ENSEMBLE EMPIRICAL MODE
DECOMPOSITION
Noise Assisted Signal Analysis (nasa)
Part II EEMD
Zhaohua Wu and N. E. Huang:
Ensemble Empirical Mode Decomposition: A Noise
Assisted Data Analysis Method. Advances in
Adaptive Data Analysis, 1, 1-41, 2009
The Ensemble Effects
The true answer is not the one
without perturbation of noise.
EXAMPLE : ORIGINAL DATA
Original Data
1
0.5
0
-0.5
-1
E1: Original Data + 0.1*RANDN/std(Oiginal data)
1
0.5
0
-0.5
-1
Mean E50 data
1
0.5
0
-0.5
-1
0
200
400
600
800
1000
1200
“LOCAL” -> “LOCAL”
IMF 4
IMF 3
IMF 2
IMF 1
Signal
Decomposition of the Dirac function
8
6
4
2
0
4
2
0
-2
-4
4
2
0
-2
-4
0
50
100
150
200
250
300
350
400
450
500
0
50
100
150
200
250
300
350
400
450
500
0
50
100
150
200
250
300
350
400
450
500
0
50
100
150
200
250
300
350
400
450
500
0
50
100
150
200
250
300
350
400
450
500
1
0
-1
0.5
0
-0.5
Ensemble EMD
Solves the mode mixing problem
utilizing the uniformly distributed
reference frame based on the white
noise
EXAMPLE : ORIGINAL DECOMP.
C1 of the Original Data
1
0
-1
C2 of the Original )
1
0
-1
C3 of the Original )
0.2
0
-0.2
Remainder of the Original )
0.2
0
-0.2
0
200
400
600
800
1000
1200
Procedures for EEMD
•
•
•
•
Add a white noise series to the targeted data;
Decompose the data with added white noise
into IMFs;
Repeat step 1 and step 2 again and again, but
with different white noise series each time; and
Obtain the (ensemble) means of
corresponding IMFs of the decompositions as
the final result.
Definition of Signal in EEMD
The signal used in EEMD is given by :
S (t) = s( t )  n j ( t ) ,
j
in which s(t) is the original signal,
n j ( t ) is th eadded noise.
Definition of IMF in EEMD
The truth defined by EEMD is given by the number of the
ensemble approaching infinite:
c (t) = lim
j
N 
in which
1
N
 c
N
k 1
j ,k
(t) +  rk ( t )

,
c j ,k (t) +  rk ( t ) is the
k-th realization of the j-th IMF in the noise
added signal, and  is the magnitude of the
added noise that is not necessarily small.
The Standard Deviation of EEMD
With the truth defined, the discrepancy, Δ, should be

 
Δ =     E cn (t)
j
j=1

t

m



- cn (t)  
j 

2
in which E{ } is the expected value as given in
Equation.
1/2
,
Effect of the White Noise
• The effects of the added white noise should decrease
following the well established statistical rule:
n 

n
, n  number of ensemble.
Data of the Noise Effects:
Dotted line = theoretical; solid line = high frequency components;
dashed line = low frequency components.
standard deviation of error vs. number of ensemble member N
-3
-5
-6
2
log (std of error)
-4
-7
-8
-9
1
2
3
4
5
6
log2(N)
7
8
9
10
11
Procedure for EEMD
Illustration
EXAMPLE : E1 DECOMP.
EXAMPLE : E10 DECOMP.
EXAMPLE : E100 DECOMP.
EXAMPLE : Intermittence DECOMP.
EXAMPLE : Difference Main IMF.
EXAMPLE : Difference Intermittent Signal.
EXAMPLE : Difference Intermittent Signal Details.
EXAMPLE : Instantaneous Frequency from Main Signal.
Summary: Numerical Data
• From the intermittency Example, we see that the
Ensemble EMD can generate IMFs with
comparable quality as the ones through the
Intermittence test.
• More ensemble in the average will improve
confidence in the EMD results.
• The main advantage of Ensemble EMD is that
we do not need to determine the ‘Intermittence
test criteria’ subjectively, which could become
impossible for complicated data.
Example I : Geophysical Data
Surface Temperature Data from Two
Difference Satellite Radiometer channels
EXAMPLE I: ORIGINAL DATA
ORIGINAL DATA ( blue: RSS-T2; red: UAH-T2 )
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
1980
1985
1990
1995
2000
EXAMPLE I: DECOMPOSITION (I)
C1 ( blue: RSS-T2; red: UAH-T2 )
0.2
0
-0.2
C2 ( blue: RSS-T2; red: UAH-T2 )
0.2
0
-0.2
C3 ( blue: RSS-T2; red: UAH-T2 )
0.5
0
-0.5
C4 ( blue: RSS-T2; red: UAH-T2 )
0.5
0
-0.5
1980
1985
1990
1995
2000
EXAMPLE I: DECOMPOSITION (II)
C5 ( blue: RSS-T2; red: UAH-T2 )
0.2
0.1
0
-0.1
-0.2
C6 ( blue: RSS-T2; red: UAH-T2 )
0.2
0.1
0
-0.1
-0.2
TREND ( blue: RSS-T2; red: UAH-T2 )
0.5
0
-0.5
1980
1985
1990
1995
2000
EXAMPLE I: NOISY DATA
(added noise std=0.1)
ORIGINAL DATA ( b: RSS-T2; r: UAH-T2; g: RSS-T2-esb01; m: UAH-T2-esb01 )
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
1980
1985
1990
1995
2000
NOISY DATA DECOMPOSITION (I)
(added noise std=0.1)
C1 ( blue: RSS-T2; red: UAH-T2 )
0.2
0
-0.2
C2 ( blue: RSS-T2; red: UAH-T2 )
0.2
0
-0.2
C3 ( blue: RSS-T2; red: UAH-T2 )
0.5
0
-0.5
C4 ( blue: RSS-T2; red: UAH-T2 )
0.5
0
-0.5
1980
1985
1990
1995
2000
NOISY DATA DECOMPOSITION (II)
(added noise std=0.1)
C5 ( blue: RSS-T2; red: UAH-T2 )
0.2
0.1
0
-0.1
-0.2
C6 ( blue: RSS-T2; red: UAH-T2 )
0.2
0.1
0
-0.1
-0.2
TREND ( blue: RSS-T2; red: UAH-T2 )
0.5
0
-0.5
1980
1985
1990
1995
2000
EXAMPLE I: NOISY DATA
(added noise std=0.2)
ORIGINAL DATA ( b: RSS-T2; r: UAH-T2; g: RSS-T2-esb02; m: UAH-T2-esb02 )
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
1980
1985
1990
1995
2000
NOISY DATA DECOMPOSITION (I)
(added noise std=0.2)
C1 ( blue: RSS-T2; red: UAH-T2 )
0.2
0
-0.2
C2 ( blue: RSS-T2; red: UAH-T2 )
0.2
0
-0.2
C3 ( blue: RSS-T2; red: UAH-T2 )
0.5
0
-0.5
C4 ( blue: RSS-T2; red: UAH-T2 )
0.5
0
-0.5
1980
1985
1990
1995
2000
NOISY DATA DECOMPOSITION (II)
(added noise std=0.2)
C5 ( blue: RSS-T2; red: UAH-T2 )
0.2
0.1
0
-0.1
-0.2
C6 ( blue: RSS-T2; red: UAH-T2 )
0.2
0.1
0
-0.1
-0.2
TREND ( blue: RSS-T2; red: UAH-T2 )
0.5
0
-0.5
1980
1985
1990
1995
2000
NOISY DATA DECOMPOSITION (II)
(RSS_T2)
Original data ( b: 0.0; r: 0.1; g: 0.2; m: 0.4; k: 1.0 )
1
0.5
0
-0.5
C1 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4; k: 1.0 )
0.2
0
-0.2
C2 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4; k: 1.0 )
0.2
0
-0.2
C3 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4; k: 1.0 )
0.2
0
-0.2
1980
1985
1990
1995
2000
NOISY DATA DECOMPOSITION (II)
(RSS_T2)
C4 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4; k: 1.0 )
1
0.5
0
-0.5
C5 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4; k: 1.0 )
0.2
0
-0.2
C6 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4; k: 1.0 )
0.2
0
-0.2
TREND ( b: 0.0; r: 0.1; g: 0.2; m: 0.4; k: 1.0 )
0.5
0
-0.5
1980
1985
1990
1995
2000
NOISY DATA DECOMPOSITION (II)
(UAH_T2)
C4 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4; k: 1.0 )
1
0.5
0
-0.5
C1 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4; k: 1.0 )
0.2
0
-0.2
C2 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4; k: 1.0 )
0.2
0
-0.2
TREND ( b: 0.0; r: 0.1; g: 0.2; m: 0.4; k: 1.0 )
0.5
0
-0.5
1980
1985
1990
1995
2000
NOISY DATA DECOMPOSITION (II)
(UAH_T2)
C4 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4; k: 1.0 )
1
0.5
0
-0.5
C5 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4; k: 1.0 )
0.2
0
-0.2
C6 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4; k: 1.0 )
0.2
0
-0.2
TREND ( b: 0.0; r: 0.1; g: 0.2; m: 0.4; k: 1.0 )
0.5
0
-0.5
1980
1985
1990
1995
2000
EXAMPLE I: CORR. COEF.’s
CORRELATION COEFFICIENTS ( r: rss00~rss04; b: uaht00~uaht04 )
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
CORRELATION COEFFICIENTS ( r: uaht00~rss04; b: uaht04~rss00 )
1
0.8
0.6
0.4
0.2
0
CORRELATION COEFFICIENTS ( r: uaht00~rss00; b: uaht04~rss04 )
1
0.8
0.6
0.4
0.2
0
6
7
Summary: Radiometer Data
• Data from the two different channels should reflect a similar overall
structure, especially for medium and long wave length.
• Straightforward sifting will have severe mode mixing for medium
scale IMFs.
• It is impossible to select the proper scales for the ‘Intermittence test’
to separate the modes.
• Ensemble EMD provided an automatic dyadic filter to separate the
modes.
• Ensemble EMD especially effective when the data contain
intermittent signal as in UAH case as shown by the correlation
coefficients between RSS and UAH series.
Example II : Geophysical Data
SOI and the Sea Surface temperature
at Nino 34
EXAMPLE II: ORIGINAL DATA
ORIGINAL DATA ( blue: CTI; red: SOI ) corrcoef=-0.5661
5
0
-5
1880
1900
1920
1940
1960
1980
2000
ENLARGEMENT
5
0
-5
1970
1975
1980
1985
1990
1995
2000
EXAMPLE II: DECOMPOSITION (I)
C1 ( blue: SOI; red: CTI )
2
0
-2
C2 ( blue: SOI; red: CTI )
2
0
-2
C3 ( blue: SOI; red: CTI )
2
0
-2
C4 ( blue: SOI; red: CTI )
2
0
-2
1880
1900
1920
1940
1960
1980
2000
EXAMPLE II: DECOMPOSITION (II)
C5 ( blue: SOI; red: CTI )
1
0
-1
C6 ( blue: SOI; red: CTI )
1
0
-1
C7 ( blue: SOI; red: CTI )
1
0
-1
C8 ( blue: SOI; red: CTI )
0.5
0
-0.5
1880
1900
1920
1940
1960
1980
2000
EXAMPLE II: DECOMPOSITION (III)
C9 ( blue: SOI; red: CTI )
0.2
0
-0.2
TREND ( blue: SOI; red: CTI )
0.2
0
-0.2
1880
1900
1920
1940
1960
1980
2000
EXAMPLE II: NOISY DATA
(added noise std=0.4)
ORIGINAL DATA ( b: CTI; r: SOI; g: CTI-esb04; m: SOI-esb04 )
5
0
-5
1880
1900
1920
1940
1960
1980
2000
ENLARGEMENT
5
0
-5
1970
1975
1980
1985
1990
1995
2000
EXAMPLE II: DECOMPOSITION (I)
(added noise std=0.4)
C1 ( blue: SOI; red: CTI )
2
0
-2
C2 ( blue: SOI; red: CTI )
2
0
-2
C3 ( blue: SOI; red: CTI )
2
0
-2
C4 ( blue: SOI; red: CTI )
2
0
-2
1880
1900
1920
1940
1960
1980
2000
EXAMPLE II: DECOMPOSITION (II)
(added noise std=0.4)
C5 ( blue: SOI; red: CTI )
1
0
-1
C6 ( blue: SOI; red: CTI )
1
0
-1
C7 ( blue: SOI; red: CTI )
1
0
-1
C8 ( blue: SOI; red: CTI )
0.5
0
-0.5
1880
1900
1920
1940
1960
1980
2000
EXAMPLE II: DECOMPOSITION (III)
(added noise std=0.4)
C9 ( blue: SOI; red: CTI )
0.2
0
-0.2
TREND ( blue: SOI; red: CTI )
0.2
0
-0.2
1880
1900
1920
1940
1960
1980
2000
CTI: DECOMPOSITION (I)
Original data ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
5
0
-5
C1 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
2
0
-2
C2 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
2
0
-2
C3 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
2
0
-2
1880
1900
1920
1940
1960
1980
2000
CTI: DECOMPOSITION (II)
C4 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
2
0
-2
C5 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
1
0
-1
C6 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
1
0
-1
C7 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
1
0
-1
1880
1900
1920
1940
1960
1980
2000
CTI: DECOMPOSITION (III)
C8 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
0.2
0
-0.2
C9 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
0.2
0
-0.2
TREND ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
0.2
0
-0.2
1880
1900
1920
1940
1960
1980
2000
SOI: DECOMPOSITION (I)
Original data ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
5
0
-5
C1 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
2
0
-2
C2 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
2
0
-2
C3 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
2
0
-2
1880
1900
1920
1940
1960
1980
2000
SOI: DECOMPOSITION (II)
C4 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
2
0
-2
C5 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
1
0
-1
C6 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
1
0
-1
C7 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
1
0
-1
1880
1900
1920
1940
1960
1980
2000
SOI: DECOMPOSITION (III)
C8 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
0.2
0
-0.2
C9 ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
0.2
0
-0.2
TREND ( b: 0.0; r: 0.1; g: 0.2; m: 0.4 )
0.2
0
-0.2
1880
1900
1920
1940
1960
1980
2000
EXAMPLE II: CORR. COEF.’s
CORRELATION COEFFICIENTS ( r: soi00~soi04; b: cti00~cti04 )
1
0.5
0
CORRELATION COEFFICIENTS ( r: cti00~soi04; b: cti04~soi00 )
1
0.5
0
-0.5
-1
CORRELATION COEFFICIENTS ( r: cti00~soi00; b: cti04~soi04 )
1
0.5
0
-0.5
-1
0
1
2
3
4
5
6
7
8
9
10
11
Summary : CTI and SOI
• Straightforward sift produced low correlated
IMFs from the two difference time series.
• Correlation test indicates that the Ensemble
EMD can separate the time scales in the IMFs,
and improve the correlation significantly.
Application of EEMD to Sound
March 30, 2005
Data : Hello
Data : Hello c3y
Data : IMF100
Hilbert Spectrum : IMF100 (500, 12000; 7x7^3)
Hilbert Spectrum : c3y (500, 12000; 7x7^3)
Hilbert Spectrum : IMF100 (500, 12000; 7x7^3)
Hilbert Spectrum : c3y (500, 12000; 7x7^3)
Summary
• True IMFs can be derived from adding
finite amplitude of noise, rather than the
case with infinitesimal noises.
• Ensemble EMD indeed enables the
signals of similar scale collated together.
• No need for a priori criteria for
intermittency.
Summary
• Sum of IMFs may not be an IMF.
• As the components produced by EEMD
are the averaged values of many IMFs,
they might not be IMFs: some of the
component might have multi-extrema.
More stringent stoppage criteria and/or
trials in the ensemble can improve the
situation.
Data
Stoppage Criteria : S=1
Stoppage Criteria : S=10
Stoppage Criteria : S=1 Detail
Stoppage Criteria : S=10 Detail
OI vs. Noise Level
OI vs. Stoppage Criteria
Orthogonal Index of S-Number & Noise Level
Orthogonal Index
0.5
0.11
0.45
0.1
0.4
0.09
0.35
Noise Level
0.08
0.3
0.07
0.25
0.06
0.2
0.05
0.15
0.04
0.03
0.1
0.02
0.05
1
0.01
2
3
4
5
6
S-Number
7
8
9
10
Orthogonal Index
0.5
0.14
0.45
0.12
0.4
0.1
Noise Level
0.35
0.08
0.3
0.25
0.06
0.2
0.04
0.15
0.02
0.1
0.05
248 16 32
0
64
128
Fixed Sifting Number
256
Same as last slide, but with log2 x-axis
Orthogonal Index
0.5
0.14
0.45
0.12
0.4
0.1
Noise Level
0.35
0.08
0.3
0.25
0.06
0.2
0.04
0.15
0.02
0.1
0.05
1
0
2
3
4
5
6
Fixed Sifting Number: Power of 2
7
8
It looks like the fixed-sifting number method can give better orthogonal
index for the same noise-level. But there are several shortcomings for
the fixed-sifting number procedure:
1) un-adaptive IMF’s number. This must be determined by user.
2) Too much computations. For the same case, the fixed sifting number
method requires extremely more computations.
Observations
•
•
•
•
It looks like the fixed-sifting number
method can give better orthogonal index
for the same noise-level.
Fixed sifting number method requires unadaptive IMF’s number. This must be
determined by user.
Too much computations. For the same
case, the fixed sifting number method
requires extremely more computations.
All EEM results might not be IMFs.
Latest Development
• It has been noted that the EEMD could
implemented with the added noise used in
pairs: once with plus sign, and once with
minus sign.
• This pair-wise noise addition could reduce
the noise in the final re-constitution of the
signal.
Conclusions
• Ensemble EMD enables the EMD method to be a
truly dyadic filter bank.
• By adding finite noise, the Ensemble EMD eliminates
mode mixing in most cases automatically.
• Ensemble EMD, utilized the scale separation
principle of EMD, represents a major improvement of
the EMD method.
• The true IMF components should be the results of
the Ensemble EMD rather than from the raw data.
• Sum of IMF is not necessarily an IMF; therefore, the
ensemble EMD results might not be IMFs.
EEMD
Is equivalent to an artificial
ensemble mean of EMDs.