A Plea for Adaptive Data Analysis: An Introduction to HHT for Nonlinear and Nonstationary Data - PowerPoint PPT Presentation

About This Presentation
Title:

A Plea for Adaptive Data Analysis: An Introduction to HHT for Nonlinear and Nonstationary Data

Description:

A Plea for Adaptive Data Analysis: An Introduction to HHT for Nonlinear and Nonstationary Data Norden E. Huang Research Center for Adaptive Data Analysis – PowerPoint PPT presentation

Number of Views:289
Avg rating:3.0/5.0
Slides: 57
Provided by: spaceUst
Category:

less

Transcript and Presenter's Notes

Title: A Plea for Adaptive Data Analysis: An Introduction to HHT for Nonlinear and Nonstationary Data


1
A 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

2
Data 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

3
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.

4
Data Analysis
  • There are, unfortunately, tensions between
    sciences and mathematics.
  • Data analysis is too important to be left to the
    mathematicians.
  • Why?!

5
Different Paradigms Mathematics vs.
Science/Engineering
  • Mathematicians
  • Absolute proofs
  • Logic consistency
  • Mathematical rigor
  • Scientists/Engineers
  • Agreement with observations
  • Physical meaning
  • Working Approximations

6
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.

7
Hilbert Transform Definition
8
The Traditional View of the Hilbert Transform
for Data Analysis
9
Traditional Viewa la Hahn (1995) Data LOD
10
Traditional Viewa la Hahn (1995) Hilbert
11
The Empirical Mode Decomposition Method and
Hilbert Spectral AnalysisSifting
12
Empirical Mode Decomposition Methodology Test
Data
13
Empirical Mode Decomposition Methodology data
and m1
14
Empirical Mode DecompositionSifting to get one
IMF component
15
The Stoppage Criteria
The Cauchy type criterion when SD is small than
a pre-set value, where
Or, simply pre-determine the number of iterations.
16
Empirical Mode Decomposition Methodology IMF
c1
17
Empirical Mode DecompositionSifting to get all
the IMF components
18
Definition of Instantaneous Frequency
19
The 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)?
20
The combination of Hilbert Spectral Analysis and
Empirical Mode Decomposition has been designated
by NASA as
  • HHT
  • (HHT vs. FFT)

21
Comparison between FFT and HHT
22
Comparisons Fourier, Hilbert Wavelet
23
Speech Analysis Hello Data
24
Four comparsions D
25
An Example of Sifting
26
Length Of Day Data
27
LOD IMF
28
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

29
LOD Data c12
30
LOD Data Sum c11-12
31
LOD Data sum c10-12
32
LOD Data c9 - 12
33
LOD Data c8 - 12
34
LOD Detailed Data and Sum c8-c12
35
LOD Data c7 - 12
36
LOD Detail Data and Sum IMF c7-c12
37
LOD Difference Data sum all IMFs
38
Traditional Viewa la Hahn (1995) Hilbert
39
Mean Annual Cycle Envelope 9 CEI Cases
40
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

41
Hilberts View on Nonlinear Data
42
Duffing Type WaveData x cos(wt0.3 sin2wt)
43
Duffing Type WavePerturbation Expansion
44
Duffing Type WaveWavelet Spectrum
45
Duffing Type WaveHilbert Spectrum
46
Duffing Type WaveMarginal Spectra
47
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

48
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

49
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

50
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
51
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 problems
  • And need a lot of helps!

52
National 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.

55
(No Transcript)
56
VOLUME 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.
.
Write a Comment
User Comments (0)
About PowerShow.com