The Empirical Mode Decomposition Method

Sifting

Goal of Data Analysis

• To define time scale or frequency.• To define energy density.• To define joint frequency-energy distribution

as a function of time.

• To do this, we need a AM-FM decomposition of the signal. X(t) = A(t) cosθ(t) , where A(t) defines local energy and θ(t) defines the local frequency.

Need for Decomposition

• Hilbert Transform (and all other IF computation methods) only offers meaningful Instantaneous Frequency for IMFs.

• For complicate data, there should be more than one independent component at any given time.

• The decomposition should be adaptive in order to study data from nonstationary and nonlinear processes.

• Frequency space operations are difficult to track temporal changes.

Why Hilbert Transform not enough?

Even though mathematicians told us that the Hilbert transform exists

for all functions of Lp-class.

Problems on Envelope

A seemingly simple proposition but it is not so easy.

Two examples

For t = 0 to 4096;

t tX 1 sin + 0.2 sin ;

16 128

t tX 2 0.2 sin + sin .

16 128

Data set 1

t tX 1 sin + 0.2 sin

16 128

Data X1

Data X1 Hilbert Transform

Data X1 Envelopes

Observations

• None of the two envelopes seem to make sense:• The Hilbert transformed amplitude oscillates too

much.• The line connecting the local maximum is almost

the tracing of the data.• It turns out that, though Hilbert transform exists,

the simple Hilbert transform does not make sense.

• For envelopes, the necessary condition for Hilbert transformed amplitude to make sense is for IMF.

Data X1 IMF

Data x1 IMF1

Data x2 IMF2

Observations

• For each IMF, the envelope will make sense.

• For complicate data, we have to decompose it before attempting envelope construction.

• To be able to determine the envelope is equivalent to AM & FM decomposition.

Data set 2

t tX 2 0.2 sin + sin

16 128

Data X2

Data X2 Hilbert Transform

Data X2 Envelopes

Observations

• Even for this well behaved function, the amplitude from Hilbert transform does not serve as an envelope well. One of the reasons is that the function has two spectrum lines.

• Complications for more complex functions are many.

• The empirical envelope seems reasonable.

Empirical Mode Decomposition

• Mathematically, there are infinite number of ways to decompose a functions into a complete set of components.

• The ones that give us more physical insight are more significant.

• In general, the few the number of representing components, the higher the information content.

• The adaptive method will represent the characteristics of the signal better.

• EMD is an adaptive method that can generate infinite many sets of IMF components to represent the original data.

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 : h2 & m3

Empirical Mode Decomposition: Methodology : h3 & m4

Empirical Mode Decomposition: Methodology : h2 & h3

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

.

Empirical Mode DecompositionSifting : to get one IMF component

1 1

1 2 2 1 2

k 1 k k 1 2 k

k 1

x( t ) m h ,

h m h x( t ) ( m m )

.....

.....

h m h x( t ) ( m

h c .

m ... m ) .

Empirical Mode Decomposition: Methodology : IMF c1

Definition of the Intrinsic Mode Function

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 DecompositionSifting : to get all the IMF components

k

1

1 1

k

1 j1

jj 1

x( t ) c r ,

c x( t

m

) m

.r

.

Empirical Mode DecompositionSifting : to get all the IMF components

1 2 2

k

2 1 2 1 2

pk

2 j 2 j1 1

j1

r c r ,

c r r r m

.c m m

Empirical Mode DecompositionSifting : to get all the IMF components

n

j1

pk k

j j 2 j1 1 1

x( t ) c

x( t ) m m m ....

x( t ) .

Empirical Mode Decomposition: Methodology : data & r1

Empirical Mode Decomposition: Methodology : data, h1 & r1

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 .

Definitions of Frequency

j

Ti

T

0

t

0

C

j

4. Dynamic System through Hamiltonian :

H( p,q,t ) and A( t )

1. Fourier Analysis :

F( ) x( t ) e dt .

2. Wavelet Analysi

5

pdq ;

H.

. Tea

s

3. Wigner Ville An

ger Energy Operator

al

6. Period between zero c

ysis

ros sin gs and

A

ji ( t ) jj

j

extrema

7. HHT Analysis (Hilbert and Quadrature) :

dx( t ) a( t ) e .

dt

The Effects of Sifting

• The first effect of sifting is to eliminate the riding waves : to make the number of extrema equals to that of zero-crossing.

• The second effect of sifting is to make the envelopes symmetric. The consequence is to make the amplitudes of the oscillations more even.

Singularity points for Instantaneous Frequency

1

2 2

yAs tan ,

x

d yd 1 dy dxdt x

x y .dt a dt dty

1x

Therefore , when the amplitude, a , becomes zero,

IF becomes sin gular .

Critical Parameters for EMD

• The maximum number of sifting allowed to extract an IMF, N.

• The criterion for accepting a sifting component as an IMF, the Stoppage criterion S.

• Therefore, the nomenclature for the IMF are CE(N, S) : for extrema sifting CC(N, S) : for curvature sifting

The 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. If the mean is smaller than a pre-assigned value.

C. 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 )

Curvature Sifting

Hidden Scales

Empirical Mode Decomposition: Methodology : Test Data

Hidden ScalesThe present sifting is based on extrema:

x'(t) = 0.

But there are scales where

x"(t) = 0, x'''(t) = 0, ....

In fact, there are infinite many such critical points. in fact

Taylor series expansion

2

gives us

x"( a ) x'''( a )x(t) = x(a) + x '(a)(t-a) + ( t a ) ( t a ) ...

2! 3!If we know the derivative of all order, we would be able to

define the whole function. Where should we stop?

Hidden Scales

2 3 / 2

We stop at curvature: Firest compute the curvature,

x" c ,

( 1 x' )

Then then find and connect these extrema in sifting.

Our justification is simple: if x is position, second derivative

is accelerati

on. In Newtonian mechanics, beyond acceleration

there is no more physical law governing the variable.

Observations

• If we decide to use curvature, we have to be careful for what we ask for.

• For example, the Duffing pendulum would produce more than one components.

• Therefore, curvature sifting is used sparsely. It is useful in the first couple of components to get rid of noises.

Intermittence Test

To alleviate the Mode Mixing

Sifting with Intermittence Test

• To avoid mode mixing, we have to institute a special criterion to separate oscillation of different time scales into different IMF components.

• The criteria is to select time scale so that oscillations with time scale shorter than this pre-selected criterion is not included in the IMF.

Intermittence Sifting : Data

Intermittence Sifting : IMF

Intermittence Sifting : Hilbert Spectra

Intermittence Sifting : Hilbert Spectra (Low)

Intermittence Sifting : Marginal Spectra

Intermittence Sifting : Marginal spectra (Low)

Intermittence Sifting : Marginal spectra (High)

Critical Parameters for Sifting

• Because of the inclusion of intermittence test there will be one set of intermittence criteria.

• Therefore, the Nomenclature for IMF here are

CEI(N,S: n1, n2, …)CCI(N, S: n1, n2, …)

with n1, n2 as the intermittence test criteria.

The mathematical Requirements for Basis

The traditional Views

IMF as Adaptive Basis

According to the established mathematical paradigm, we should check the following properties of the basis:

• Convergence• completeness• orthogonality• Uniqueness

Convergence

Convergence Problem

• Given an arbitrary number, ε, there always exists a large finite number N, such that Nth envelope mean, mN , satisfies | mN | ≤ε:

thn

n

Given , n, such that the n envelope mean, m ,

satisfies m every where.

Convergence Problem

• Given an arbitrary number, ε, there always exists a large finite number N, such that N- th sifting satisfies

th

th

Given , n, such that the difference between n and

( n 1 ) , trials is less than every where.

T2

N 1 Nt 0

T2

N 1t 0

h ( t ) h ( t )SD

h ( t )

Convergence• There is another convergence problem: we have

only finite number of components.

• Complete proof for convergence is underway.

• We can prove the convergence under simplified condition of linear segment fitting for sifting.

• Empirically, we found all cases converge in finite steps. The finite component, n, is less than or equal to log2N, with N as the total number of data points.

Convergence

• The necessary condition for convergence is that the mean line should have less extrema than the original data.

• This might not be true if we use the middle points and a single spline; the procedure might not converge.

Completeness

Completeness

• Completeness is given by the algebraic equation

• Therefore, the sum of IMF can be as close to the original data as required.

• Completeness is given.

n n 1

j n jj 1 j 1

x( t ) c r c .

Orthogonality

Orthogonality

• Definition: Two vectors x and y are orthogonal if their inner product is zero.

x ∙y = (x1 y1 + x2 y2 + x3 y3 + …) = 0.

The need for an orthogonality check

• Orthogonal is required for:

jn

2 2j i j

n i j

2i j

i j

i ji j

x( t ) c ( t )

x ( t ) c ( t ) c ( t )c ( t ) .

If c ( t )c ( t ) o, x ( t ) could be negative.

Therefore ,we require c ( t )c ( t ) o, the orthogonal condition.

Orthogonality• Orthogonality is a requirement for any linear

decomposition.• For a nonlinear decomposition, as EMD, the

orthogonality should not be a requirement, for nonlinear waves of different scale could share the same harmonics.

• Fortunately, the EMD is basically a Reynolds type decomposition , U = <U> + u’, orthogonality is always approximately satisfied to the degree of nonlinearity.

• Orthogonality Index should be checked for each cases as a goodness of decomposition confirmation.

Orthogonality Index

T

i jt 1

ij T T2 2

i jt 1 t 1

T

i ji j t 1

2

c ( t )c ( t )1

OI .T

c ( t ) c ( t )

c ( t )c ( t )1

OI .T 2 x ( t )

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

Uniqueness

Uniqueness• EMD, with different critical parameters, can

generate infinite sets of IMFs.• The result is unique only with respect to the

critical parameters and sifting method selected; therefore, all results should be properly named according to the nomenclature scheme proposed above.

• The present sifting is based on cubic spline. Different spline fitting in the sifting procedure will generate different results.

• The ensemble of IMF sets offers a Confidence Limit as function of time and frequency.

Some Tricks in Sifting

Some Tricks in Sifting

• Sometimes straightforward application of sifting will not generate good results.

• Invoking intermittence criteria is an alternative to get physically meaningful IMF components.

• By adding low level noise can improve the sifting.

• By using curvature may also help.

An Example

Adding Noise of small amplitude only,

A prelude to the true Ensemble EMD

Data: 2 Coincided Waves

IMF from Data of 2 Coincided Waves

Data: 2 Coincided Waves + NoiseThe Amplitude of the noise is 1/1000

IMF form Data 2 Coincided Waves + Noise

IMF c1 and Component2 : 2 Coincided Waves

IMF c2+c3 and Component1 : 2 Coincided Waves

A Flow Chart

DataData IMFsifting

With Intermittence

Hilbert Spectrum

IF

Marginal Spectrum

OI

CL

Ensemble EMD