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An Introduction to HilbertHuang Transform: A Plea for Adaptive Data Analysis Norden E' Huang Researc

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Title: An Introduction to HilbertHuang Transform: A Plea for Adaptive Data Analysis Norden E' Huang Researc


1
An Introduction to Hilbert-Huang TransformA
Plea for Adaptive Data AnalysisNorden E.
HuangResearch Center for Adaptive Data
AnalysisNational Central University
2
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.
  • 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.

3
Data Analysis
  • Why we do it?
  • How did we do it?
  • What should we do?

4
Why?
5
Why do we have to analyze data?
  • Data are the only connects we have with the
    reality
  • data analysis is the only means we can find the
    truth and deepen our understanding of the
    problems.

6
Ever since the advance of computer and sensor
technology, there is an explosion of very
complicate data. The situation has changed
from a thirsty for data to that of drinking from
a fire hydrant.
7
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 our data.

8
Data and Data Analysis
  • Data Analysis is the key step in converting the
    facts into the edifice of science.
  • It infuses meanings to the cold numbers, and lets
    data telling their own stories and singing their
    own songs.

9
Science vs. Philosophy
  • Data and Data Analysis are what separate science
    from philosophy
  • With data we are talking about sciences
  • Without data we can only discuss philosophy.

10
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.

11
Data Analysis
  • Data analysis is too important to be left to the
    mathematicians.
  • Why?!

12
Different Paradigms IMathematics vs.
Science/Engineering
  • Mathematicians
  • Absolute proofs
  • Logic consistency
  • Mathematical rigor
  • Scientists/Engineers
  • Agreement with observations
  • Physical meaning
  • Working Approximations

13
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

14
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

15
How?
16
Data Processing vs. Analysis
  • All traditional data analysis methods are
    really for data processing. They are either
    developed by or established according to
    mathematicians rigorous rules. Most of the
    methods consist of standard algorithms, which
    produce a set of simple parameters.
  • They can only be qualified as data processing,
    not really data analysis.
  • Data processing produces mathematical meaningful
    parameters data analysis reveals physical
    characteristics of the underlying processes.

17
Data Processing vs. Analysis
  • In pursue of mathematic rigor and certainty,
    however, we lost sight of physics and are forced
    to idealize, but also deviate from, the reality.
  • As a result, we are forced to live in a
    pseudo-real world, in which all processes are
  • Linear and Stationary

18
????
  • Trimming the foot to fit the shoe.

19
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

20
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

21
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.

22
The Real World
  • Mathematics are well and good but nature keeps
    dragging us around by the nose.
  • Albert Einstein

23
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.

24
What?
25
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 methods and let the data sing, and not
to force the data to fit preconceived modes.
26
How to define nonlinearity?
  • Based on Linear Algebra nonlinearity is defined
    based on input vs. output.
  • But in reality, such an approach is not
    practical. The alternative is to define
    nonlinearity based on data characteristics.

27
Characteristics of Data from Nonlinear Processes
28
Duffing Pendulum
x
29
Hilbert Transform Definition
30
Hilbert Transform Fit
31
Conformation to reality rather then to Mathematics
  • We do not have to apologize, we should use
    methods that can analyze data generated by
    nonlinear and nonstationary processes.
  • That means we have to deal with the intrawave
    frequency modulations, intermittencies, and
    finite rate of irregular drifts. Any method
    satisfies this call will have to be adaptive.

32
The Traditional Approach of Hilbert Transform
for Data Analysis
33
Traditional Approacha la Hahn (1995) Data LOD
34
Traditional Approacha la Hahn (1995) Hilbert
35
Traditional Approacha la Hahn (1995) Phase
Angle
36
Traditional Approacha la Hahn (1995) Phase
Angle Details
37
Traditional Approacha la Hahn (1995)
Frequency
38
Why the traditional approach does not work?
39
Hilbert Transform a cos ? b Data
40
Hilbert Transform a cos ? b Phase Diagram
41
Hilbert Transform a cos ? b Phase Angle
Details
42
Hilbert Transform a cos ? b Frequency
43
The Empirical Mode Decomposition Method and
Hilbert Spectral AnalysisSifting
44
Empirical Mode Decomposition Methodology Test
Data
45
Empirical Mode Decomposition Methodology data
and m1
46
Empirical Mode Decomposition Methodology data
h1
47
Empirical Mode Decomposition Methodology h1
m2
48
Empirical Mode Decomposition Methodology h3
m4
49
Empirical Mode Decomposition Methodology h4
m5
50
Empirical Mode DecompositionSifting to get one
IMF component
51
Two Stoppage Criteria S and SD
  • 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. SD is small than a pre-set value, where

52
Empirical Mode Decomposition Methodology IMF
c1
53
Definition of the Intrinsic Mode Function (IMF)

54
Empirical Mode DecompositionSifting to get all
the IMF components
55
Empirical Mode Decomposition Methodology data
r1
56
Empirical Mode Decomposition Methodology data
and m1
57
Empirical Mode Decomposition Methodology
data, r1 and m1
58
Empirical Mode Decomposition Methodology IMFs
59
Definition of Instantaneous Frequency
60
Definition of Frequency
Given the period of a wave as T the frequency
is defined as
61
Equivalence
  • The definition of frequency is equivalent to
    defining velocity as
  • Velocity Distance / Time

62
Instantaneous Frequency
63
The combination of Hilbert Spectral Analysis and
Empirical Mode Decomposition is designated as
  • HHT
  • (HHT vs. FFT)

64
Jean-Baptiste-Joseph Fourier
  • 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
65
Comparison between FFT and HHT
66
Comparisons Fourier, Hilbert Wavelet
67
An Example of Sifting
68
Length Of Day Data
69
LOD IMF
70
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

71
LOD Data c12
72
LOD Data Sum c11-12
73
LOD Data sum c10-12
74
LOD Data c9 - 12
75
LOD Data c8 - 12
76
LOD Detailed Data and Sum c8-c12
77
LOD Data c7 - 12
78
LOD Detail Data and Sum IMF c7-c12
79
LOD Difference Data sum all IMFs
80
Traditional Viewa la Hahn (1995) Hilbert
81
Mean Annual Cycle Envelope 9 CEI Cases
82
Mean Hilbert Spectrum All CEs
83
Tidal Machine
84
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

85
Hilberts View on Nonlinear Data
86
Duffing Type WaveData x cos(wt0.3 sin2wt)
87
Duffing Type WavePerturbation Expansion
88
Duffing Type WaveWavelet Spectrum
89
Duffing Type WaveHilbert Spectrum
90
Duffing Type WaveMarginal Spectra
91
Duffing Equation
92
Duffing Equation Data
93
Duffing Equation IMFs
94
Duffing Equation Hilbert Spectrum
95
Duffing Equation Detailed Hilbert Spectrum
96
Duffing Equation Wavelet Spectrum
97
Duffing Equation Hilbert Wavelet Spectra
98
Speech Analysis
  • Nonlinear and nonstationary data

99
Speech Analysis Hello Data
100
Four comparsions D
101
Global Temperature Anomaly
  • Annual Data from 1856 to 2003

102
Global Temperature Anomaly 1856 to 2003
103
IMF Mean of 10 Sifts CC(1000, I)
104
Statistical Significance Test
105
Data and Trend C6
106
Rate of Change Overall Trends EMD and Linear
107
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

108
Comparisons
109
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.

110
  • 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, (in press)
  • Defined statistical significance and
    predictability.
  • 2004 On the Instantaneous Frequency, Proc. Roy.
    Soc. London, (Under review)
  • Removal of the limitations posted by Bedrosian
    and Nuttall theorems for instantaneous Frequency
    computations.

111
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 Gravity Wave
  • (NASA Goddard)
  • Financial market data analysis
  • (NCU)

112
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
  • in the January 2008

113
Oliver Heaviside1850 - 1925
Why should I refuse a good dinner simply because
I don't understand the digestive processes
involved.
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