MATH 3290 Mathematical Modeling Tutorial on the Empirical Mode Decomposition Method (EMD)

MATH 3290 Mathematical Modeling

Tutorial on the Empirical Mode Decomposition Method

(EMD)

First, review of the procedure of EMD method

The main idea of the EMD method is

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 : SD

Standard Deviation is small than a pre-set value, where

T2

k 1 kt 0

T2

k 1t 0

h ( t ) h ( t )SD

h ( t )

Stoppage Criteria

• It is critical that we use the correct stoppage criterion.

• Over shifting, we can prove that the envelopes defined has to be a straight line.

• If the data is not monotonically increasing or decreasing, the straight lines would be horizontal lines.

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: data

Empirical Mode Decomposition: IMFs and residue

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

Hilbert-Huang Transform

(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 .

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 traditional established mathematical paradigm is based on Integral transform. But integral transform suffers the limitation of the uncertainty principle.

Global Temperature Anomaly

Annual Data from 1856 to 2003

Global Temperature Anomaly 1856 to 2003

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

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 a scientifically meaningful way to analyze data.

It is a way to find out the underlying physical processes; therefore, it is indispensable in scientific research.

It is physical, direct, and simple.

History of EMD & 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, 460, 1597-1611.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 (Advances in Adaptive Data Analysis. Advances in Adaptive data Analysis, 1, 177-229).

2009: Multi-Dimensional Ensemble Empirical Mode Decomposition. Advances in Adaptive Data Analysis (Advances in Adaptive Data Analysis. Advances in Adaptive data Analysis, 1, 339-372).

2010: The Time-Dependent Intrinsic Correlation based on the Empirical Mode Decomposition (Advances in Adaptive Data Analysis. Advances in Adaptive data Analysis, 2, 233-265).

2010: On Hilbert Spectral Analysis (to appear in AADA).

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)

Reference:• Huang, M. L. Wu, S. R. Long, S. S. Shen, W. D. Qu, P. Gloersen, and K. L.

Fan (1998)The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. Roy. Soc. Lond., 454A, 903-993.

• Flandrin, P., G. Rilling, and P. Gonçalves (2004) Empirical mode decomposition as a filter bank. IEEE Signal Proc Lett., 11, 112-114.

• Research Center for Adaptive Data Analysis, National Central University

http://rcada.ncu.edu.tw/research1.htm