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

Norden E. HuangResearch Center for Adaptive Data Analysis

National Central University

Data Processing and Data Analysis

• Processing [proces < L. Processus < 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.

Data Analysis

• Why we do it?

• How did we do it?

• What should we do?

Why?

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.

Ever 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.

Henri 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.

Data 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.

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.

Scientific 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.

Data Analysis

Data analysis is too important to be left to the mathematicians.

Why?!

Different Paradigms IMathematics vs. Science/Engineering

• Mathematicians

• Absolute proofs

• Logic consistency

• Mathematical rigor

• Scientists/Engineers

• Agreement with observations

• Physical meaning

• Working Approximations

Different 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

Rigor 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

How?

Data Processing vs. Analysis

All traditional ‘data analysis’ methods are really for ‘data processing’. They are either developed by or established according to mathematician’s 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.

Data 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

削足適履

Trimming the foot to fit the shoe.

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

Available 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

Typical 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.

The Real World

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

Albert Einstein

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.

What?

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.

The Job of a Scientist

How 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.

Characteristics of Data from Nonlinear Processes

32

2

2

22

d xx cos t

dt

d xx cos t

dt

Spring with positiondependent cons tan t ,

int ra wave frequency mod ulation;

therefore ,we need ins tan

x

1

taneous frequenc

x

y.

Duffing Pendulum

2

22( co .) s1

d xx tx

dt

x

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

Hilbert Transform Fit

Conformation 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.

The Traditional Approach of Hilbert Transform for Data Analysis

Traditional Approacha la Hahn (1995) : Data LOD

Traditional Approacha la Hahn (1995) : Hilbert

Traditional Approacha la Hahn (1995) : Phase Angle

Traditional Approacha la Hahn (1995) : Phase Angle Details

Traditional Approacha la Hahn (1995) : Frequency

Why the traditional approach does not work?

Hilbert Transform a cos + b : Data

Hilbert Transform a cos + b : Phase Diagram

Hilbert Transform a cos + b : Phase Angle Details

Hilbert Transform a cos + b : Frequency

The Empirical Mode Decomposition Method and Hilbert Spectral Analysis

Sifting

Empirical Mode Decomposition: Methodology : Test Data

Empirical Mode Decomposition: Methodology : data and m1

Empirical Mode Decomposition: Methodology : data & h1

Empirical Mode Decomposition: Methodology : h1 & m2

Empirical Mode Decomposition: Methodology : h3 & m4

Empirical Mode Decomposition: Methodology : h4 & m5

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

.

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, whereT

2

k 1 kt 0

T2

k 1t 0

h ( t ) h ( t )SD

h ( t )

Empirical Mode Decomposition: Methodology : IMF c1

Definition of the Intrinsic Mode Function (IMF)

Any function having the same numbers of

zero cros sin gs and extrema,and also having

symmetric envelopes defined by local max ima

and min ima respectively is defined as an

Intrinsic Mode Function( IMF ).

All IMF enjoys good Hilbert Transfo

i ( t )

rm :

c( t ) a( t )e

Empirical Mode DecompositionSifting : to get all the IMF components

1 1

1 2 2

n 1 n n

n

j nj 1

x( t ) c r ,

r c r ,

x( t ) c r

. . .

r c r .

.

Empirical Mode Decomposition: Methodology : data & r1

Empirical Mode Decomposition: Methodology : data and m1

Empirical Mode Decomposition: Methodology : data, r1 and m1

Empirical Mode Decomposition: Methodology : IMFs

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 taneous Frequency .

Definition of Frequency

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

1.

T

Equivalence :

The definition of frequency is equivalent to defining velocity as

Velocity = Distance / Time

Instantaneous Frequency

distanceVelocity ; mean velocity

time

dxNewton v

dt

1Frequency ; mean frequency

period

dHH

So that both v and

T defines the p

can appear in differential equations.

hase functiondt

The combination of Hilbert Spectral Analysis and

Empirical Mode Decomposition is designated as

HHT

(HHT vs. FFT)

Jean-Baptiste-Joseph Fourier

1807 “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.’

1817 Elected to Académie des Sciences

1822 Appointed as Secretary of Math Section

paper published

Fourier’s work is a great mathematical poem. Lord Kelvin

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 .

Comparisons: Fourier, Hilbert & Wavelet

An Example of Sifting

Length Of Day Data

LOD : IMF

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

LOD : Data & c12

LOD : Data & Sum c11-12

LOD : Data & sum c10-12

LOD : Data & c9 - 12

LOD : Data & c8 - 12

LOD : Detailed Data and Sum c8-c12

LOD : Data & c7 - 12

LOD : Detail Data and Sum IMF c7-c12

LOD : Difference Data – sum all IMFs

Traditional Viewa la Hahn (1995) : Hilbert

Mean Annual Cycle & Envelope: 9 CEI Cases

Mean Hilbert Spectrum : All CEs

Tidal Machine

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

a posteriori:

Complete

Convergent

Orthogonal

Unique

Hilbert’s View on Nonlinear Data

Duffing Type Wave

Data: x = cos(wt+0.3 sin2wt)

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 solutionof Duffing equation .

Duffing Type WaveWavelet Spectrum

Duffing Type WaveHilbert Spectrum

Duffing Type WaveMarginal Spectra

Duffing Equation

23

2.

Solved with for t 0 to 200 with

1

0.1

od

0.04 Hz

Initial condition :

[ x( o ) ,

d xx x c

x'( 0 ) ] [1

os t

, 1]

3

t

e2

d

tb

Duffing Equation : Data

Duffing Equation : IMFs

Duffing Equation : Hilbert Spectrum

Duffing Equation : Detailed Hilbert Spectrum

Duffing Equation : Wavelet Spectrum

Duffing Equation : Hilbert & Wavelet Spectra

Speech Analysis

Nonlinear and nonstationary data

Speech Analysis Hello : Data

Four comparsions D

Global Temperature Anomaly

Annual Data from 1856 to 2003

Global Temperature Anomaly 1856 to 2003

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

Statistical Significance Test

Data and Trend C6

Rate of Change Overall Trends : EMD and Linear

What 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

Comparisons

Fourier Wavelet Hilbert

Basis a priori a priori Adaptive

Frequency Convolution: Global Convolution: 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 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.

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.

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)

Advances 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 NCUThomas Yizhao Hou, CALTECH

in the January 2008

Oliver Heaviside1850 - 1925

Why should I refuse a good dinner simply because I don't understand the digestive processes involved.