An Introduction to HHT for Nonlinear and Nonstationary Time Series Analysis: A Plea for Adaptive...
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Transcript of An Introduction to HHT for Nonlinear and Nonstationary Time Series Analysis: A Plea for Adaptive...
An Introduction to HHT
for Nonlinear and Nonstationary Time Series Analysis:
A Plea for Adaptive Data Analysis
Norden E. HuangResearch Center for Adaptive Data Analysis
National Central University
What is data?
• Data (plural of Datum) [Latin: data – what is given] Information; facts, evidence, records, statistics, etc. from which conclusions can be formed.
Information in a form suitable for storing and processing by a computer.
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!
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 ProcessingData 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 AnalysisData Analysis >>>> Physical understandings
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.
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.
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.
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.
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.
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.
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
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
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 Methods for 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.
The Real World
Mathematics are well and good but nature keeps dragging us around by the nose.
Albert Einstein
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 methods and let the data sing, and not to force the data to fit preconceived modes.
The Job of a Scientist
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.
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.
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
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.
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
.
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 )
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 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 .
Equivalence :
The definition of frequency is equivalent to defining velocity as
Velocity = Distance / Time
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 .
Fourier analysis is incapable of representing any variation in temporal.
It is not even capable to separate noise from delta functions!
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
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
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 .
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
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
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.
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.
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 (in press)
2009: Multi-Dimensional Ensemble Empirical Mode Decomposition. Advances in Adaptive Data Analysis (Accepted; Patent Pending)
Current Efforts and Applications
• Non-destructive Evaluation for Structural Health Monitoring – (DOT, NSWC, DFRC/NASA, KSC/NASA Shuttle, THSR)
• Vibration, speech, and acoustic signal analyses– (FBI, and DARPA)
• Earthquake Engineering– (DOT)
• Bio-medical applications– (Harvard, Johns Hopkins, UCSD, NIH, NTU, VHT, AS)
• Climate changes– (NASA Goddard, NOAA, CCSP)
• Cosmological Gravity Wave– (NASA Goddard)
• Financial market data analysis– (NCU)
• Theoretical foundations– (Princeton University and Caltech)
The Idea behind EMD• To be able to analyze data from the
nonstationary and nonlinear processes and reveal their physical meaning, the method has to be Adaptive.
• Adaptive requires a posteriori (not a priori) basis. But the present established mathematical paradigm is based on a priori basis.
• Only a posteriori basis could fit the varieties of nonlinear and nonstationary data without resorting to the mathematically necessary (but physically nonsensical) harmonics.
The Idea behind EMD
• The method has to be local.
• Locality requires differential operation to define properties of a function.
• Take frequency, for example. The present established mathematical paradigm is based on Integral transform. But integral transform suffers the limitation of the uncertainty principle.
John von Neumann
By and large it is uniformly true that in mathematics there is a time lapse between a mathematical discovery and the moment it becomes useful; and that this lapse can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful.
Norden E. Huang : paraphrase
By and large it is usually true that in science there is a time lapse between the discovery of a useful method and the moment it becomes mathematically proved; and that this lapse can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function with a firm direction: always reference to usefulness, and with strong desire to do things which are useful.
On Calculus
“Newton and Leibniz's approach to the calculus fell well short of later standards of rigor. We now see their "proof" as being in truth mostly a heuristic hodgepodge mainly grounded in geometric intuition.” Wikipedia
dxx
dt
fdx F
1643-1727 1646-1716
On Calculus• George Berkeley, in a tract called The Analyst
and elsewhere, ridiculed this and other aspects of the early calculus, pointing out that natural science grounded in the calculus required just as big of a leap of faith as theology grounded in Christian revelation.
• Modern, rigorous calculus only emerged in the
19th century, thanks to the efforts of Augustin Louis Cauchy, Bernhard Riemann, Karl Weierstrass, and others, who based their work on the definition of a limit and on a precise understanding of real numbers.
Jean-Baptiste-Joseph Fourier (1768-1830)
1807 “On the Propagation of Heat in Solid Bodies”
1812 Grand Prize of Paris Institute
“Théorie analytique de la chaleur”
‘... manner in which the author arrives at these equations is not exempt of difficulties and that the 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 Académie
paper published
Fourier’s work is a great mathematical poem. Lord Kelvin
Fourier Transform
• Michel Plancherel (16 January 1885 to 4 March 1967) : Plancherel, Michel (1910) "Contribution a l'etude de la representation d'une fonction arbitraire par les integrales définies," Rendiconti del Circolo Matematico di Palermo, vol. 30, pages 298-335.
• In which he proved the convergence of the Fourier transform by excluding denumerable number of discontinuity points in the function. He then proved that the totality of the excluded parts have zero measure.
Oliver Heaviside1850 - 1925
Adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations (later found to be equivalent to Laplace transform),
introduced delta and step functions, and invented modern vector analysis, thereby reducing the original twenty equations in twenty unknowns down to the four differential equations in
two unknowns we now know as Maxwell’s Equations.
Why should I refuse a good dinner simply because I don't understand the digestive processes involved.
I will not wait for the Mathematician’s proof.
Historically, the proof would not change the method, but will
change the mathematician’s view.
John von NeumannAs a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from “reality,” it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l’art pour l’art. This need not be bad if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men [or women] with exceptionally well-developed taste. But there is a grave danger that the subject will developed along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and the discipline will become a disorganized mass of details and complexities.
HHT is at the same stage as Wavelet was in the late 1980s.
We need someone, as Ingrid Daubechies, to set the rigorous foundation.
But, in principle, HHT is much harder!
VOLUME ITECHNICAL PROPOSAL AND MANAGEMENT
APPROACHMathematical Analysis of the Empirical Mode Decomposition
Ingrid Daubechies1 and Norden Huang2
1 Program in Applied and Computational Mathematics (Princeton)2 Research Center for Adaptive Data Analysis,
(National Central University)
Since its invention by PI Huang over ten years ago, the Empirical Mode Decomposition (EMD) has been applied to a wide range of applications. The EMD is a two-stage, adaptive method that provides a nonlinear time-frequency analysis that has been remarkably successful in the analysis of nonstationary signals. It has been used in a wide range of fields, including (among many others) biology, geophysics, ocean research, radar and medicine. …….
The Battle Hymn for HHTBy Ingrid Daubechies
You can use wavelets or Fourier And find something that is useful,But if you want something new to say, You cannot be any old fool.
You then find a new kind of algorithm And let it loose on the world.Before they even know what hit’em, They’re up to their necks on work.
Rushing in to meet your frustrating challenge, They filter and stretch and squeeze.
But always some signal throws a monkey wrench, And make them huff and sneeze.
Your name is, of course, Norden Huang, Long may you live and smile!We’re here to learn and get the hang And will not quit for a very long
while.
Hilbert-Huang transform, you will triumph!
Composed on 17 December 2008 in Guangzhou, to be sung to the tune of ‘The Internationale’.
At The Second International Conference on the Advances of Hilbert-Huang Transform and its Applications.
Up Hill
Does the road wind up-hill all the way?
Yes, to the very end.
Will the day’s journey take the whole long day?
From morn to night, my friend.
--- Christina Georgina Rossetti
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
It is better lucky than smart.
Everyone needs luck sometimes: Albert Michelson, Carl Wilson, Arno
Penzias and Robert Wilson,…
I am lucky to have found this simple method.
Now we need smart people to tell us why it works and why it works so well.
Good luck to us all!!
Outline of the Course I• Introduction
• EMD and EEMD• EMD• Intermittency and confidence limit• EEMD• Orthogonality• End effects• Relationship with Fourier decomposition: a
conjecture• Trend and detrend
• HHT Operations
Outline of the Course II• Hilbert Spectral Analysis• Mathematical preliminary• End effects• Wavelet and Wigner-Ville Distribution• Instantaneous Frequency• Paradoxes of instantaneous frequency• Hilbert Spectral Representation and Marginal spectrum• Multi-dimensional EMD• Available approaches• MD-EEMD• Applications• Water wave studies• HHT based nondestructive Health monitoring• Stability spectral analysis• Global Climate Change