empirical mode decomposition an introduction

1

AN INTRODUCTION TO EMPIRICAL MODE DECOMPOSITION

By Harikrishna satish.T

M.E. Electrical engg,Jadavpur university.

2

Empirical mode decomposition

Method for processing non stationary signals and signals produced by nonlinear processes and Decomposition of the signal into a set of Intrinsic Mode Functions (IMF) which are defined as

1. In the whole set of data, the numbers of local extrema

and the numbers of zero crossings must be equal or differ by 1 at most.

2. At any time point, the mean value of the “upper envelope” (defined by the local Maxima) and the “lower envelope” (defined by the local minima) must be zero.

3

The simplest model for a signal is given by circular functions of the type

Such “Fourier modes” are of particular interest in the case of stationary signals and linear systems.

(Type I)

Why Empirical Mode Decomposition?

4

we can think of representing these signals in terms of amplitude and frequency modulated (AM–FM) components

However, many physical situations are known to undergo non stationary and/or nonlinear behaviors.

The rationale for such a modeling is to compactly encode possible non stationarities in a time variation of the amplitudes and frequencies of Fourier-like modes.

(Type II)

5

More generally, signals may also be generated by nonlinear systems for which oscillations are not necessarily associated with circular functions, thus suggesting decompositions of the following form

(Type III)

Each of the components has to have physical and mathematical meaning.

6

Empirical Mode Decomposition (EMD) is designed primarily for obtaining representations of Type II or Type III in the case of signals which are oscillatory, possibly non stationary or generated by a nonlinear system, in some automatic, fully data-driven way.

7

Main differences between EMD and traditional data analysis methods

Stationary Non-stationary Linear Non-linear Theory

Fourier

Wavelets

Time-series

EMD

Traditional methods EMD

- Not appropriate for nonlinear & non stationary signals.

- Predefined basis and/or system model.

- Distorted information extracted.

- Full theoretical basis.

- Adequate for both nonlinear & non stationary.

- Adaptive – data driven basis.

- Preserves physical meaning.

- Sharper spectrum

- Lack of theoretical analysis.

8

Basic Parts of the Empirical Mode Decomposition

• Interpolation technique (cubic spline).

• Sifting process to extract and identify intrinsic modes.

• Numerical convergence criteria (mainly to stop the iterative process of identifying every IMF as well as the whole set of IMFs)

9

The starting point of EMD is to consider oscillatory signals at the level of their local oscillations and to formalize the idea that:

“signal = fast oscillations superimposed to slow oscillations”

10

“signal = fast oscillations superimposed to slow oscillations”

Iterate on the slow oscillations component considered as a new signal.

11

Decomposition of the signal:-

Thus , we can say that the original signal is the combination of all the EMF’s decomposed and the residue.

12

• Decomposition:

• Mode: Intrinsic Mode Functions (IMF’s) -Represents the oscillation modes embedded in the data.

• Empirical: The Sifting process is essentially defined by an algorithm.EMD lacks theoretical foundations.

Empirical Mode Decomposition

13

1. identify all extrema of x(t).

2. Interpolate the local maxima to form an upper envelope u(x).

3. Interpolate the local minima to form an lower envelope l(x).

4. Calculate the mean envelope:m(t)=[u(x)+l(x)]/2.

5. Extract the mean from the signal:h(t)=x(t)-m(t)

6. Check whether h(t) satisfies the IMF condition. YES: h(t) is an IMF, stop sifting. NO: let x(t)=h(t), keep sifting.

How to find one Intrinsic Mode Functions of a signal?Sifting procedure

14

10 20 30 40 50 60 70 80 90 100 110 120

-2

-1

0

1

2

IMF 1; iteration 0Residue = s(t)

I1(t) = Residue

i = 1

k = 1

while Residue not equal zero or not

monotone

while Ii has non-negligible local

mean

U(t) = spline through local

maxima of IiL(t) = spline through local

minima of IiAv(t) = 1/2 (U(t) + L(t))

Ii(t) = Ii(t) - Av(t)

i = i + 1

end

IMFk(t) = Ii(t)

Residue = Residue - IMFk

k = k+1

end

Huang’s “Sifting Process”.

15

10 20 30 40 50 60 70 80 90 100 110 120

-2

-1

0

1

2


I1(t) = Residue

i = 1

k = 1


monotone


mean


maxima of Ii

L(t) = spline through local



i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


16

10 20 30 40 50 60 70 80 90 100 110 120

-2

-1

0

1

2


I1(t) = Residue

i = 1

k = 1


monotone


mean


maxima of Ii




i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


17

10 20 30 40 50 60 70 80 90 100 110 120

-2

-1

0

1

2


I1(t) = Residue

i = 1

k = 1


monotone


mean



minima of Ii

Av(t) = 1/2 (U(t) + L(t))


i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


18

10 20 30 40 50 60 70 80 90 100 110 120

-2

-1

0

1

2


I1(t) = Residue

i = 1

k = 1


monotone


mean



minima of Ii

Av(t) = 1/2 (U(t) + L(t))


i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


19

10 20 30 40 50 60 70 80 90 100 110 120

-2

-1

0

1

2


I1(t) = Residue

i = 1

k = 1

while Residue not equal zero or

not monotone

while Ii has non-negligible

local mean





i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


20

10 20 30 40 50 60 70 80 90 100 110 120

-2

-1

0

1

2

IMF 1; iteration 0

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


not monotone


mean





(“residue”-->)

i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


21

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

IMF 1; iteration 1

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


not monotone


local mean





i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


22

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

IMF 1; iteration 1

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


monotone


mean


maxima of Ii




i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


23

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

IMF 1; iteration 1

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


monotone


mean


maxima of Ii

L(t) = spline through local minima

of IiAv(t) = 1/2 (U(t) + L(t))


i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


24

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

IMF 1; iteration 1

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


monotone


mean

U(t) = spline through local maxima

of IiL(t) = spline through local

minima of Ii

Av(t) = 1/2 (U(t) + L(t))


i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


25

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

IMF 1; iteration 1

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


monotone


mean



minima of Ii

Av(t) = 1/2 (U(t) + L(t))


i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


26

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

IMF 1; iteration 1

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


not monotone


local mean





i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


27

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

IMF 1; iteration 1

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


not monotone


local mean





(“residue”-->)

i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


28

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

IMF 1; iteration 2

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


not monotone


local mean





i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


29

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

IMF 1; iteration 2

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


monotone


mean


maxima of Ii

L(t) = spline through local minima of

IiAv(t) = 1/2 (U(t) + L(t))


i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


30

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

IMF 1; iteration 2

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


monotone


mean


maxima of Ii




i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


31

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

IMF 1; iteration 2

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


not monotone


mean



minima of Ii

Av(t) = 1/2 (U(t) + L(t))


i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


32

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

IMF 1; iteration 2

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


monotone


mean



minima of Ii

Av(t) = 1/2 (U(t) + L(t))


i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


33

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

IMF 1; iteration 2

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


not monotone


local mean





i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


34

10 20 30 40 50 60 70 80 90 100 110 120

-1.5

-1

-0.5

0

0.5

1

1.5

IMF 1; iteration 2

10 20 30 40 50 60 70 80 90 100 110 120

-1

-0.5

0

0.5

1

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


not monotone


local mean





(“residue”-->)

i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


35

10 20 30 40 50 60 70 80 90 100 110 120

-1

-0.5

0

0.5

1

IMF 1; iteration 3

10 20 30 40 50 60 70 80 90 100 110 120

-1

-0.5

0

0.5

1

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


monotone


mean


maxima of Ii


minima of Ii

Av(t) = 1/2 (U(t) + L(t))


i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


36

10 20 30 40 50 60 70 80 90 100 110 120

-1

-0.5

0

0.5

1

IMF 1; iteration 4

10 20 30 40 50 60 70 80 90 100 110 120

-1

-0.5

0

0.5

1

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


monotone

while Ii has non-negligible local mean


maxima of Ii


minima of Ii

Av(t) = 1/2 (U(t) + L(t))


i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


37

10 20 30 40 50 60 70 80 90 100 110 120

-1

-0.5

0

0.5

1

IMF 1; iteration 5

10 20 30 40 50 60 70 80 90 100 110 120

-1

-0.5

0

0.5

1

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


monotone


mean


maxima of Ii


minima of Ii

Av(t) = 1/2 (U(t) + L(t))


i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


38

10 20 30 40 50 60 70 80 90 100 110 120

-1

-0.5

0

0.5

1

IMF 1; iteration 6

10 20 30 40 50 60 70 80 90 100 110 120

-1

-0.5

0

0.5

1

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


monotone


mean


maxima of Ii


minima of Ii

Av(t) = 1/2 (U(t) + L(t))


i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


39

10 20 30 40 50 60 70 80 90 100 110 120

-1

-0.5

0

0.5

1

IMF 1; iteration 7

10 20 30 40 50 60 70 80 90 100 110 120

-1

-0.5

0

0.5

1

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


monotone


mean


maxima of Ii


minima of Ii

Av(t) = 1/2 (U(t) + L(t))


i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


40

10 20 30 40 50 60 70 80 90 100 110 120

-1

-0.5

0

0.5

1

IMF 1; iteration 8

10 20 30 40 50 60 70 80 90 100 110 120

-1

-0.5

0

0.5

1

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


monotone


mean


maxima of Ii


minima of Ii

Av(t) = 1/2 (U(t) + L(t))


i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


41

10 20 30 40 50 60 70 80 90 100 110 120

-1

-0.5

0

0.5

1

IMF 2; iteration 0

10 20 30 40 50 60 70 80 90 100 110 120

-1

-0.5

0

0.5

1

residue

Residue = s(t)

I1(t) = Residue

i = 1

k = 1


monotone


mean





i = i + 1

end

IMFk(t) = Ii(t)


k = k+1

end


42

An example:-(puts emphasis on the potentially “nonharmonic” nature of EMD)

the signal is composed of 1. a “high frequency” triangular waveform

whose amplitude is slowly (linearly) growing.

2. a “middle frequency”sine wave whose amplitude is quickly (linearly) decaying

3. a “low frequency” triangular waveform

Both linear and nonlinear oscillations are effectively identified and separated by EMD. whereas any “harmonic” analysis (Fourier, wavelets, . . . ) would end up in the same context with a much less compact and physically less meaningful decomposition.

43

No analytic definition — The decomposition is only defined as the output of an algorithm

Some remarks on Huang’s EMD

Rationale — Intuitive, simple, local and fully data-driven.

Still lacks from solid theoretical grounds

44

Acoustics, noise, and vibration: Machine vibration analysis, Speech/sound analysis , Biomedical signal analysis.

Environmental: Oceanography, Earthquake engineering , Water and wind dynamics.

Industrial: Machine monitoring and failure prediction .

Applications of EMD

45

References:- P. FLANDRIN, P. GONCALVES, 2004 :

"Empirical Mode Decompositions as Data-Driven Wavelet-Like Expansions," Int. J. of Wavelets, Multires. and Info. Proc., Vol. 2, No. 4, pp. 477-496.

ON EMPIRICAL MODE DECOMPOSITION AND ITS ALGORITHMSGabriel Rilling∗, Patrick Flandrin∗ and Paulo Gon¸calvès∗∗∗Laboratoire de Physique (UMR CNRS 5672), École Normale Supérieure de Lyon 46, allée d’Italie 69364 Lyon Cedex 07, France. The analysis of the Empirical Mode Decomposition MethodVesselin Vatchev, USC ,November 20, 2002

http://perso.ens lyon.fr/patrick.flandrin/emd.html

http://perso.ens/

http://lyon.fr/patrick.flandrin/emd.html



empirical mode decomposition an introduction

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