Title: A Plea for Adaptive Data Analysis: An Introduction to HHT for Nonlinear and Nonstationary Data
1A Plea for Adaptive Data AnalysisAn
Introduction to HHT for Nonlinear and
Nonstationary Data
- Norden E. Huang
- Research Center for Adaptive Data Analysis
- National Central University
- Nanjing
- October 2009
2Data Processing and Data Analysis
- Processing proces lt L. Processus lt pp of
Procedere Proceed pro- forward cedere, to
go A particular method of doing something. - Data Processing gtgtgtgt 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 gtgtgtgt Physical understandings
3Scientific 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.
4Data Analysis
- There are, unfortunately, tensions between
sciences and mathematics. - Data analysis is too important to be left to the
mathematicians. - Why?!
5Different Paradigms Mathematics vs.
Science/Engineering
- Mathematicians
- Absolute proofs
- Logic consistency
- Mathematical rigor
- Scientists/Engineers
- Agreement with observations
- Physical meaning
- Working Approximations
6Motivations 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.
7Hilbert Transform Definition
8The Traditional View of the Hilbert Transform
for Data Analysis
9Traditional Viewa la Hahn (1995) Data LOD
10Traditional Viewa la Hahn (1995) Hilbert
11The Empirical Mode Decomposition Method and
Hilbert Spectral AnalysisSifting
12Empirical Mode Decomposition Methodology Test
Data
13Empirical Mode Decomposition Methodology data
and m1
14Empirical Mode DecompositionSifting to get one
IMF component
15The Stoppage Criteria
The Cauchy type criterion when SD is small than
a pre-set value, where
Or, simply pre-determine the number of iterations.
16Empirical Mode Decomposition Methodology IMF
c1
17Empirical Mode DecompositionSifting to get all
the IMF components
18Definition of Instantaneous Frequency
19The Idea and the need of Instantaneous Frequency
According to the classic wave theory, the wave
conservation law is based on a gradually changing
f(x,t) such that
Therefore, both wave number and frequency must
have instantaneous values. But how to find f(x,
t)?
20The combination of Hilbert Spectral Analysis and
Empirical Mode Decomposition has been designated
by NASA as
21Comparison between FFT and HHT
22Comparisons Fourier, Hilbert Wavelet
23Speech Analysis Hello Data
24Four comparsions D
25An Example of Sifting
26Length Of Day Data
27LOD IMF
28Orthogonality 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
29LOD Data c12
30LOD Data Sum c11-12
31LOD Data sum c10-12
32LOD Data c9 - 12
33LOD Data c8 - 12
34LOD Detailed Data and Sum c8-c12
35LOD Data c7 - 12
36LOD Detail Data and Sum IMF c7-c12
37LOD Difference Data sum all IMFs
38Traditional Viewa la Hahn (1995) Hilbert
39Mean Annual Cycle Envelope 9 CEI Cases
40Properties 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
41Hilberts View on Nonlinear Data
42Duffing Type WaveData x cos(wt0.3 sin2wt)
43Duffing Type WavePerturbation Expansion
44Duffing Type WaveWavelet Spectrum
45Duffing Type WaveHilbert Spectrum
46Duffing Type WaveMarginal Spectra
47Ensemble 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
48New 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
49What 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
50Comparisons
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
51Conclusion
- 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 problems
- And need a lot of helps!
52National Central University
- Research Center for Adaptive Data Analysis
53- 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)
54- 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.
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56VOLUME I TECHNICAL PROPOSAL AND MANAGEMENT
APPROACH Mathematical 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.
.