A Plea for Adaptive Data Analysis An Introduction to HHT

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A Plea for Adaptive Data AnalysisAn
Introduction to HHT

• Norden E. Huang
• Research Center for Adaptive Data Analysis
• National Central University

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Ever since the advance of computer, there is an
explosion of data. The situation has changed
from a thirsty for data to that of drinking from
a fire hydrant.We are drowning in data, but
thirsty for knowledge!
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Henri Poincaré

• Science is built up of facts,
• as a house is built of stones
• but an accumulation of facts is no more a science
  
• than a heap of stones is a house.
• Here facts are indeed data.

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Scientific Activities

• Collecting, analyzing, synthesizing, and
  theorizing are the core of scientific activities.
• Theory without data to prove is just hypothesis.
• Therefore, data analysis is a key link in this
  continuous loop.

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Data Analysis

• Data analysis is too important to be left to the
  mathematicians.
• Why?!

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Different Paradigms IMathematics vs.
Science/Engineering

• Mathematicians
• Absolute proofs
• Logic consistency
• Mathematical rigor

• Scientists/Engineers
• Agreement with observations
• Physical meaning
• Working Approximations

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Different Paradigms IIMathematics vs.
Science/Engineering

• Mathematicians
• Idealized Spaces
• Perfect world in which everything is known
• Inconsistency in the different spaces and the
  real world

• Scientists/Engineers
• Real Space
• Real world in which knowledge is incomplete and
  limited
• Constancy in the real world within allowable
  approximation

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Rigor vs. Reality

• As far as the laws of mathematics refer to
  reality, they are not certain and as far as they
  are certain, they do not refer to reality.
• Albert Einstein

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Data Processing and Data Analysis

• Processing proces lt L. Processus lt pp of
  Procedere Proceed pro- forward cedere, to
  go A particular method of doing something.
• Data Processing gtgtgtgt Mathematically meaningful
  parameters
• Analysis Gr. ana, up, throughout lysis, a
  loosing A separating of any whole into its
  parts, especially with an examination of the
  parts to find out their nature, proportion,
  function, interrelationship etc.
• Data Analysis gtgtgtgt Physical understandings

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Traditional Data Analysis

• All traditional data analysis methods are
  either developed by or established according to
  mathematicians rigorous rules. They are really
  data processing methods.
• In pursue of mathematic rigor and certainty,
  however, we are forced to
• idealize, but also deviate from, the reality.

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Traditional Data Analysis

• As a result, we are forced to live in a
  pseudo-real world, in which all processes are
• Linear and Stationary

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????

• Trimming the foot to fit the shoe.

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Available Data Analysis Methodsfor
Nonstationary (but Linear) time series

• Spectrogram
• Wavelet Analysis
• Wigner-Ville Distributions
• Empirical Orthogonal Functions aka Singular
  Spectral Analysis
• Moving means
• Successive differentiations

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Available Data Analysis Methodsfor Nonlinear
(but Stationary and Deterministic) time series

• Phase space method
• Delay reconstruction and embedding
• Poincaré surface of section
• Self-similarity, attractor geometry fractals
• Nonlinear Prediction
• Lyapunov Exponents for stability

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Typical Apologia

• Assuming the process is stationary .
• Assuming the process is locally stationary .
• As the nonlinearity is weak, we can use
  perturbation approach .
• Though we can assume all we want, but
• the reality cannot be bent by the assumptions.

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????

• Stealing the bell with muffed ears

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Motivations for alternatives Problems for
Traditional Methods

• Physical processes are mostly nonstationary
• Physical Processes are mostly nonlinear
• Data from observations are invariably too short
• Physical processes are mostly non-repeatable.
• ? Ensemble mean impossible, and temporal mean
  might not be meaningful for lack of stationarity
  and ergodicity. Traditional methods are
  inadequate.

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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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Characteristics of Data from Nonlinear Processes
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Duffing Pendulum
x
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Duffing Equation Data
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Hilbert Transform Definition
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Hilbert Transform Fit
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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
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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 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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Definition of Frequency
Given the period of a wave as T the frequency
is defined as
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Instantaneous Frequency
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The combination of Hilbert Spectral Analysis and
Empirical Mode Decomposition is designated as

• HHT
• (HHT vs. FFT)

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Jean-Baptiste-Joseph Fourier

1. On the Propagation of Heat in Solid Bodies

1812 Grand Prize of Paris Institute
Théorie analytique de la chaleur ... the
manner in which the author arrives at these
equations is not exempt of difficulties and that
his analysis to integrate them still leaves
something to be desired on the score of
generality and even rigor.

• Elected to Académie des Sciences
• Appointed as Secretary of Math Section
• paper published

Fouriers work is a great mathematical
poem. Lord Kelvin
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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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An Example of Sifting
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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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Traditional Viewa la Hahn (1995) Hilbert
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Mean Annual Cycle Envelope 9 CEI Cases
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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
• a posteriori
• Complete
• Convergent
• Orthogonal
• Unique

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Hilberts View on Nonlinear Data
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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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Duffing Type WaveMarginal Spectra
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Duffing Equation
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Duffing Equation Data
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Duffing Equation IMFs
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Duffing Equation Hilbert Spectrum
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Duffing Equation Detailed Hilbert Spectrum
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Duffing Equation Wavelet Spectrum
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Duffing Equation Hilbert Wavelet Spectra
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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 the only scientifically
  meaningful way to analyze data.
• It is the only 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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Current Applications

• Non-destructive Evaluation for Structural Health
  Monitoring
• (DOT, NSWC, and DFRC/NASA, KSC/NASA Shuttle)
• Vibration, speech, and acoustic signal analyses
• (FBI, MIT, and DARPA)
• Earthquake Engineering
• (DOT)
• Bio-medical applications
• (Harvard, UCSD, Johns Hopkins)
• Global Primary Productivity Evolution map from
  LandSat data
• (NASA Goddard, NOAA)
• Cosmological Gravitational Wave
• (NASA Goddard)
• Financial market data analysis
• (NCU)
• Geophysical and Climate studies
• (COLA, NASA, NCU)

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Outstanding Mathematical Problems

• Adaptive data analysis methodology in general
• Nonlinear system identification methods
• Prediction problem for nonstationary processes
• (end effects)
• Optimization problem (the best IMF selection
• and uniqueness. Is there a unique solution?)
• Spline problem (best spline implement of HHT,
• convergence and 2-D)
• Approximation problem (Hilbert transform
• and quadrature)

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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, 1597-1611
• Defined statistical significance and
  predictability.
• 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
• 2008 On Ensemble Empirical Mode Decomposition.
  Advances in Adaptive Data Analysis (in press)
• 2008 On instantaneosu Frequency. Advances in
  Adaptive Data Analysis (Accepted)

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Advances in Adaptive data Analysis Theory and
Applications

• A new journal to be published by
• the World Scientific
• Under the joint Co-Editor-in-Chief
• Norden E. Huang, RCADA NCU
• Thomas Yizhao Hou, CALTECH
• To be launched in the March 2008

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Thanks!
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Milankovitch Time scales Temperature Data from
Vostok Ice Core

• Data length 3311 points covering 422,766 Years BP

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Milutin Milankovitch 1879-1958
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How the Sun Affects Climate Solar and
Milankovitch Cycles


                                                                                     The
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Milankovitch Cycles
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A Truly Nonlinear World
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Data and Even Spaced Spline at Dt20 Year
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Data and CKY11 with Error Bounds
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Data and CKY10 with Error Bounds
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Data and CKY9 with Error Bounds
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Data and CKY8 with Error Bounds
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Data and Sum CKY 11 to 13 100K
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Data and Sum CKY 10 to 13 40K
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Data and Sum CKY 9 to 13 25K
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Data and Sum CKY 8 to 13 10K
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Data and Sum CKY 1 7 Less than 10K
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Hilbert Spectrum CKY 811
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Marginal Hilbert Spectrum CKY 811