Hahn (1995) Phase
Angle Details
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Traditional Approacha la Hahn (1995)
Frequency
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Why the traditional approach does not work?
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Hilbert Transform a cos ? b Data
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Hilbert Transform a cos ? b Phase Diagram
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Hilbert Transform a cos ? b Phase Angle
Details
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Hilbert Transform a cos ? b Frequency
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The Empirical Mode Decomposition Method and
Hilbert Spectral AnalysisSifting
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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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The Stoppage Criteria
The Cauchy type criterion when SD is small than
a pre-set value, where
Or, simply pre-determine the number of iterations.
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Effects of Sifting

• To remove ridding waves
• To reduce amplitude variations
• The systematic study of the stoppage criteria
  leads to the conjecture connecting EMD and
  Fourier Expansion.

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Empirical Mode Decomposition Methodology IMF
c1
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Definition of the Intrinsic Mode Function (IMF)
a necessary condition only!

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Empirical Mode Decomposition Methodology
data, r1 and m1
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Empirical Mode DecompositionSifting to get all
the IMF components
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Definition of Instantaneous Frequency
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An Example of Sifting Time-Frequency Analysis
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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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LOD Data c12
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LOD Data Sum c11-12
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LOD Data sum c10-12
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LOD Data c9 - 12
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LOD Data c8 - 12
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LOD Detailed Data and Sum c8-c12
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LOD Data c7 - 12
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LOD Detail Data and Sum IMF c7-c12
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LOD Difference Data sum all IMFs
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Properties of EMD Basis

• The Adaptive Basis based on and derived from the
  data by the empirical method satisfy nearly all
  the traditional requirements for basis
  empirically and a posteriori
• Complete
• Convergent
• Orthogonal
• Unique

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The combination of Hilbert Spectral Analysis and
Empirical Mode Decomposition has been designated
by NASA as

• HHT
• (HHT vs. FFT)

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Comparison between FFT and HHT
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Comparisons Fourier, Hilbert Wavelet
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Speech Analysis Hello Data
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Four comparsions D
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Traditional Viewa la Hahn (1995) Hilbert
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Mean Annual Cycle Envelope 9 CEI Cases
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Mean Hilbert Spectrum All CEs
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For quantifying nonlinearity we need
instantaneous frequency.

• For

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How to define Nonlinearity?

• How to quantify it through data alone?

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The term, Nonlinearity, has been loosely used,
most of the time, simply as a fig leaf to cover
our ignorance.

• Can we be more precise?

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How is nonlinearity defined?

• Based on Linear Algebra nonlinearity is defined
  based on input vs. output.
• But in reality, such an approach is not
  practical natural system are not clearly
  defined inputs and out puts are hard to
  ascertain and quantify. Furthermore, without the
  governing equations, the order of nonlinearity is
  not known.
• In the autonomous systems the results could
  depend on initial conditions rather than the
  magnitude of the inputs.
• The small parameter criteria could be misleading
  sometimes, the smaller the parameter, the more
  nonlinear.

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Linear Systems

• Linear systems satisfy the properties of
  superposition and scaling. Given two valid inputs
  to a system H,
• as well as their respective outputs
• then a linear system, H, must satisfy
• for any scalar values a and ß.

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How is nonlinearity defined?

• Based on Linear Algebra nonlinearity is defined
  based on input vs. output.
• But in reality, such an approach is not
  practical natural system are not clearly
  defined inputs and out puts are hard to
  ascertain and quantify. Furthermore, without the
  governing equations, the order of nonlinearity is
  not known.
• In the autonomous systems the results could
  depend on initial conditions rather than the
  magnitude of the inputs.
• The small parameter criteria could be misleading
  sometimes, the smaller the parameter, the more
  nonlinear.

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How should nonlinearity be defined?

• The alternative is to define nonlinearity based
  on data characteristics Intra-wave frequency
  modulation.
• Intra-wave frequency modulation is known as the
  harmonic distortion of the wave forms. But it
  could be better measured through the deviation of
  the instantaneous frequency from the mean
  frequency (based on the zero crossing period).

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Characteristics of Data from Nonlinear Processes
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Duffing Pendulum
x
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Duffing Equation Data
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Hilberts View on Nonlinear DataIntra-wave
Frequency Modulation
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A simple mathematical model
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Duffing Type WaveData x cos(wt0.3 sin2wt)
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Duffing Type WavePerturbation Expansion
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Duffing Type WaveWavelet Spectrum
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Duffing Type WaveHilbert Spectrum
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Degree of nonlinearity
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Degree of Nonlinearity

• DN is determined by the combination of d?
  precisely with Hilbert Spectral Analysis. Either
  of them equals zero means linearity.
• We can determine d and ? separately
• ? can be determined from the instantaneous
  frequency modulations relative to the mean
  frequency.
• d can be determined from DN with known ?.
• NB from any IMF, the value of d? cannot be
  greater than 1.
• The combination of d and ? gives us not only
  the Degree of Nonlinearity, but also some
  indications of the basic properties of the
  controlling Differential Equation, the Order of
  Nonlinearity.

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Stokes Models
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Data and IFs C1
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Summary Stokes I
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Lorenz Model

• Lorenz is highly nonlinear it is the model
  equation that initiated chaotic studies.
• Again it has three parameters. We decided to fix
  two and varying only one.
• There is no small perturbation parameter.
• We will present the results for ?28, the classic
  chaotic case.

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Phase Diagram for ro28
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X-Component

• DN10.5147
• CDN0.5027

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Data and IF
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Spectra data and IF
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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, quantifying
Non-stationary no yes Yes, quantifying
Uncertainty yes yes no
Harmonics yes yes no
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How to define Trend ?

• Parametric or Non-parametric?
• Intrinsic vs. extrinsic approach?

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The State-of-the arts TrendOne economists
trend is another economists cycle Watson
Engle, R. F. and Granger, C. W. J. 1991 Long-run
Economic Relationships. Cambridge University
Press.

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Philosophical Problem Anticipated
??????? ???????                        ???
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On Definition Without a proper
definition, logic discourse would be
impossible.Without logic discourse, nothing
can be accomplished. Confucius
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Definition of the Trend
Within the given data span, the trend is an
intrinsically fitted monotonic function, or a
function in which there can be at most one
extremum. The trend should be an intrinsic and
local property of the data it is determined by
the same mechanisms that generate the data.
Being local, it has to associate with a local
length scale, and be valid only within that
length span, and be part of a full wave
length. The method determining the trend should
be intrinsic. Being intrinsic, the method for
defining the trend has to be adaptive. All
traditional trend determination methods are
extrinsic.
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Algorithm for Trend

• Trend should be defined neither parametrically
  nor non-parametrically.
• It should be the residual obtained by removing
  cycles of all time scales from the data
  intrinsically.
• Through EMD.

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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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Mean IMF
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STD IMF
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Statistical Significance Test
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Data and Trend C6
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Rate of Change Overall Trends EMD and Linear
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Conclusion

• With EMD, we can define the true instantaneous
  frequency and extract trend from any data.
• We can also talk about nonlinearity
  quantitatively.
• Among other applications, the degree of
  nonlinearity could be used to set an objective
  criterion in structural health monitoring and to
  quantify the degree of nonlinearity in natural
  phenomena the trend could be used in financial
  as well as natural sciences.
• These are all possible because of adaptive data
  analysis method.

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The Job of a Scientist
The job of a scientist is to listen carefully to
nature, not to tell nature how to
behave. Richard Feynman To listen is to
use adaptive method and let the data sing, and
not to force the data to fit preconceived modes.
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All these results depends on adaptive approach.

• Thanks

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What This Means

• EMD separates scales in physical space it
  generates an extremely sparse representation for
  any given data.
• Added noises help to make the decomposition more
  robust with uniform scale separations.
• Instantaneous Frequency offers a total different
  view for nonlinear data instantaneous frequency
  needs no harmonics and is unlimited by
  uncertainty principle.
• Adaptive basis is indispensable for nonstationary
  and nonlinear data analysis
• EMD establishes a new paradigm of data analysis

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Outstanding Mathematical Problems

• Mathematic rigor on everything we do. (tighten
  the definitions of IMF,.)
• Adaptive data analysis (no a priori basis
  methodology in general)
• 3. Prediction problem for nonstationary processes
  
• (end effects)
• 4. Optimization problem (the best stoppage
  criterion and the uniqueness of the
  decomposition)
• 5. Convergence problem (best spline implement,
• and 2-D)

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Conclusion

• Adaptive method is the only scientifically
  meaningful way to analyze nonlinear and
  nonstationary data.
• It is the only way to find out the underlying
  physical processes therefore, it is
  indispensable in scientific research.
• EMD is adaptive It is physical, direct, and
  simple.
• But, we have a lot of problems
• And need a lot of helps!

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Up Hill

• Does the road wind up-hill all the way?
• Yes, to the very end.
• Will the days journey take the whole long day?
• From morn to night, my friend.
• --- Christina
  Georgina Rossetti

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A less poetic paraphrase

• There is no doubt that our road will be long and
  that our climb will be steep.
• But, anything is possible.
• --- Barack Obama
• 18 Jan 2009,
  Lincoln Memorial

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• History of 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, A460, 1179-1611.
• 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 , 2, 177-229.
• 2010 Multi-Dimensional Ensemble Empirical Mode
  Decomposition. Advances in Adaptive Data Analysis
  3, 339-372.
• 2011 Degree of Nonlinearity. Patent and Paper

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