norden e. huang research center for adaptive data analysis
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An Introduction to HHT: Instantaneous Frequency, Trend, Degree of Nonlinearity and Non- stationarity. Norden E. Huang Research Center for Adaptive Data Analysis Center for Dynamical Biomarkers and Translational Medicine NCU, Zhongli , Taiwan, China. - PowerPoint PPT PresentationTRANSCRIPT
An Introduction to HHT: Instantaneous Frequency, Trend,
Degree of Nonlinearity and Non-stationarity
Norden E. Huang
Research Center for Adaptive Data AnalysisCenter for Dynamical Biomarkers and Translational Medicine
NCU, Zhongli, Taiwan, China
OutlineRather than the implementation details,
I will talk about the physics of the method.
• What is frequency?
• How to quantify the degree of nonlinearity?
• How to define and determine trend?
What is frequency?
It seems to be trivial.
But frequency is an important parameter for us to understand many physical phenomena.
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
Other Definitions of Frequency : For any data from linear Processes
j
Ti t
j 0
1. Fourier Analysis :
F( ) x( t ) e dt .
2. Wavelet Analysis
3. Wigner Ville Analysis
Definition of Power Spectral Density
Since a signal with nonzero average power is not square integrable, the Fourier transforms do not exist in this case.
Fortunately, the Wiener-Khinchin Theroem provides a simple alternative. The PSD is the Fourier transform of the auto-correlation function, R(τ), of the signal if the signal is treated as a wide-sense stationary random process:
2i 2S( ) R( )e d S( )d ( t )
Fourier Spectrum
Problem with Fourier Frequency• Limited to linear stationary cases: same
spectrum for white noise and delta function.
• Fourier is essentially a mean over the whole domain; therefore, information on temporal (or spatial) variations is all lost.
• Phase information lost in Fourier Power spectrum: many surrogate signals having the same spectrum.
Surrogate Signal:Non-uniqueness signal vs. Power Spectrum
I. Hello
The original data : Hello
The surrogate data : Hello
The Fourier Spectra : Hello
The Importance of Phase
To utilize the phase to define
Instantaneous Frequency
Prevailing Views onInstantaneous Frequency
The term, Instantaneous Frequency, should be banished forever from the dictionary of the communication engineer.
J. Shekel, 1953
The uncertainty principle makes the concept of an Instantaneous Frequency impossible.
K. Gröchennig, 2001
The Idea and the need of Instantaneous Frequency
k , ;
t
tk
0 .
According to the classic wave theory, the wave conservation law is based on a gradually changing φ(x,t) such that
Therefore, both wave number and frequency must have instantaneous values and differentiable.
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 ) ,
wherey( t )
a( t ) x y and ( t ) tan .x( t )
a( t ) e
Hilbert Transform : Definition
The Traditional View of the Hilbert Transform for Data Analysis
Traditional Viewa la Hahn (1995) : Data LOD
Traditional Viewa la Hahn (1995) : Hilbert
Traditional Approacha la Hahn (1995) : Phase Angle
Traditional Approacha la Hahn (1995) : Phase Angle Details
Traditional Approacha la Hahn (1995) : Frequency
The Real World
Mathematics are well and good but nature keeps dragging us around by the nose.
Albert Einstein
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
(Other alternatives, e.g., Nonlinear Matching Pursuit)
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 )
Or, simply pre-determine the number of iterations.
Empirical Mode Decomposition: Methodology : IMF c1
Definition of the Intrinsic Mode Function (IMF): a necessary condition only!
Any function having the same numbers ofzero cros sin gs and extrema,and also havingsymmetric envelopes defined by local max imaand min ima respectively is defined as anIntrinsic 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 ModeFunnction, 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 .
An Example of Sifting &
Time-Frequency Analysis
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
Mean Hilbert Spectrum : All CEs
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 empirically and a posteriori:
CompleteConvergentOrthogonalUnique
The combination of Hilbert Spectral Analysis and Empirical Mode Decomposition has been
designated by NASA as
HHT
(HHT vs. FFT)
Comparison between FFT and HHT
j
jt
i tj
j
i ( )d
jj
1. FFT :
x( t ) a e .
2. HHT :
x( t ) a ( t ) e .
Comparisons: Fourier, Hilbert & Wavelet
Surrogate Signal:Non-uniqueness signal vs. Spectrum
I. Hello
The original data : Hello
The IMF of original data : Hello
The surrogate data : Hello
The IMF of Surrogate data : Hello
The Hilbert spectrum of original data : Hello
The Hilbert spectrum of Surrogate data : Hello
To quantify nonlinearity we also need instantaneous frequency.
Additionally
How to define Nonlinearity?
How to quantify it through data alone (model independent)?
The term, ‘Nonlinearity,’ has been loosely used, most of the time, simply
as a fig leaf to cover our ignorance.
Can we be more precise?
How is nonlinearity defined?Based on Linear Algebra: nonlinearity is defined based on input vs. output.
But in reality, such an approach is not practical: natural system are not clearly defined; inputs and out puts are hard to ascertain and quantify. Furthermore, without the governing equations, the order of nonlinearity is not known.
In the autonomous systems the results could depend on initial conditions rather than the magnitude of the ‘inputs.’
The small parameter criteria could be misleading: sometimes, the smaller the parameter, the more nonlinear.
Linear Systems
Linear systems satisfy the properties of superposition and scaling. Given two valid inputs to a system H,
as well as their respective outputs
then a linear system, H, must satisfy
for any scalar values α and β.
x ( t ) and x ( t )1 2
y ( t ) H{ x ( t )} and y (t) = H{ x ( t )}
1 1
2 2
y ( t ) y ( t ) H{ x ( t ) x ( t )} 1 1 1 2
How is nonlinearity defined?Based on Linear Algebra: nonlinearity is defined based on input vs. output.
But in reality, such an approach is not practical: natural system are not clearly defined; inputs and out puts are hard to ascertain and quantify. Furthermore, without the governing equations, the order of nonlinearity is not known.
In the autonomous systems the results could depend on initial conditions rather than the magnitude of the ‘inputs.’
The small parameter criteria could be misleading: sometimes, the smaller the parameter, the more nonlinear.
How should nonlinearity be defined?
The alternative is to define nonlinearity based on data characteristics: Intra-wave frequency modulation.
Intra-wave frequency modulation is known as the harmonic distortion of the wave forms. But it could be better measured through the deviation of the instantaneous frequency from the mean frequency (based on the zero crossing period).
Characteristics of Data from Nonlinear Processes
32
2
2
22
d x x cos tdt
d x x cos tdt
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 .) s1d x
x txdt
x
Duffing Equation : Data
Hilbert’s View on Nonlinear DataIntra-wave Frequency Modulation
A simple mathematical model
x( t ) cos t sin 2 t
Duffing Type WaveData: x = cos(wt+0.3 sin2wt)
Duffing Type WavePerturbation Expansion
For 1 , we can have
cos t cos sin 2 t sin t sin sin 2 t
cos t sin t sin 2 t ....
This is very similar to the solutionof D
x( t ) cos t sin 2 t
1 cos t cos 3
uffing equ
t ....2
atio
2
n .
Duffing Type WaveWavelet Spectrum
Duffing Type WaveHilbert Spectrum
Degree of nonlinearity
1/ 22
z
z
Let us consider a generalized intra-wave frequency modulation model as:
d x( t ) cos( t sin t )
IF IFDN (Degree of Nolinearity ) should be
I
cos t IF=
.2
Depending on
d
F
t
1t
he value of , we can have either a up-down symmetric or a asymmetric wave form.
Degree of Nonlinearity• DN is determined by the combination of δη precisely with
Hilbert Spectral Analysis. Either of them equals zero means linearity.
• We can determine δ and η separately:– η can be determined from the instantaneous
frequency modulations relative to the mean frequency.
– δ can be determined from DN with known η. NB: from any IMF, the value of δη cannot be greater than 1.
• The combination of δ and η gives us not only the Degree of Nonlinearity, but also some indications of the basic properties of the controlling Differential Equation, the Order of Nonlinearity.
Stokes Models
22
2
2 with ; =0.1.
25
: is positive ranging from 0.1 to 0.375; beyond 0.375, there is no solution.
: is negative ranging from 0.1 to 0.391
d xx
;
Stokes I
Stokes II
x cos t dt
beyond 0.391, there is no solution.
Data and IFs : C1
0 20 40 60 80 100-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Time: Second
Freq
uenc
y : H
z
Stokes Model c1: e=0.375; DN=0.2959
IFqIFzDataIFq-IFz
Summary Stokes I
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Epsilon
Com
bine
d D
egre
e of
Sta
tiona
rity
Summary Stokes I
C1C2CDN
Lorenz Model• Lorenz is highly nonlinear; it is the model
equation that initiated chaotic studies.
• Again it has three parameters. We decided to fix two and varying only one.
• There is no small perturbation parameter.
• We will present the results for ρ=28, the classic chaotic case.
Phase Diagram for ro=28
010
2030
4050
-20-10
010
20-30
-20
-10
0
10
20
30
x
Lorenz Phase : ro=28, sig=10, b=8/3
y
z
X-Component
DN1=0.5147
CDN=0.5027
Data and IF
0 5 10 15 20 25 30-10
0
10
20
30
40
50
Time : second
Freq
uenc
y : H
z
Lorenz X : ro=28, sig=10, b=8/3
IFqIFzDataIFq-IFz
ComparisonsFourier 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, quantifying
Non-stationary no yes Yes, quantifying
Uncertainty yes yes no
Harmonics yes yes no
How to define and determine Trend ?
Parametric or Non-parametric?Intrinsic vs. extrinsic approach?
The State-of-the arts: Trend
“One economist’s trend is another economist’s cycle”
Watson : Engle, R. F. and Granger, C. W. J. 1991 Long-run Economic Relationships. Cambridge University Press.
Philosophical Problem Anticipated
名不正則言不順言不順則事不成
——孔夫子
Definition of the TrendProceeding Royal Society of London, 1998
Proceedings of National Academy of Science, 2007
Within the given data span, the trend is an intrinsically fitted monotonic function, or a function in which there can be at most one extremum.
The trend should be an intrinsic and local property of the data; it is determined by the same mechanisms that generate the data.
Being local, it has to associate with a local length scale, and be valid only within that length span, and be part of a full wave length.
The method determining the trend should be intrinsic. Being intrinsic, the method for defining the trend has to be adaptive.
All traditional trend determination methods are extrinsic.
Algorithm for Trend
• Trend should be defined neither parametrically nor non-parametrically.
• It should be the residual obtained by removing cycles of all time scales from the data intrinsically.
• Through EMD.
Global Temperature Anomaly
Annual Data from 1856 to 2003
GSTA
IMF Mean of 10 Sifts : CC(1000, I)
Statistical Significance Test
IPCC Global Mean Temperature Trend
Comparison between non-linear rate with multi-rate of IPCC
1840 1860 1880 1900 1920 1940 1960 1980 2000 2020-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0.025Various Rates of Global Warming
Time : Year
War
min
g R
ate:
o C/Y
r
Blue shadow and blue line are the warming rate of non-linear trend.Magenta shadow and magenta line are the rate of combination of non-linear trend and AMO-like components.Dashed lines are IPCC rates.
Conclusion• With HHT, we can define the true instantaneous
frequency and extract trend from any data.
• We can quantify the degree of nonlinearity. Among the applications, the degree of nonlinearity could be used to set an objective criterion in biomedical and structural health monitoring, and to quantify the degree of nonlinearity in natural phenomena.
• We can also determine the trend, which could be used in financial as well as natural sciences.
• These are all possible because of adaptive data analysis method.
History of HHT1998: 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, 1179-1611.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.
2009: On Ensemble Empirical Mode Decomposition. Adv. Adaptive data Anal., 1, 1-41
2009: On instantaneous Frequency. Adv. Adaptive Data Anal., 2, 177-229.
2010: The Time-Dependent Intrinsic Correlation Based on the Empirical Mode Decomposition. Adv. Adaptive Data Anal., 2. 233-266.
2010: Multi-Dimensional Ensemble Empirical Mode Decomposition. Adv. Adaptive Data Anal., 3, 339-372.
2011: Degree of Nonlinearity. (Patent and Paper)2012: Data-Driven Time-Frequency Analysis (Applied and
Computational Harmonic Analysis, T. Hou and Z. Shi)
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)
Outstanding Mathematical Problems
1. Mathematic rigor on everything we do. (tighten the definitions of IMF,….)
2. Adaptive data analysis (no a priori basis methodology in general)
3. Prediction problem for nonstationary processes (end effects)
4. Optimization problem (the best stoppage criterion and the unique decomposition ….)
5. Convergence problem (finite step of sifting, best spline implement, …)
Do mathematical results make physical sense?
Do mathematical results make physical sense?
Good math, yes; Bad math ,no.
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
All these results depends on adaptive approach.
Thanks