A Plea for Adaptive Data Analysis:Instantaneous Frequencies and Trends

For Nonstationary Nonlinear Data

Norden E. HuangResearch Center for Adaptive Data Analysis

National Central UniversityZhongli, Taiwan, China

Hot Topic Conference, 2011

In search of frequency I found the trend and other information

Instantaneous Frequencies and Trends

For Nonstationary Nonlinear Data

Prevailing Views onInstantaneous Frequency

The term, Instantaneous Frequency, should be banished forever from the dictionary of the communication engineer.

J. Shekel, 1953

The uncertainty principle makes the concept of an Instantaneous Frequency impossible.

K. Gröchennig, 2001

How to define frequency?

It seems to be trivial.

But frequency is an important parameter for us to understand many physical phenomena.

Definition of Frequency

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

1

T.

Traditional Definition of Frequency

• frequency = 1/period.

• Definition too crude• Only work for simple sinusoidal waves• Does not apply to nonstationary processes• Does not work for nonlinear processes • Does not satisfy the need for wave equations

The Idea and the need of Instantaneous Frequency

k , ;

t

t

k0 .

According to the classic wave theory, the wave conservation law is based on a gradually changing φ(x,t) such that

Therefore, both wave number and frequency must have instantaneous values and differentiable.

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

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

The Traditional View of the Hilbert Transform for Data Analysis

Traditional Viewa la Hahn (1995) : Data LOD

Traditional Viewa 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

.

The Stoppage Criteria

The Cauchy type criterion: when SD is small than a pre-set value, where

T2

k 1 kt 0

T2

k 1t 0

h ( t ) h ( t )SD

h ( t )

Or, simply pre-determine the number of iterations.

Effects of Sifting

• To remove ridding waves

• To reduce amplitude variations

• The systematic study of the stoppage criteria leads to the conjecture connecting EMD and Fourier Expansion.

Empirical Mode Decomposition: Methodology : IMF c1

Definition of the Intrinsic Mode Function (IMF): a necessary condition only!

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 Decomposition: Methodology : data, r1 and m1

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 .

.

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 .

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

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

The combination of Hilbert Spectral Analysis and Empirical Mode Decomposition has been

designated by NASA as

HHT

(HHT vs. FFT)

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

Traditional Viewa la Hahn (1995) : Hilbert

Mean Annual Cycle & Envelope: 9 CEI Cases

Mean Hilbert Spectrum : All CEs

For quantifying nonlinearity we need instantaneous frequency.

For

How to define Nonlinearity?

How to quantify it through data alone?

The term, ‘Nonlinearity,’ has been loosely used, most of the time, simply

as a fig leaf to cover our ignorance.

Can we be more precise?

How is nonlinearity defined?

Based on Linear Algebra: nonlinearity is defined based on input vs. output.

But in reality, such an approach is not practical: natural system are not clearly defined; inputs and out puts are hard to ascertain and quantify. Furthermore, without the governing equations, the order of nonlinearity is not known.

In the autonomous systems the results could depend on initial conditions rather than the magnitude of the ‘inputs.’

The small parameter criteria could be misleading: sometimes, the smaller the parameter, the more nonlinear.

Linear Systems

Linear systems satisfy the properties of superposition and scaling. Given two valid inputs to a system H,

as well as their respective outputs

then a linear system, H, must satisfy

for any scalar values α and β.

x ( t ) and x ( t )1 2

y ( t ) H{ x ( t )} and

y (t) = H{ x ( t )}

1 1

2 2

y ( t ) y ( t ) H{ x ( t ) x ( t )} 1 1 1 2

How is nonlinearity defined?

Based on Linear Algebra: nonlinearity is defined based on input vs. output.

But in reality, such an approach is not practical: natural system are not clearly defined; inputs and out puts are hard to ascertain and quantify. Furthermore, without the governing equations, the order of nonlinearity is not known.

In the autonomous systems the results could depend on initial conditions rather than the magnitude of the ‘inputs.’

The small parameter criteria could be misleading: sometimes, the smaller the parameter, the more nonlinear.

How should nonlinearity be defined?

The alternative is to define nonlinearity based on data characteristics: Intra-wave frequency modulation.

Intra-wave frequency modulation is known as the harmonic distortion of the wave forms. But it could be better measured through the deviation of the instantaneous frequency from the mean frequency (based on the zero crossing period).

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

Hilbert’s View on Nonlinear DataIntra-wave Frequency Modulation

A simple mathematical model

x( t ) cos t sin 2 t

Duffing Type Wave

Data: x = cos(wt+0.3 sin2wt)

Duffing Type WavePerturbation Expansion

For 1 , we can have

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

cos t sin t sin 2 t ....

This is very similar to the solutionof D

x( t ) cos t sin 2 t

1 cos t cos 3

uffing equ

t ....2

atio

2

n .

Duffing Type WaveWavelet Spectrum

Duffing Type WaveHilbert Spectrum

Degree of nonlinearity

1 / 22

z

z

Let us consider a generalized intra-wave frequency modulation model

as:

d x( t ) cos( t sin t )

IF IFDN (Degree of Nolinearity ) should be

I

cos t IF=

.2

Depending on

d

F

t

1t

he value of , we can have either a up-down symmetric

or a asymmetric wave form.

Degree of Nonlinearity

• DN is determined by the combination of δη precisely with Hilbert Spectral Analysis. Either of them equals zero means linearity.

• We can determine δ and η separately:– η can be determined from the instantaneous frequen

cy modulations relative to the mean frequency.– δ can be determined from DN with known η.

NB: from any IMF, the value of δη cannot be greater than 1.

• The combination of δ and η gives us not only the Degree of Nonlinearity, but also some indications of the basic properties of the controlling Differential Equation, the Order of Nonlinearity.

Stokes Models

22

2

2 with ; =0.1.

25

: is positive ranging from 0.1 to 0.375;

beyond 0.375, there is no solution.

: is negative ranging from 0.1 to 0.391

d xx

;

Stokes I

Stokes II

x cos t dt

beyond 0.391, there is no solution.

Data and IFs : C1

0 20 40 60 80 100 120 140 160 180 200-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time: Second

Fre

qu

ency

: H

z

Stokes Model c1: e=0.375; DN=0.2959

IFqIFzDataIFq-IFz

Summary Stokes I

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Epsilon

Co

mb

ined

Deg

ree

of

Sta

tio

nar

ity

Summary Stokes I

C1C2CDN

Lorenz Model

• Lorenz is highly nonlinear; it is the model equation that initiated chaotic studies.

• Again it has three parameters. We decided to fix two and varying only one.

• There is no small perturbation parameter.

• We will present the results for ρ=28, the classic chaotic case.

Phase Diagram for ro=28

010

2030

4050

-20

-10

0

10

20-30

-20

-10

0

10

20

30

x

Lorenz Phase : ro=28, sig=10, b=8/3

y

z

X-Component

DN1=0.5147

CDN=0.5027

Data and IF

0 5 10 15 20 25 30 35 40 45 50-10

0

10

20

30

40

50

Time : second

Fre

qu

ency

: H

z

Lorenz X : ro=28, sig=10, b=8/3

IFqIFzDataIFq-IFz

Spectra data and IF

10-1

100

101

102

10-4

10-3

10-2

10-1

100

101

102

Frequency : Hz

Sp

ectr

al D

ensi

ty

Spectra of Data and IF : Lorenz 28

DataFrequency

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, quantifying

Non-stationary no yes Yes, quantifying

Uncertainty yes yes no

Harmonics yes yes no

How to define Trend ?

Parametric or Non-parametric?

Intrinsic vs. extrinsic approach?

The State-of-the arts: Trend

“One economist’s trend is another economist’s cycle”

Watson : Engle, R. F. and Granger, C. W. J. 1991 Long-run Economic Relation

ships. Cambridge University Press.

Philosophical Problem Anticipated

名不正則言不順

言不順則事不成 

                     ——孔夫子

On Definition

Without a proper definition, logic discourse would be impossible.

Without logic discourse, nothing can be accomplished.

Confucius

Definition of the Trend

Within the given data span, the trend is an intrinsically fitted monotonic function, or a function in which there can be at most one extremum.

The trend should be an intrinsic and local property of the data; it is determined by the same mechanisms that generate the data.

Being local, it has to associate with a local length scale, and be valid only within that length span, and be part of a full wave length.

The method determining the trend should be intrinsic. Being intrinsic, the method for defining the trend has to be adaptive.

All traditional trend determination methods are extrinsic.

Algorithm for Trend

• Trend should be defined neither parametrically nor non-parametrically.

• It should be the residual obtained by removing cycles of all time scales from the data intrinsically.

• Through EMD.

Global Temperature Anomaly

Annual Data from 1856 to 2003

Global Temperature Anomaly 1856 to 2003

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

Mean IMF

STD IMF

Statistical Significance Test

Data and Trend C6

Rate of Change Overall Trends : EMD and Linear

Conclusion• With EMD, we can define the true instantaneous

frequency and extract trend from any data.

• We can also talk about nonlinearity quantitatively.

• Among other applications, the degree of nonlinearity could be used to set an objective criterion in structural health monitoring and to quantify the degree of nonlinearity in natural phenomena; the trend could be used in financial as well as natural sciences.

• These are all possible because of adaptive data analysis method.

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 method and let the data sing, and not to force the data to fit preconceived modes.

The Job of a Scientist

All these results depends on adaptive approach.

Thanks

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

Outstanding Mathematical Problems

1. Mathematic rigor on everything we do. (tighten the definitions of IMF,….)

2. Adaptive data analysis (no a priori basis methodology in general)

3. Prediction problem for nonstationary processes (end effects)

4. Optimization problem (the best stoppage criterion and the uniqueness of the decomposition)

5. Convergence problem (best spline implement, and 2-D)

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 problemsAnd need a lot of helps!

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

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 decompositio

n 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, A460, 1179-1611.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 , 2, 177-229.

2010: Multi-Dimensional Ensemble Empirical Mode Decomposition. Advances in Adaptive Data Analysis 3, 339-372.

2011: Degree of Nonlinearity. Patent and Paper