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An Introduction to Hilbert-Huang Transform:A Plea for Adaptive Data Analysis
Norden E. HuangResearch Center for Adaptive Data Analysis
National Central University
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Data Processing and Data Analysis
• Processing [proces < L. Processus < 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.
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Data Analysis
• Why we do it?
• How did we do it?
• What should we do?
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Why?
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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.
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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.
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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 our data.
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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.
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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.
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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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How?
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Data Processing vs. Analysis
All traditional ‘data analysis’ methods are really for ‘data processing’. They are either developed by or established according to mathematician’s 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.
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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
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削足適履
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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The Real World
Mathematics are well and good but nature keeps dragging us around by the nose.
Albert Einstein
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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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What?
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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.
The Job of a Scientist
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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.
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Characteristics of Data from Nonlinear Processes
32
2
2
22
d xx cos t
dt
d xx cos t
dt
Spring with positiondependent cons tan t ,
int ra wave frequency mod ulation;
therefore ,we need ins tan
x
1
taneous frequenc
x
y.
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Duffing Pendulum
2
22( co .) s1
d xx tx
dt
x
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p
2 2 1 / 2 1
i ( t )
For any x( t ) L ,
1 x( )y( t ) d ,
t
then, x( t )and y( t ) form the analytic pairs:
z( t ) x( t ) i y( t ) ,
where
y( t )a( t ) x y and ( t ) tan .
x( t )
a( t ) e
Hilbert Transform : Definition
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Hilbert Transform Fit
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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.
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The Traditional Approach of Hilbert Transform for Data Analysis
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Traditional Approacha la Hahn (1995) : Data LOD
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Traditional Approacha 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 Analysis
Sifting
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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
1 1
1 2 2
k 1 k k
k 1
x( t ) m h ,
h m h ,
.....
.....
h m h
.h c
.
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Two 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. SD is small than a pre-set value, whereT
2
k 1 kt 0
T2
k 1t 0
h ( t ) h ( t )SD
h ( t )
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Empirical Mode Decomposition: Methodology : IMF c1
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Definition of the Intrinsic Mode Function (IMF)
Any function having the same numbers of
zero cros sin gs and extrema,and also having
symmetric envelopes defined by local max ima
and min ima respectively is defined as an
Intrinsic Mode Function( IMF ).
All IMF enjoys good Hilbert Transfo
i ( t )
rm :
c( t ) a( t )e
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Empirical Mode DecompositionSifting : to get all the IMF components
1 1
1 2 2
n 1 n n
n
j nj 1
x( t ) c r ,
r c r ,
x( t ) c r
. . .
r c r .
.
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Empirical Mode Decomposition: Methodology : data & r1
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Empirical Mode Decomposition: Methodology : data and m1
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Empirical Mode Decomposition: Methodology : data, r1 and m1
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Empirical Mode Decomposition: Methodology : IMFs
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Definition of Instantaneous Frequency
i ( t )
t
The Fourier Transform of the Instrinsic Mode
Funnction, c( t ), gives
W ( ) a( t ) e dt
By Stationary phase approximation we have
d ( t ),
dt
This is defined as the Ins tan taneous Frequency .
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Definition of Frequency
Given the period of a wave as T ; the frequency is defined as
1.
T
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Equivalence :
The definition of frequency is equivalent to defining velocity as
Velocity = Distance / Time
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Instantaneous Frequency
distanceVelocity ; mean velocity
time
dxNewton v
dt
1Frequency ; mean frequency
period
dHH
So that both v and
T defines the p
can appear in differential equations.
hase functiondt
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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
1807 “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.’
1817 Elected to Académie des Sciences
1822 Appointed as Secretary of Math Section
paper published
Fourier’s work is a great mathematical poem. Lord Kelvin
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Comparison between FFT and HHT
j
j
t
i t
jj
i ( )d
jj
1. FFT :
x( t ) a e .
2. HHT :
x( t ) a ( t ) e .
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Comparisons: Fourier, Hilbert & Wavelet
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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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Mean Hilbert Spectrum : All CEs
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Tidal Machine
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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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Hilbert’s View on Nonlinear Data
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Duffing Type Wave
Data: x = cos(wt+0.3 sin2wt)
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Duffing Type WavePerturbation Expansion
For 1 , we can have
x( t ) cos t sin 2 t
cos t cos sin 2 t sin t sin sin 2 t
cos t sin t sin 2 t ....
1 cos t cos 3 t ....2 2
This is very similar to the solutionof Duffing equation .
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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
23
2.
Solved with for t 0 to 200 with
1
0.1
od
0.04 Hz
Initial condition :
[ x( o ) ,
d xx x c
x'( 0 ) ] [1
os t
, 1]
3
t
e2
d
tb
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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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Speech Analysis
Nonlinear and nonstationary data
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Speech Analysis Hello : Data
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Four comparsions D
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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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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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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 Convolution: Global Convolution: 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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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.
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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 Gravity Wave
– (NASA Goddard)
• Financial market data analysis
– (NCU)
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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 NCUThomas Yizhao Hou, CALTECH
in the January 2008
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Oliver Heaviside1850 - 1925
Why should I refuse a good dinner simply because I don't understand the digestive processes involved.