An Introduction to HHT

for Nonlinear and Nonstationary Time Series Analysis:

A Plea for Adaptive Data Analysis

Norden E. HuangResearch Center for Adaptive Data Analysis

National Central University

What is data?

• Data (plural of Datum) [Latin: data – what is given] Information; facts, evidence, records, statistics, etc. from which conclusions can be formed.

Information in a form suitable for storing and processing by a computer.

Data

• “In God we trust”; everyone else has to show data.

•“我們相信上帝” ; 其他人都得有數據 .

The 21st century is a century dominated by data of all forms.

Data, data everywhere!

GoogleTrend© : Happy

GoogleTrend© : Love

love  

GoogleTrend© : Crisis

crisis  1.00  

crisis  

Ever since the advance of computer, there is an explosion of data.

The situation has changed from a thirsty for data to that of drinking from a fire hydrant.

We are drowning in data,

but thirsty for knowledge!

Data Processing and Data Analysis

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

• Data ProcessingData Processing >>>> 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 AnalysisData Analysis >>>> Physical understandings

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

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 Methods for 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.

掩耳盜鈴

Stealing the bell with muffed ears

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

Duffing Equation : Data

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

Speech Analysis Hello : Data

Four comparsions D

Fourier analysis is incapable of representing any variation in temporal.

It is not even capable to separate noise from delta functions!

Noise and Delta Functions

Movies

An Example of Sifting &

Time-Frequency Analysis

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 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

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.

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.The correct adaptive trend determination method

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 (in press)

2009: Multi-Dimensional Ensemble Empirical Mode Decomposition. Advances in Adaptive Data Analysis (Accepted; Patent Pending)

Current Efforts and Applications

• Non-destructive Evaluation for Structural Health Monitoring – (DOT, NSWC, DFRC/NASA, KSC/NASA Shuttle, THSR)

• Vibration, speech, and acoustic signal analyses– (FBI, and DARPA)

• Earthquake Engineering– (DOT)

• Bio-medical applications– (Harvard, Johns Hopkins, UCSD, NIH, NTU, VHT, AS)

• Climate changes– (NASA Goddard, NOAA, CCSP)

• Cosmological Gravity Wave– (NASA Goddard)

• Financial market data analysis– (NCU)

• Theoretical foundations– (Princeton University and Caltech)

HHT looks wonderfully simple,

but

The Devil is in the details.

The Idea behind EMD• To be able to analyze data from the

nonstationary and nonlinear processes and reveal their physical meaning, the method has to be Adaptive.

• Adaptive requires a posteriori (not a priori) basis. But the present established mathematical paradigm is based on a priori basis.

• Only a posteriori basis could fit the varieties of nonlinear and nonstationary data without resorting to the mathematically necessary (but physically nonsensical) harmonics.

The Idea behind EMD

• The method has to be local.

• Locality requires differential operation to define properties of a function.

• Take frequency, for example. The present established mathematical paradigm is based on Integral transform. But integral transform suffers the limitation of the uncertainty principle.

John von Neumann

By and large it is uniformly true that in mathematics there is a time lapse between a mathematical discovery and the moment it becomes useful; and that this lapse can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful.

Norden E. Huang : paraphrase

By and large it is usually true that in science there is a time lapse between the discovery of a useful method and the moment it becomes mathematically proved; and that this lapse can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function with a firm direction: always reference to usefulness, and with strong desire to do things which are useful.

On Calculus

“Newton and Leibniz's approach to the calculus fell well short of later standards of rigor. We now see their "proof" as being in truth mostly a heuristic hodgepodge mainly grounded in geometric intuition.” Wikipedia

dxx

dt

fdx F

1643-1727 1646-1716

On Calculus• George Berkeley, in a tract called The Analyst

and elsewhere, ridiculed this and other aspects of the early calculus, pointing out that natural science grounded in the calculus required just as big of a leap of faith as theology grounded in Christian revelation.

• Modern, rigorous calculus only emerged in the

19th century, thanks to the efforts of Augustin Louis Cauchy, Bernhard Riemann, Karl Weierstrass, and others, who based their work on the definition of a limit and on a precise understanding of real numbers.

Jean-Baptiste-Joseph Fourier (1768-1830)

1807 “On the Propagation of Heat in Solid Bodies”

1812 Grand Prize of Paris Institute

“Théorie analytique de la chaleur”

‘... manner in which the author arrives at these equations is not exempt of difficulties and that the 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 Académie

paper published

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

Fourier Transform

• Michel Plancherel (16 January 1885 to 4 March 1967) : Plancherel, Michel (1910) "Contribution a l'etude de la representation d'une fonction arbitraire par les integrales définies," Rendiconti del Circolo Matematico di Palermo, vol. 30, pages 298-335.

• In which he proved the convergence of the Fourier transform by excluding denumerable number of discontinuity points in the function. He then proved that the totality of the excluded parts have zero measure.

Oliver H

eaviside

Oliver Heaviside1850 - 1925

Adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations (later found to be equivalent to Laplace transform),

introduced delta and step functions, and invented modern vector analysis, thereby reducing the original twenty equations in twenty unknowns down to the four differential equations in

two unknowns we now know as Maxwell’s Equations.

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

I will not wait for the Mathematician’s proof.

Historically, the proof would not change the method, but will

change the mathematician’s view.

John von NeumannAs a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from “reality,” it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l’art pour l’art. This need not be bad if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men [or women] with exceptionally well-developed taste. But there is a grave danger that the subject will developed along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and the discipline will become a disorganized mass of details and complexities.

I prefer the Heaviside principle.

For the time being.

HHT is at the same stage as Wavelet was in the late 1980s.

We need someone, as Ingrid Daubechies, to set the rigorous foundation.

But, in principle, HHT is much harder!

VOLUME ITECHNICAL PROPOSAL AND MANAGEMENT

APPROACHMathematical 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. …….

The Battle Hymn for HHTBy Ingrid Daubechies

You can use wavelets or Fourier And find something that is useful,But if you want something new to say, You cannot be any old fool.

You then find a new kind of algorithm And let it loose on the world.Before they even know what hit’em, They’re up to their necks on work.

Rushing in to meet your frustrating challenge, They filter and stretch and squeeze.

But always some signal throws a monkey wrench, And make them huff and sneeze.

Your name is, of course, Norden Huang, Long may you live and smile!We’re here to learn and get the hang And will not quit for a very long

while.

Hilbert-Huang transform, you will triumph!

Composed on 17 December 2008 in Guangzhou, to be sung to the tune of ‘The Internationale’.

At The Second International Conference on the Advances of Hilbert-Huang Transform and its Applications.

Up Hill

Does the road wind up-hill all the way?

Yes, to the very end.

Will the day’s journey take the whole long day?

From morn to night, my friend.

--- Christina Georgina Rossetti

A less poetic paraphrase

• There is no doubt that our road will be long and that our climb will be steep.

……

• But, anything is possible.

--- Barack Obama

18 Jan 2009, Lincoln Memorial

It is better lucky than smart.

Everyone needs luck sometimes: Albert Michelson, Carl Wilson, Arno

Penzias and Robert Wilson,…

I am lucky to have found this simple method.

Now we need smart people to tell us why it works and why it works so well.

Good luck to us all!!

Outline of the Course I• Introduction

• EMD and EEMD• EMD• Intermittency and confidence limit• EEMD• Orthogonality• End effects• Relationship with Fourier decomposition: a

conjecture• Trend and detrend

• HHT Operations

Outline of the Course II• Hilbert Spectral Analysis• Mathematical preliminary• End effects• Wavelet and Wigner-Ville Distribution• Instantaneous Frequency• Paradoxes of instantaneous frequency• Hilbert Spectral Representation and Marginal spectrum• Multi-dimensional EMD• Available approaches• MD-EEMD• Applications• Water wave studies• HHT based nondestructive Health monitoring• Stability spectral analysis• Global Climate Change