The Ensemble Empirical Mode Decompositionhelper.ipam.ucla.edu/publications/ada2013/ada2013_10887.pdf · Imagined Story (physical hypothesis to be tested): The brown bat first stretched

The Ensemble Empirical Mode Decomposition:A Noise-Assisted Data Analysis Method

Zhaohua Wu1, Norden E. Huang2, 3, and Xianyao Chen3

1Department of Earth, Ocean, and Atmospheric Science Florida State University

2Research Center for Adaptive Data AnalysisNational Central University, Taiwan

3The First Institute of OceanographyState Oceanic Administration, China

Jan 28, 2013 IPAM, UCLA

OUTLINE

• The Empirical Mode Decomposition (EMD)

• The Ensemble EMD (EEMD)

• Multi-dimensional EEMD

A SAMPLE INPUT

Jan. 28, 2013 IPAM/UCLA

0 50 100 150 200 250 300 350 400-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Task: To decompose this recorded vocal signal into “physically” meaningful components.

SOME CRETERIA FOR DECOMPOSITION

• From Ingrid Daubechies (Dec. 2008)

Jan. 28, 2013 IPAM, UCLA


“NATURAL” WAVEFORM

( ) ( )

⋅ ∫

t

tdtA

0

cos ττω


EMPIRICAL MODE DECOMP.

receiver

signal source 1

signal source 2

0 100 200 300 400 500 600-6

-4

-2

0

2

4

0 100 200 300 400 500 600-4

-2

0

2

4

0 100 200 300 400 500 600-4

-2

0

2

4

Overall Signal

Norden E Huang


EMPIRICAL MODE DECOMPOSITION1 1

1 2 2

k 1 k k

k 1

x( t ) m h ,h m h ,

.....

.....h m h

.h c

.−

− =

− =

− =

=⇒

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 .

.

−

=

− =

− =

−⇒ =

− =

∑0 50 100 150 200 250 300

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2


SIFTING

-2

0

2INPUT

1 1

1 2 2

k 1 k k

k 1

x( t ) m h ,h m h ,

.....

.....h m h

.h c

.−

− =

− =

− =

=⇒


SIFTING

-2

0

2INPUT

1 1

1 2 2

k 1 k k

k 1

x( t ) m h ,h m h ,

.....

.....h m h

.h c

.−

− =

− =

− =

=⇒


SIFTING

-2

0

2INPUT

1 1

1 2 2

k 1 k k

k 1

x( t ) m h ,h m h ,

.....

.....h m h

.h c

.−

− =

− =

− =

=⇒


SIFTING

-2

0

2INPUT

1 1

1 2 2

k 1 k k

k 1

x( t ) m h ,h m h ,

.....

.....h m h

.h c

.−

− =

− =

− =

=⇒

Black lineBlue line


SIFTING

-2

0

2INPUT

-2

0

2AFTER SIFTING ONE TIME

1 1

1 2 2

k 1 k k

k 1

x( t ) m h ,h m h ,

.....

.....h m h

.h c

.−

− =

− =

− =

=⇒

Blue lineBlack line


SIFTING

-2

0

2INPUT

-2

0


1 1

1 2 2

k 1 k k

k 1

x( t ) m h ,h m h ,

.....

.....h m h

.h c

.−

− =

− =

− =

=⇒


SIFTING

-2

0

2INPUT

-2

0


1 1

1 2 2

k 1 k k

k 1

x( t ) m h ,h m h ,

.....

.....h m h

.h c

.−

− =

− =

− =

=⇒


SIFTING

-2

0

2INPUT

-2

0


1 1

1 2 2

k 1 k k

k 1

x( t ) m h ,h m h ,

.....

.....h m h

.h c

.−

− =

− =

− =

=⇒


SIFTING

-2

0

2INPUT

-2

0


-2

0

2AFTER SIFTING TWO TIME

1 1

1 2 2

k 1 k k

k 1

x( t ) m h ,h m h ,

.....

.....h m h

.h c

.−

− =

− =

− =

=⇒


SIFTING

-2

0

2INPUT

-2

0


-2

0

2AFTER SIFTING TWO TIME

-2

0

2AFTER SIFTING N TIME

IMF

1

1 1

1 2 2

k 1 k k

k 1

x( t ) m h ,h m h ,

.....

.....h m h

.h c

.−

− =

− =

− =

=⇒


SIFTING

-2

0

2INPUT

-2

0

2

IMF

1

-2

0

2

IMF

2

-2

0

2

RE

MA

IND

ER

1 1

1 2 2

k 1 k k

k 1

x( t ) m h ,h m h ,

.....

.....h m h

.h c

.−

− =

− =

− =

=⇒

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 .

.

−

=

− =

− =

−⇒ =

− =

∑

DECOMPOSITION OF BAT VOICE


decomposition of bat voice

TIME-FREQUENCY-AMPLITUDE DIAGRAM OF BAT VOICE








PHYSICAL MEANING


Imagined Story (physical hypothesis to be tested):

The brown bat first stretched tightly its vocal cord, and then let it relax. After some time, an extra “finger” touched the cord, like a violinist plays his/her music, and a second tone came. The vocal cord continued to relax and both tones shifted to lower frequency.





SCALE MIXING PROBLEM

-2

0

2INPUT

0.8

1

1.2

Identifying Maximums

-2

0

2Sifting

200 400 600 800 1000 1200-2

0

2After Sifting Once

time

a

b

c

d



-1

0

1

C1 of the Original Data

-1

0

1

C2 of the Original

-0.2

0

0.2C3 of the Original

0 200 400 600 800 1000 1200-0.2

0

0.2Remainder of the Original

a

b

c

d


IMPLICATION OF SCALE MIXING


PHYSICAL UNIQUENESS

• The Physical Uniqueness (P-U)the decompositions of a data set and of the same data set with added noise perturbation of small but not infinitesimal amplitude bear little quantitative and no qualitative change

• Does P-U Matter in Data Analysis?– Yes, since a data set from real world always contains random

noise

• Does a method currently available satisfy P-U– Fourier Transform does– Wavelet decomposition does– EMD often does not, and is not stable and hard to interpret


AGAIN, WHAT IS DATA ?

• Definition– A collection or representation of facts, concepts, or instructions

in a manner suitable for communication, interpretation, analysis, or processing

data = facts + distortion

X(t) = S(t) + N(t)• Observations

– Observation I

X1(t) = X(t) + N1(t)– Observation II

X2(t) = X(t) + N2(t)


DECOMPOSITION OF DIRAC DELTA DUNCTION

0 50 100 150 200 250 300 350 400 450 50002468

Signa

l

Decomposition of the Dirac function

0 50 100 150 200 250 300 350 400 450 500-4-2024

IMF 1

0 50 100 150 200 250 300 350 400 450 500-4-2024

IMF 2

0 50 100 150 200 250 300 350 400 450 500

-101

IMF 3

0 50 100 150 200 250 300 350 400 450 500

-0.50

0.5

IMF 4

EMD is, in this case, an adaptive wavelet.


DECOMPOSITION OF NOISE

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000-0.4-0.200.2R

-0.50

0.5

C9

-0.50

0.5

C8

-0.50

0.5

C7

-101

C6

-101

C5

-202

C4

-202

C3

-505

C2

-505

C1

-100

10

Raw

Data


PERIODS OF WHITE NOISE COMPONENTS

• A Million Data Points

Period Doubling !!!!!

Mode 1 2 3 4 5 6 7 8 9

White Noise

# peaks 347042 168176 83456 41632 20877 10471 5290 2658 1348

period 2.881 5.946 11.98 24.02 47.90 95.50 189.0 376.2 741.8


FOURIER SPECTRA OF IMFs


DISTRIBUTIONS OF IMFs

• Normal Distribution

-1 0 10

5000

-1 -0.5 0 0.5 10

5000

-0.5 0 0.50

5000

-0.5 0 0.50

5000

-0.4 -0.2 0 0.2 0.40

5000

-0.2 0 0.20

5000

-0.2 -0.1 0 0.1 0.20

5000

-0.1 0 0.10

5000

mode 2 mode 3

mode 4 mode 5

mode 6 mode 7

mode 8 mode 9


USE NOISE TO ASSISTS DECOMPOSITION

• Two qualities – The true signals in data should not be affected by the

observations

– White noise, as a dyadic filter bank in EMD, should provide some control of the width of spectral window of real data decomposition, consequently robust decomposition

• One wishful thinking– adding noise to the targeted data during data analysis

could be helpful — Noise-Assisted Data Analysis (NADA) ?!



-2

0

2INPUT

0.8

1

1.2

Identifying Maximums

-2

0

2Sifting

200 400 600 800 1000 1200-2

0

2After Sifting Once

time

a

b

c

d


NADA — PRELIMINARY TEST (I)

-1

0

1

E1 Data (blue) and the Original Data (red)

-0.2

0

0.2C1 of E1

-0.2

0

0.2C2 of E1

0 200 400 600 800 1000 1200-0.2

0

0.2C3 of E1


NADA — PRELIMINARY TEST (II)

-0.2

0

0.2C4 of E1 Data

-1

0

1

C5 (blue) of E1 and the original data (red)

-0.1

0

0.1C6 of E1 )

0 200 400 600 800 1000 1200-0.1

0

0.1Remainder of E1 )


NOISE-ASSISTED DATA ANALYSIS• Ensemble EMD

– STEP 1: add a noise series to the targeted data

– STEP 2: decompose the data with added noise into IMFs

– STEP 3: repeat STEP 1 and STEP 2 again and again, but with different noise series each time

– STEP 4: obtain the (ensemble) means of corresponding IMFs of the decompositions as the final result

• Effects– In the mean IMFs, the added noise canceled with each other

– The mean IMFs stays within the natural filter period windows (significantly reducing the chance of scale mixing and preserving dyadic property)


EEMD — NADA (I)

-1

0

1

E50 Data (blue) and the Original Data (red)

-0.1

0

0.1C1 of E50

-0.1

0

0.1C2 of E50

0 200 400 600 800 1000 1200-0.1

0

0.1C3 of E50


EEMD — NADA (II)

-0.1

0

0.1C4 of E50 Data

-1

0

1

C5 (blue) of E50 and the original data (red)

-0.1

0

0.1C6 of E50 )

0 200 400 600 800 1000 1200-0.1

0

0.1Remainder of E50 )


DEMONSTRATION OF STABILITY


VOICE: WPD

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000

Wavelet Decomposition

Time: second*22050

S

C7

C6

C5

C4

C3

C2

C1

R

C8


VOICE: EMD

0 2000 4000 6000 8000 10000 12000

Time: Second / 22050

EMD

S

R

C8

C7

C6

C5

C4

C3

C2

C1


VOICE: EEMD

0 2000 4000 6000 8000 10000 12000

Time: Second / 22050

Ensemble EMD: 100 Trials

S

C7

C6

C5

C4

C3

C2

C1

R

C8


0 2000 4000 6000 8000 10000 12000

Time: Second*22050

VOICES

HELLO

C3 of EEMD

C2 of EMD

C5 of WPD

VOICES


EXTENSION TO 2D

• Find extrema in 2D

• Replace 1D envelopes with 2D membranes


DIFFICULTIES IN DIRECT EXTENSION

• 2D: extrema to membranes– Is a saddle point a maximum or minimum?

– Should the ridge (trough) be considered a line of maximum (minimum)

– Scale mixing problem

– Scale unconvertible in temporal-spatial data

• 3D: …– No known surface fitting


DIRECT 2D EEMD

• 2D EEMD– Add noise to 2D data, and decompose noise

added 2D data– Repeat the processes many times– Taking ensemble mean

• Problems– Computationally demanding– Extrema definition difficulty remains– Extension to multi-dimensional EMD

unimaginable


MULTI-DIMENSIONAL EMD/EEMD TEMPORAL-SPATIAL 2D





EVOLUTION OF ENSO

20

TRIGGER OF ENSOFirst Interannual Component (2-3yr)

Order of grid points along the path


SPATIAL 2D EMD


SCHEMATIC OF DECOMPOSITION

C1 C2 C3


SCHEMATIC OF DECOMPOSITION

C1 C2 C3

C1,1

C1,2 C2,2 C2,3

C2,1C3,1


DECOMPOSITION EXAMPLE


MINIMUM SCALE STRATEGY

Among all the components resulted from applying of EEMD in two orthogonal directions, the components that have approximately the same minimal scales are combined to one component


MINIMUM SCALE STRATEGY


THE FINAL COMPONENTS


OCEAN COLOR PICTURE

Original Image and Residuals of Different Levels

Components

Mar. 6, 2012 IPAM, UCLA

EXTENSION TO MULTI-DIMENSIONAL EEMD

COMPUTATIONAL SPEED: O(NlogN), N the total data points


PHYSICAL CONSTRAINTS

1. Later evolution can not change the past

2. What matter to a dynamic system’s future evolution are its initial condition boundary condition, and external forcing

A CB Dt


TEMPORAL LOCALITY

Suppose that the data BC contains physically meaningful oscillation (signal) and an analysis method extracts that oscillation. If the data is extended to AD and the same method is applied to AD, the physically meaningful oscillation within BC should not be changed.

A CB Dt

When a scientific data analysis method is designed, “temporal locality” should be checked.


CONCLUSION

Noise is a

KEY

to unlock thesignals in data

Documents

The Ensemble Empirical Mode Decompositionhelper.ipam.ucla.edu/publications/ada2013/ada2013_10887.pdf · Imagined Story (physical hypothesis to be tested): The brown bat first stretched