Title: An Introduction to HilbertHuang Transform: A Plea for Adaptive Data Analysis Norden E' Huang Researc
1An Introduction to Hilbert-Huang TransformA
Plea for Adaptive Data AnalysisNorden E.
HuangResearch Center for Adaptive Data
AnalysisNational Central University
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. - 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.
3Data Analysis
- Why we do it?
- How did we do it?
- What should we do?
4Why?
5Why do we have to analyze data?
- Data are the only connects we have with the
reality - data analysis is the only means we can find the
truth and deepen our understanding of the
problems.
6Ever since the advance of computer and sensor
technology, there is an explosion of very
complicate data. The situation has changed
from a thirsty for data to that of drinking from
a fire hydrant.
7Henri Poincaré
-
- Science is built up of facts,
- as a house is built of stones
- but an accumulation of facts is no more a science
- than a heap of stones is a house.
- Here facts are indeed our data.
8Data and Data Analysis
- Data Analysis is the key step in converting the
facts into the edifice of science. -
- It infuses meanings to the cold numbers, and lets
data telling their own stories and singing their
own songs.
9Science vs. Philosophy
- Data and Data Analysis are what separate science
from philosophy - With data we are talking about sciences
- Without data we can only discuss philosophy.
10Scientific Activities
- Collecting, analyzing, synthesizing, and
theorizing are the core of scientific activities. -
- Theory without data to prove is just hypothesis.
-
- Therefore, data analysis is a key link in this
continuous loop.
11Data Analysis
-
- Data analysis is too important to be left to the
mathematicians. - Why?!
12Different Paradigms IMathematics vs.
Science/Engineering
- Mathematicians
- Absolute proofs
- Logic consistency
- Mathematical rigor
- Scientists/Engineers
- Agreement with observations
- Physical meaning
- Working Approximations
13Different Paradigms IIMathematics vs.
Science/Engineering
- Mathematicians
- Idealized Spaces
- Perfect world in which everything is known
- Inconsistency in the different spaces and the
real world
- Scientists/Engineers
- Real Space
- Real world in which knowledge is incomplete and
limited - Constancy in the real world within allowable
approximation
14Rigor vs. Reality
- As far as the laws of mathematics refer to
reality, they are not certain and as far as they
are certain, they do not refer to reality. - Albert Einstein
-
15How?
16Data Processing vs. Analysis
- All traditional data analysis methods are
really for data processing. They are either
developed by or established according to
mathematicians rigorous rules. Most of the
methods consist of standard algorithms, which
produce a set of simple parameters. - They can only be qualified as data processing,
not really data analysis. - Data processing produces mathematical meaningful
parameters data analysis reveals physical
characteristics of the underlying processes.
17Data Processing vs. Analysis
- In pursue of mathematic rigor and certainty,
however, we lost sight of physics and are forced
to idealize, but also deviate from, the reality. - As a result, we are forced to live in a
pseudo-real world, in which all processes are - Linear and Stationary
18????
- Trimming the foot to fit the shoe.
19Available Data Analysis Methodsfor Nonstationary
(but Linear) time series
- Spectrogram
- Wavelet Analysis
- Wigner-Ville Distributions
- Empirical Orthogonal Functions aka Singular
Spectral Analysis - Moving means
- Successive differentiations
20Available Data Analysis Methodsfor Nonlinear
(but Stationary and Deterministic) time series
- Phase space method
- Delay reconstruction and embedding
- Poincaré surface of section
- Self-similarity, attractor geometry fractals
- Nonlinear Prediction
- Lyapunov Exponents for stability
21Typical Apologia
- Assuming the process is stationary .
- Assuming the process is locally stationary .
- As the nonlinearity is weak, we can use
perturbation approach . - Though we can assume all we want, but
- the reality cannot be bent by the assumptions.
22The Real World
-
- Mathematics are well and good but nature keeps
dragging us around by the nose. - Albert Einstein
23Motivations 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.
24What?
25The Job of a Scientist
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 methods and let the data sing, and not
to force the data to fit preconceived modes.
26How to define nonlinearity?
- Based on Linear Algebra nonlinearity is defined
based on input vs. output. - But in reality, such an approach is not
practical. The alternative is to define
nonlinearity based on data characteristics.
27Characteristics of Data from Nonlinear Processes
28Duffing Pendulum
x
29Hilbert Transform Definition
30Hilbert Transform Fit
31Conformation to reality rather then to Mathematics
- We do not have to apologize, we should use
methods that can analyze data generated by
nonlinear and nonstationary processes. -
- That means we have to deal with the intrawave
frequency modulations, intermittencies, and
finite rate of irregular drifts. Any method
satisfies this call will have to be adaptive.
32The Traditional Approach of Hilbert Transform
for Data Analysis
33Traditional Approacha la Hahn (1995) Data LOD
34Traditional Approacha la Hahn (1995) Hilbert
35Traditional Approacha la Hahn (1995) Phase
Angle
36Traditional Approacha la Hahn (1995) Phase
Angle Details
37Traditional Approacha la Hahn (1995)
Frequency
38Why the traditional approach does not work?
39Hilbert Transform a cos ? b Data
40Hilbert Transform a cos ? b Phase Diagram
41Hilbert Transform a cos ? b Phase Angle
Details
42Hilbert Transform a cos ? b Frequency
43The Empirical Mode Decomposition Method and
Hilbert Spectral AnalysisSifting
44Empirical Mode Decomposition Methodology Test
Data
45Empirical Mode Decomposition Methodology data
and m1
46Empirical Mode Decomposition Methodology data
h1
47Empirical Mode Decomposition Methodology h1
m2
48Empirical Mode Decomposition Methodology h3
m4
49Empirical Mode Decomposition Methodology h4
m5
50Empirical Mode DecompositionSifting to get one
IMF component
51Two Stoppage Criteria S and SD
- The S number S is defined as the consecutive
number of siftings, in which the numbers of
zero-crossing and extrema are the same for these
S siftings. - B. SD is small than a pre-set value, where
52Empirical Mode Decomposition Methodology IMF
c1
53Definition of the Intrinsic Mode Function (IMF)
54Empirical Mode DecompositionSifting to get all
the IMF components
55Empirical Mode Decomposition Methodology data
r1
56Empirical Mode Decomposition Methodology data
and m1
57Empirical Mode Decomposition Methodology
data, r1 and m1
58Empirical Mode Decomposition Methodology IMFs
59Definition of Instantaneous Frequency
60Definition of Frequency
Given the period of a wave as T the frequency
is defined as
61Equivalence
- The definition of frequency is equivalent to
defining velocity as - Velocity Distance / Time
62Instantaneous Frequency
63The combination of Hilbert Spectral Analysis and
Empirical Mode Decomposition is designated as
64Jean-Baptiste-Joseph Fourier
- On the Propagation of Heat in Solid Bodies
1812 Grand Prize of Paris Institute
Théorie analytique de la chaleur ... the
manner in which the author arrives at these
equations is not exempt of difficulties and that
his analysis to integrate them still leaves
something to be desired on the score of
generality and even rigor.
- Elected to Académie des Sciences
- Appointed as Secretary of Math Section
- paper published
Fouriers work is a great mathematical
poem. Lord Kelvin
65Comparison between FFT and HHT
66Comparisons Fourier, Hilbert Wavelet
67An Example of Sifting
68Length Of Day Data
69LOD IMF
70Orthogonality 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
71LOD Data c12
72LOD Data Sum c11-12
73LOD Data sum c10-12
74LOD Data c9 - 12
75LOD Data c8 - 12
76LOD Detailed Data and Sum c8-c12
77LOD Data c7 - 12
78LOD Detail Data and Sum IMF c7-c12
79LOD Difference Data sum all IMFs
80Traditional Viewa la Hahn (1995) Hilbert
81Mean Annual Cycle Envelope 9 CEI Cases
82Mean Hilbert Spectrum All CEs
83Tidal Machine
84Properties 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 - a posteriori
- Complete
- Convergent
- Orthogonal
- Unique
85Hilberts View on Nonlinear Data
86Duffing Type WaveData x cos(wt0.3 sin2wt)
87Duffing Type WavePerturbation Expansion
88Duffing Type WaveWavelet Spectrum
89Duffing Type WaveHilbert Spectrum
90Duffing Type WaveMarginal Spectra
91Duffing Equation
92Duffing Equation Data
93Duffing Equation IMFs
94Duffing Equation Hilbert Spectrum
95Duffing Equation Detailed Hilbert Spectrum
96Duffing Equation Wavelet Spectrum
97Duffing Equation Hilbert Wavelet Spectra
98Speech Analysis
- Nonlinear and nonstationary data
99Speech Analysis Hello Data
100Four comparsions D
101Global Temperature Anomaly
- Annual Data from 1856 to 2003
102Global Temperature Anomaly 1856 to 2003
103IMF Mean of 10 Sifts CC(1000, I)
104Statistical Significance Test
105Data and Trend C6
106Rate of Change Overall Trends EMD and Linear
107What This Means
- Instantaneous Frequency offers a total different
view for nonlinear data instantaneous frequency
with no need for harmonics and unlimited by
uncertainty. - Adaptive basis is indispensable for nonstationary
and nonlinear data analysis - HHT establishes a new paradigm of data analysis
108Comparisons
109Conclusion
- Adaptive method is the only scientifically
meaningful way to analyze data. - It is the only way to find out the underlying
physical processes therefore, it is
indispensable in scientific research. - It is physical, direct, and simple.
110- 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, (in press) - Defined statistical significance and
predictability. - 2004 On the Instantaneous Frequency, Proc. Roy.
Soc. London, (Under review) - Removal of the limitations posted by Bedrosian
and Nuttall theorems for instantaneous Frequency
computations.
111Current Applications
- Non-destructive Evaluation for Structural Health
Monitoring - (DOT, NSWC, and DFRC/NASA, KSC/NASA Shuttle)
- Vibration, speech, and acoustic signal analyses
- (FBI, MIT, and DARPA)
- Earthquake Engineering
- (DOT)
- Bio-medical applications
- (Harvard, UCSD, Johns Hopkins)
- Global Primary Productivity Evolution map from
LandSat data - (NASA Goddard, NOAA)
- Cosmological Gravity Wave
- (NASA Goddard)
- Financial market data analysis
- (NCU)
112Advances in Adaptive data Analysis Theory and
Applications
- A new journal to be published by
- the World Scientific
- Under the joint Co-Editor-in-Chief
- Norden E. Huang, RCADA NCU
- Thomas Yizhao Hou, CALTECH
- in the January 2008
113Oliver Heaviside1850 - 1925
Why should I refuse a good dinner simply because
I don't understand the digestive processes
involved.