properties of emd basis the adaptive basis based on and derived from the data by the empirical...
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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 empirically and a posteriori:
Complete
Convergent
Orthogonal
Unique
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
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
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 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
Hilbert’s View on Nonlinear DataIntra-wave Frequency Modulation
A simple mathematical model
x( t ) cos t sin 2 t
Duffing Type Wave
Data: 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
Fre
qu
en
cy
: 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
Co
mb
ined
Deg
ree
of
Sta
tio
nar
ity
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
0
10
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
Fre
qu
en
cy
: H
z
Lorenz X : ro=28, sig=10, b=8/3
IFqIFzDataIFq-IFz
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, 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:
oC
/Yr
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 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, 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
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