The Empirical Mode Decomposition Method Sifting. Goal of Data Analysis To define time scale or...
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The Empirical Mode Decomposition Method
Sifting
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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.
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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.
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Why Hilbert Transform not enough?
Even though mathematicians told us that the Hilbert transform exists
for all functions of Lp-class.
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Problems on Envelope
A seemingly simple proposition but it is not so easy.
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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
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Data set 1
t tX 1 sin + 0.2 sin
16 128
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Data X1
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Data X1 Hilbert Transform
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Data X1 Envelopes
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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.
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Data X1 IMF
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Data x1 IMF1
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Data x2 IMF2
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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.
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Data set 2
t tX 2 0.2 sin + sin
16 128
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Data X2
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Data X2 Hilbert Transform
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Data X2 Envelopes
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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.
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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.
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Empirical Mode Decomposition: Methodology : Test Data
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Empirical Mode Decomposition: Methodology : data and m1
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Empirical Mode Decomposition: Methodology : data & h1
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Empirical Mode Decomposition: Methodology : h1 & m2
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Empirical Mode Decomposition: Methodology : h2 & m3
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Empirical Mode Decomposition: Methodology : h3 & m4
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Empirical Mode Decomposition: Methodology : h2 & h3
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Empirical Mode Decomposition: Methodology : h4 & m5
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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
.
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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 ) .
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Empirical Mode Decomposition: Methodology : IMF c1
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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
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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 .
.
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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
.
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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
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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 ) .
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Empirical Mode Decomposition: Methodology : data & r1
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Empirical Mode Decomposition: Methodology : data, h1 & r1
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Empirical Mode Decomposition: Methodology : IMFs
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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 .
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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
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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.
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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 .
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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
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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 )
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Curvature Sifting
Hidden Scales
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Empirical Mode Decomposition: Methodology : Test Data
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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?
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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.
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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.
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Intermittence Test
To alleviate the Mode Mixing
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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.
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Intermittence Sifting : Data
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Intermittence Sifting : IMF
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Intermittence Sifting : Hilbert Spectra
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Intermittence Sifting : Hilbert Spectra (Low)
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Intermittence Sifting : Marginal Spectra
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Intermittence Sifting : Marginal spectra (Low)
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Intermittence Sifting : Marginal spectra (High)
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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.
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The mathematical Requirements for Basis
The traditional Views
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IMF as Adaptive Basis
According to the established mathematical paradigm, we should check the following properties of the basis:
• Convergence• completeness• orthogonality• Uniqueness
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Convergence
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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.
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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 )
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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.
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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.
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Completeness
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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 .
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Orthogonality
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Orthogonality
• Definition: Two vectors x and y are orthogonal if their inner product is zero.
x ∙y = (x1 y1 + x2 y2 + x3 y3 + …) = 0.
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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.
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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.
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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 )
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Length Of Day Data
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LOD : IMF
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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
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Uniqueness
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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.
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Some Tricks in Sifting
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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.
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An Example
Adding Noise of small amplitude only,
A prelude to the true Ensemble EMD
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Data: 2 Coincided Waves
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IMF from Data of 2 Coincided Waves
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Data: 2 Coincided Waves + NoiseThe Amplitude of the noise is 1/1000
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IMF form Data 2 Coincided Waves + Noise
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IMF c1 and Component2 : 2 Coincided Waves
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IMF c2+c3 and Component1 : 2 Coincided Waves
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A Flow Chart
DataData IMFsifting
With Intermittence
Hilbert Spectrum
IF
Marginal Spectrum
OI
CL
Ensemble EMD