an adaptive data analysis method with some biomedical applications norden e. huang research center...
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An Adaptive Data Analysis Method with Some Biomedical Applications
Norden E. HuangResearch Center for Adaptive Data Analysis
National Central University
2008
Data and Data Analysis
Data are the our only connection to the reality.
We are overwhelmed by the quantity of the data,
But underserved by the quality of the analysis.
Data Processing and Data Analysis
• Processing [proces < L. Processus < pp of Procedere = Proceed: pro- forward + cedere, to go] : A particular method of doing something.
• Data Processing >>>> 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 >>>> Physical understandings
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.
* Unexamined facts (data) are useless!
Scientific Activities
Collecting, analyzing, synthesizing, and theorizing are the core of scientific activities.
Data are our only connection to reality; they are also what separate science from philosophy.
Therefore, data analysis is a key link in this continuous loop.
Data Analysis
Data analysis is too important to be left to the mathematicians.
Why?!
Different Paradigms IMathematics vs. Science/Engineering
• Mathematicians
• Absolute proofs
• Logic consistency
• Mathematical rigor
• Scientists/Engineers
• Agreement with observations
• Physical meaning
• Working Approximations
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
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
Traditional Data Analysis
In pursue of mathematic rigor and certainty, however, we 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
削足適履
Trimming the foot to fit the shoe.
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
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
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.
掩耳盜鈴
Stealing the bell with muffed ears
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.
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.
The Job of a Scientist
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.
Duffing Pendulum
2
22( co .) s1
d xx tx
dt
x
Duffing Equation : Data
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
Hilbert Transform Fit
The Traditional View of the Hilbert Transform
for Data Analysis
Traditional Viewa la Hahn (1995) : Data LOD
Traditional Viewa la Hahn (1995) : Hilbert
Why the traditional approach does not work?
Hilbert Transform a cos + b : Data
Hilbert Transform a cos + b : Phase Diagram
Hilbert Transform a cos + b : Phase Angle Details
Hilbert Transform a cos + b : Frequency
The Empirical Mode Decomposition Method and
Hilbert Spectral Analysis
Sifting
Empirical Mode Decomposition: Methodology : Test Data
Empirical Mode Decomposition: Methodology : data and m1
Empirical Mode Decomposition: Methodology : data & h1
Empirical Mode Decomposition: Methodology : h1 & m2
Empirical Mode Decomposition: Methodology : h3 & m4
Empirical Mode Decomposition: Methodology : h4 & m5
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
.
The Stoppage Criteria
The Cauchy type criterion:
when SD is small than a pre-set value, where
T2
k 1 kt 0
T2
k 1t 0
h ( t ) h ( t )SD
h ( t )
Empirical Mode Decomposition: Methodology : IMF c1
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
Empirical Mode Decomposition: Methodology : data, r1 and m1
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 .
.
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 .
Definition of Frequency
Given the period of a wave as T ; the frequency is defined as
1.
T
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
The combination of Hilbert Spectral Analysis and
Empirical Mode Decomposition is designated as
HHT
(HHT vs. FFT)
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
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 .
Comparisons: Fourier, Hilbert & Wavelet
Speech Analysis Hello : Data
Four comparsions D
An Example of Sifting
Length Of Day Data
LOD : IMF
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
LOD : Data & c12
LOD : Data & Sum c11-12
LOD : Data & sum c10-12
LOD : Data & c9 - 12
LOD : Data & c8 - 12
LOD : Detailed Data and Sum c8-c12
LOD : Data & c7 - 12
LOD : Detail Data and Sum IMF c7-c12
LOD : Difference Data – sum all IMFs
Traditional Viewa la Hahn (1995) : Hilbert
Mean Annual Cycle & Envelope: 9 CEI Cases
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
Hilbert’s View on Nonlinear Data
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
Duffing Equation : Data
Duffing Equation : IMFs
Duffing Equation : Hilbert Spectrum
Duffing Equation : Detailed Hilbert Spectrum
Duffing Equation : Wavelet Spectrum
Duffing Equation : Hilbert & Wavelet Spectra
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
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
Application of Hilbert Huang Transform in Cerebral Blood
Flow Regulation
Multimodal pressure-flow method to assess dynamics of cerebral auto-regulation
By Dr. Lo Meng-Yzung of RCADA, NCU and the Harvard Medical School, Division of Inter-disciplinary Medicine.
Valsalva Maneuver Dynamics
Time (sec)
20 30 40 50 60 70 80
mm
Hg
40
60
80
100
120
140
160
180
Arterial Blood PressureHHT residual
II
I
III
IVI. Expiration - mechanicalI. Expiration - mechanical
II. reduced venous return, BP falls
II. reduced venous return, BP falls
III. Inspiration - mechanicalIII. Inspiration - mechanical
IV. increased cardiac output and increased peripheral resistance
IV. increased cardiac output and increased peripheral resistance
Valsalva Maneuver Dynamics
Blood Pressure
Blood Flow Velocity – Right Middle Cerebral Artery
Blood Flow Velocity – Left Middle Cerebral Artery
Empirical Mode Decomposition
Original blood pressure waveform
Key mode of blood pressure waveform during Valsalva maneuver
Blood pressure versus blood flow velocityTemporal (time) Relationship
Blood pressure versus blood flow velocityPhase Relationship
Control Stroke
EEMD can easily identify blood pressure oscillations induced by the spontaneous breathing (0.4~0.15 Hz , i.e. period 2.5~7 sec)
Cerebral Auto-regulation in stroke patients
86
Hu K, Peng CK, Lo MT, Zhao P, LaRose S, Novak P, Selim M, Lipsitz LA, Novak V (2008)
Assessment of Cerebral Autoregulation from Spontaneous Blood Pressure and Cerebral Blood Flow Fluctuations.
Stroke (abstract, accepted for publication)
Cerebral Auto-regulation in diabetes
87
Hu K, Peng CK, Huang NE, et al. (2008)
Altered phase interactions between spontaneous blood pressure and flow fluctuations in Type 2 diabetes mellitus: Nonlinear assessment of cerebral autoregulation. Physica A 387 2279–2292.
Compare with Transfer Function Analysis (Fourier based method) for Diabetes with Valsalva maneuver.
0 50 100 150 200 250 300 350 400 450-2
-1
0
1
2simulated BP
0 50 100 150 200 250 300 350 400 450-2
-1
0
1
2simulated BFV
Identify the epoch with unwanted interference
0 50 100 150 200 250 300 350 400 450-4
-2
0
2
4phase difference
0 50 100 150 200 250 300 350 400 450-1
0
1
2annotation
Nonlinear pressure-flow relationship is able to detect asymmetry of brain blood circulation associated with midline shift
Dr. Marek Czosnyka, UK neuroscience
Association between the midline shift and left-right difference in BP-BFV phase shift
The difference in BP-BFV phase shifts between the right and left hemispheres was positively correlated to midline shift of the brain
Image Analysis
We have extended HHT to multi-dimensional data; A patent is being filed.
Image analysis by bidimensional empirical mode decompositionJ.C. Nunes*, Y. Bouaoune, E. Delechelle, O. Niang, Ph. Bunel
Image and Vision Computing, 2003
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.
But, we have only started and what we have done is only a scratch of the surface.
Many of the most significant and interesting challenges of the modern world require boundary-crossing collaborations among scientists and scholars with widely different
fields of expertise.
Allison Richard
Vice Chancellor, Cambridge University
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)
History of HHT1998: The Empirical Mode Decomposition Method and the Hilbert Spectrum for No
n-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 spe
ctral 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, 1597-1611Defined 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
2009: On Ensemble Empirical Mode Decomposition. Advances in Adaptive Data Analysis 1, 1-41.
2009: On instantaneous Frequency. Advances in Adaptive Data Analysis (Accepted)
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
To be launched in the March 2008