A Plea for Adaptive Data Analysis: An Introduction to HHT for Nonlinear and Nonstationary Data...
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Transcript of A Plea for Adaptive Data Analysis: An Introduction to HHT for Nonlinear and Nonstationary Data...
A Plea for Adaptive Data Analysis:An Introduction to HHT for Nonlinear and
Nonstationary Data
Norden E. HuangResearch Center for Adaptive Data Analysis
National Central University
NanjingOctober 2009
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
Scientific Activities
Collecting and analyzing data, synthesizing and theorizing the analyzed results are the core of scientific activities.
Therefore, data analysis is a key link in this continuous loop.
Data Analysis
There are, unfortunately, tensions between sciences and mathematics.
Data analysis is too important to be left to the mathematicians.
Why?!
Different Paradigms Mathematics vs. Science/Engineering
• Mathematicians
• Absolute proofs
• Logic consistency
• Mathematical rigor
• Scientists/Engineers
• Agreement with observations
• Physical meaning
• Working Approximations
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.
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
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 DecompositionSifting : to get all the IMF components
1 1
1
n
n
jj
2 2
n
1
n
n
1
x( t ) c r ,
r c r ,
r
r
. . .
r c .
c .x( t )
Definition of Instantaneous Frequency
i ( t )
t
The Fourier Transform of the Instrinsic Mode
Funnction, c( t ), gives
W ( ) a( t ) e dt
By Sta
d ( t ),
d
tionary phase approximation we have
This is defined as the Ins tan taneous Frequency .
t
The Idea and the need of Instantaneous Frequency
k , ;t
k0 .
t
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. But how to find φ(x, t)?
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 .
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 empirically and a posteriori:
Complete
Convergent
Orthogonal
Unique
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 .
Ensemble EMDNoise Assisted Signal Analysis (nasa)
Utilizing the uniformly distributed reference frame based on the white noise to eliminate the mode mixing
Enable EMD to apply to function with spiky or flat portion
The true result of EMD is the ensemble of infinite trials.
Wu and Huang, Adv. Adapt. Data Ana., 2009
New Multi-dimensional EEMD
• Extrema defined easily• Computationally inexpensive, relatively• Ensemble approach removed the Mode
Mixing• Edge effects easier to fix in each 1D slice• Results are 2-directional
Wu, Huang and Chen, AADA, 2009
What This Means• EMD separates scales in physical space; it generates
an extremely sparse representation for any given data.
• Added noises help to make the decomposition more robust with uniform scale separations.
• Instantaneous Frequency offers a total different view for nonlinear data: instantaneous frequency needs no harmonics and is unlimited by uncertainty principle.
• Adaptive basis is indispensable for nonstationary and nonlinear data analysis
• EMD 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 nonlinear and nonstationary data.
It is the only way to find out the underlying physical processes; therefore, it is indispensable in scientific research.
EMD is adaptive; It is physical, direct, and simple.
But, we have a lot of problemsAnd need a lot of helps!
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.
1999: A New View of Nonlinear Water Waves – The Hilbert Spectrum, Ann. Rev. Fluid Mech. 31, 417-457.
2003: A confidence Limit for the Empirical mode decomposition and the Hilbert spectral analysis, Proc. of Roy. Soc. London, A459, 2317-2345.
2004: A Study of the Characteristics of White Noise Using the Empirical Mode Decomposition Method, Proc. Roy. Soc. London, (in press)
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. Advances in Adaptive Data Analysis. Advances in Adaptive data Analysis, 1, 1-41
2009: On instantaneous Frequency. Advances in Adaptive Data Analysis 1, 177-229.
2009: Multi-Dimensional Ensemble Empirical Mode Decomposition. Advances in Adaptive Data Analysis, 1, 339-372.
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. …….