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An Introduction to Hilbert-Huang Transform:A Plea for Adaptive Data Analysis

Norden E. HuangResearch Center for Adaptive Data Analysis

National Central University

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Data Processing and Data Analysis

Processing [proces < L. Processus < pp ofProcedere = Proceed: pro- forward + cedere, togo] : A particular method of doing something.

Analysis [Gr. ana, up, throughout + lysis, aloosing] : A separating of any whole into its parts,especially with an examination of the parts tofind out their nature, proportion, function,interrelationship etc.

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Data Analysis

Why we do it?

How did we do it?

What should we do?

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Why?

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Why 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.

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Ever since the advance of computer andsensor 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.

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Henri Poincar

Science is built up of facts*,as a house is built of stones;

but an accumulation of facts is no more a sciencethan a heap of stones is a house.

* Here facts are indeed our data.

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Data and Data Analysis

Data Analysis is the key step in convertingthe 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.

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Science 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.

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Scientific Activities

Collecting, analyzing, synthesizing, andtheorizing are the core of scientific activities.

Theory without data to prove is just hypothesis.

Therefore, data analysis is a key link in thiscontinuous loop.

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Data Analysis

Data analysis is too important to be left tothe mathematicians.

Why?!

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Different Paradigms IMathematics vs. Science/Engineering

Mathematicians

Absolute proofs

Logic consistency

Mathematical rigor

Scientists/Engineers

Agreement with observations

Physical meaning

Working Approximations

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Different Paradigms IIMathematics vs. Science/Engineering

Mathematicians

Idealized Spaces

Perfect world in which

everything is known

Inconsistency in the differentspaces and the real world

Scientists/Engineers

Real Space

Real world in which knowledge is

incomplete and limited

Constancy in the real world withinallowable approximation

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Rigor vs. Reality

As far as the laws of mathematics refer toreality, they are not certain; and as far asthey are certain, they do not refer to reality.

Albert Einstein

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How?

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Data Processing vs. Analysis

All traditional data analysis methods are really for dataprocessing. They are either developed by or establishedaccording to mathematicians rigorous rules. Most of the

methods consist of standard algorithms, which produce aset of simple parameters.

They can only be qualified as data processing, not reallydata analysis.

Data processing produces mathematical meaningfulparameters; data analysis reveals physical characteristicsof the underlying processes.

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Data Processing vs. Analysis

In pursue of mathematic rigor and certainty,however, we lost sight of physics and are forcedto idealize, but also deviate from, the reality.

As a result, we are forced to live in a pseudo-realworld, in which all processes are

Linear and Stationary

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Trimming the foot to fit the shoe.

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Available 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

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Available Data Analysis Methodsfor Nonlinear (but Stationary and Deterministic)

time series

Phase space method Delay reconstruction and embedding

Poincar surface of section Self-similarity, attractor geometry & fractals

Nonlinear Prediction

Lyapunov Exponents for stability

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Typical Apologia

Assuming the process is stationary .

Assuming the process is locally stationary .

As the nonlinearity is weak, we can use perturbationapproach .

Though we can assume all we want, but

the reality cannot be bent by the assumptions.

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The Real World

Mathematics are well and good but naturekeeps dragging us around by the nose.

Albert Einstein

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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 notbe meaningful for lack of stationarity and ergodicity.

Traditional methods are inadequate.

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What?

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The job of a scientist is to listen carefully tonature, 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.

The Job of a Scientist

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How to define nonlinearity?

Based on Linear Algebra: nonlinearity isdefined based on input vs. output.

But in reality, such an approach is notpractical. The alternative is to definenonlinearity based on data characteristics.

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Characteristics of Data fromNonlinear Processes

32

2

2

2

2

d x x cos t

dt

d x x cos t

dt

Spring with position dependent cons tan t ,int ra wave frequency mod ulation;

therefore , we need ins tan

x

1

tan eous frequenc

x

y .

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Duffing Pendulum

2

2

2( co .) s1

d xx tx

dt

x

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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

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Hilbert Transform Fit

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Conformation to reality rather then toMathematics

We do not have to apologize, we should usemethods that can analyze data generated by

nonlinear and nonstationary processes.

That means we have to deal with the intrawave

frequency modulations, intermittencies, andfinite rate of irregular drifts. Any methodsatisfies this call will have to be adaptive.

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The Traditional Approach ofHilbert Transform for Data Analysis

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Traditional Approacha la Hahn (1995) : Data LOD

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Traditional Approacha la Hahn (1995) : Hilbert

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Traditional Approacha la Hahn (1995) : Phase Angle

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Traditional Approacha la Hahn (1995) : Phase Angle Details

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Traditional Approacha la Hahn (1995) : Frequency

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Why the traditional approachdoes not work?

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Hilbert Transform a cos+ b:Data

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Hilbert Transform a cos+ b:Phase Diagram

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Hilbert Transform a cos+ b:Phase Angle Details

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Hilbert Transform a cos+ b:Frequency

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The Empirical Mode Decomposition

Method and Hilbert Spectral Analysis

Sifting

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Empirical Mode Decomposition:Methodology : Test Data

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Empirical Mode Decomposition:Methodology : data and m1

E i i l M d D iti

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Empirical Mode Decomposition:Methodology : data & h1

Empirical Mode Decomposition:

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Empirical Mode Decomposition:Methodology : h1 & m2


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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

.

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Two Stoppage Criteria : S and SD

A. 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

T 2

k 1 k

t 0

T2

k 1

t 0

h ( t ) h ( t )

SD

h ( t )


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Empirical Mode Decomposition:Methodology : IMF c1

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Definition ofthe Intrinsic Mode Function (IMF)

Any functionhaving the same numbersof

zero cros sin gsand extrema,and alsohaving

symmetric envelopesdefined by local max ima

and min ima respectively isdefined as an

Intrinsic Mode Function( IMF ).

All IMF enjoys good Hilbert Transfo

i ( t )

rm :

c( t ) a( t )e

E i i l M d D i i

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Empirical Mode DecompositionSifting : to get all the IMF components

1 1

1 2 2

n 1 n n

n

j n

j 1

x( t ) c r ,

r c r ,

x( t ) c r

. . .

r c r .

.


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Empirical Mode Decomposition:Methodology : data & r1

Em i i l M d D m iti

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Empirical Mode Decomposition:Methodology : data and m1


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Empirical Mode Decomposition:Methodology : data, r1 and m1


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Empirical Mode Decomposition:Methodology : IMFs

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Definition of Instantaneous Frequency

i ( t )

t

The Fourier Transform of the Instrinsic Mode

Funnction, c( t ), gives

W ( ) a( t ) e dt

By Stationary phase approximation we have

d ( t ) ,dt

This is defined as the Ins tan tan eous Frequency .

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Definition of Frequency

Given the period of a wave as T; the frequency is

defined as

1.

T

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Equivalence :

The definition of frequency is equivalent todefining velocity as

Velocity = Distance / Time

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Instantaneous Frequency

distanceVelocity ; mean velocity

time

dx

Newton v dt

1 Frequency ; mean frequency

period

dHH

So that both v and

T defines the p

can appear in differential equations.

hase functiondt

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The combination of Hilbert Spectral Analysis and

Empirical Mode Decomposition is designated as

HHT(HHT vs. FFT)

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Jean-Baptiste-Joseph Fourier

1807 On the Propagation of Heat in Solid Bodies

1812 Grand Prize of Paris Institute

Thorie analytique de la chaleur

... the manner in which the author arrives atthese equations is not exempt of difficulties andthat his analysis to integrate them still leavessomething to be desired on the score of generality

and even rigor.

1817 Elected to Acadmie des Sciences

1822 Appointed as Secretary of Math Section

paper published

Fouriers work is a great mathematical poem.

Lord Kelvin

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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 .

C i

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Comparisons:Fourier, Hilbert & Wavelet

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An Example of Sifting

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Length Of Day Data

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LOD : IMF

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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

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LOD : Data & c12

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LOD : Data & Sum c11-12

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LOD : Data & sum c10-12

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LOD : Data & c9 - 12

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LOD : Detailed Data and Sum c8-c12

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LOD : Detail Data and Sum IMF c7-c12

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LOD : Difference Data sum all IMFs

T di i l Vi

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Traditional Viewa la Hahn (1995) : Hilbert

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Mean Annual Cycle & Envelope: 9 CEICases

Mean Hilbert Spectrum : All CEs

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Mean Hilbert Spectrum : All CEs

Tidal Machine

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P ti f EMD B i

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Properties of EMD Basis

The Adaptive Basis based on and derived fromthe data by the empirical method satisfy nearlyall the traditional requirements for basis

a posteriori:CompleteConvergent

OrthogonalUnique

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Hilberts View onNonlinear Data

Duffing Type Wave

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Duffing Type WaveData:x = cos(wt+0.3 sin2wt)

D ffi T W

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Duffing Type WavePerturbation Expansion

For 1 , we can have

x( t ) cos t sin 2 t

cos t cos sin 2 t sin t sin sin 2 t

cos t sin t sin 2 t ....

1 cos t cos 3 t ....2 2

This is very similar to the solution of Duffing equation .

D ffi T W

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Duffing Type WaveWavelet Spectrum

D ffi T W

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Duffing Type WaveHilbert Spectrum

D ffi g T W

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Duffing Type WaveMarginal Spectra

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Duffing Equation

23

2.

Solved with for t 0 to 200 with

1

0.1

od

0.04 Hz

Initial condition :

[ x( o ) ,

d x x x c

x'( 0 ) ] [ 1

os t

, 1 ]

3

t

e2

d

tb

Duffing Equation : Data

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Duffing Equation : Data

Duffing Equation : IMFs

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Duffing Equation : IMFs

Duffing Equation : Hilbert Spectrum

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Duffing Equation : Hilbert Spectrum

Duffing Equation : Detailed Hilbert Spectrum

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Duffing Equation : DetailedHilbert Spectrum

Duffing Equation : Wavelet Spectrum

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u g quat o a e et Spect u

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Duffing Equation : Hilbert & Wavelet Spectra

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Speech Analysis

Nonlinear and nonstationary data

Speech Analysis

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Speech AnalysisHello : Data

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Four comparsions D

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Global Temperature Anomaly

Annual Data from 1856 to 2003

Global Temperature Anomaly 1856 to 2003

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Global Temperature Anomaly 1856 to 2003

IMF Mean of 10 Sifts : CC(1000, I)

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Statistical Significance Test

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Data and Trend C6

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Rate of Change Overall Trends : EMD and Linear

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What This Means

Instantaneous Frequency offers a total differentview for nonlinear data: instantaneousfrequency with no need for harmonics andunlimited by uncertainty.

Adaptive basis is indispensable fornonstationary and nonlinear data analysis

HHT establishes a new paradigm of dataanalysis

Comparisons

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Comparisons

Fourier Wavelet Hilbert

Basis a priori a priori Adaptive

Frequency Convolution:Global

Convolution:Regional

Differentiation:

LocalPresentation Energy-frequency Energy-time-

frequencyEnergy-time-frequency

Nonlinear no no yes

Non-stationary no yes yesUncertainty yes yes no

Harmonics yes yes no

Conclusion

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Conclusion

Adaptive method is the only scientificallymeaningful way to analyze data.

It is the only way to find out the underlyingphysical processes; therefore, it isindispensable in scientific research.

It is physical, direct, and simple.

History of HHT

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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 determiningthe 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 Hilbertspectral 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 ModeDecomposition 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 forinstantaneous Frequency computations.

Current Applications

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Current 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)

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Advances in Adaptive data Analysis:

Theory and ApplicationsA new journal to be published by

the World Scientific

Under the joint Co-Editor-in-Chief

Norden E. Huang, RCADA NCUThomas Yizhao Hou, CALTECH

in the January 2008

Oliver Heaviside

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Oliver Heaviside1850 - 1925

Why should I refuse a good dinnersimply because I don't understandthe digestive processes involved.

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