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

Surrogate Signal:Non-uniqueness signal vs. Spectrum

I. Hello

The original data : Hello

The IMF of original data : Hello

The surrogate data : Hello

The IMF of Surrogate data : Hello

The Hilbert spectrum of original data : Hello

The Hilbert spectrum of Surrogate data : Hello

To quantify nonlinearity we also need instantaneous frequency.

Additionally

How to define Nonlinearity?

How to quantify it through data alone (model independent)?

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

frequency 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 80 100-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time: Second

Fre

qu

en

cy

: 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 30-10

0

10

20

30

40

50

Time : second

Fre

qu

en

cy

: H

z

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

IFqIFzDataIFq-IFz

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 and determine 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

Relationships. Cambridge University Press.

Philosophical Problem Anticipated

名不正則言不順言不順則事不成 

                      ——孔夫子

Definition of the TrendProceeding Royal Society of London, 1998

Proceedings of National Academy of Science, 2007

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

GSTA

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

Statistical Significance Test

IPCC Global Mean Temperature Trend

Comparison between non-linear rate with multi-rate of IPCC

1840 1860 1880 1900 1920 1940 1960 1980 2000 2020-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025Various Rates of Global Warming

Time : Year

War

min

g R

ate:

oC

/Yr

Blue shadow and blue line are the warming rate of non-linear trend.Magenta shadow and magenta line are the rate of combination of non-linear trend and AMO-like components.Dashed lines are IPCC rates.

Conclusion• With HHT, we can define the true instantaneous

frequency and extract trend from any data.

• We can quantify the degree of nonlinearity. Among the applications, the degree of nonlinearity could be used to set an objective criterion in biomedical and structural health monitoring, and to quantify the degree of nonlinearity in natural phenomena.

• We can also determine the trend, which could be used in financial as well as natural sciences.

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

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, 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.

2009: On Ensemble Empirical Mode Decomposition. Adv. Adaptive data Anal., 1, 1-41

2009: On instantaneous Frequency. Adv. Adaptive Data Anal., 2, 177-229.

2010: The Time-Dependent Intrinsic Correlation Based on the Empirical Mode Decomposition. Adv. Adaptive Data Anal., 2. 233-266.

2010: Multi-Dimensional Ensemble Empirical Mode Decomposition. Adv. Adaptive Data Anal., 3, 339-372.

2011: Degree of Nonlinearity. (Patent and Paper)

2012: Data-Driven Time-Frequency Analysis (Applied and Computational Harmonic Analysis, T. Hou and Z. Shi)

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)

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 unique decomposition ….)

5. Convergence problem (finite step of sifting, best spline implement, …)

Do mathematical results make physical sense?

Do mathematical results make physical sense?

Good math, yes; Bad math ,no.

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