[Beamer] A New Wavelet-Based Mode Decomposition for Oscillating Signals and Comparison with the Empirical Mode Decomposition

A New Wavelet-Based Mode Decomposition for

Oscillating Signals and Comparison with the

Empirical Mode Decomposition

Adrien DELIÈGE University of Liège, Belgium

ITNG 2016 - 13th International Conference on Information Technology: New Generations Las Vegas, Nevada – April 2016

Joint work with S. NICOLAY

Introduction

Decomposingtime series into severalmodeshas become more and more

popular andusefulin signal analysis.

Methods such as EMD or SSA (among others) have been successfully

applied inmedicine, finance, climatology, ...

Old but gold: Fouriertransform allows to decompose a signal as

f

(

t

_{) ≈}

K

∑

k=1

ckcos(

ω

kt

+

φ

k

).

Problem: oftentoo many componentsin the decomposition.

Idea: Considering theamplitudes and frequencies as functions of tto

decrease the number of terms.

Development of an empirical wavelet mode decomposition(EWMD)

method which can be used in numerous ways.

Edit: EWMD becomesWIME(wavelet-induced mode extraction).

Table of contents

1 Methods: EMD and WIME

2 Reconstruction skills

3 _{Period detection skills}

4 _{Real-life data}

5 _{Recent improvements}

Methods: EMD and WIME

Table of contents

1 Methods: EMD and WIME

2 Reconstruction skills

3 _{Period detection skills}

4 _{Real-life data}

5 _{Recent improvements}

6 Conclusion

Methods: EMD and WIME

Empirical Mode Decomposition (EMD) in a nutshell

1. Given a signal f , compute m the mean of its upper and lowerenvelopes.

2. Compute h

=

f

₋

m andrepeatsteps 1.and 2.with h instead of funtilh is

“stableenough” and becomes an “intrinsic mode component” (IMF), f1.

3. Repeatthe whole process with f

₋

f1instead of f .

4. The procedure alwaysstopsand the IMFs successively extracted

reconstructf accurately.

Maindrawbacksinclude: lack of solid mathematical theoretical background, high sensitivity to noise, mode-mixing problems.

Methods: EMD and WIME

Empirical Wavelet Mode Decomposition (EWMD) – WIME

Thewavelet usedin this study is the function

ψ

(

t

) =

e iΩt 2

√

2

π

e −(2Ω8tΩ+2π)2



eπΩt

₊₁



with

Ω =

π

p

2/ln 2, which is similar to theMorletwavelet. TheFourier

transform of

ψ

is given by

ˆ

ψ

(

ω

) =

sin



πω

2Ω



e−(ω−Ω) 2 2

.

Thewavelet transformof the signal is computed as:

Wf

(

a

,

t

) =

Z f

(

x

)¯

ψ



x

₋

t a



dx a

,

where

ψ

¯

is the complex conjugate of

ψ

, t

_{∈ R}

stands for the location/time

parameter and a

>

0 denotes thescaleparameter.

Methods: EMD and WIME

Empirical Wavelet Mode Decomposition (EWMD) – WIME

a) Perform the continuouswavelet transformWf

(

a

,

t

)

of f .

b) Compute thewavelet spectrum

Λ

associated to f :

Λ(

a

) =

E

_|

Wf

(

a

, .)

_|

where E denotes the mean over time. Then look for thescalea∗at which

Λ

reaches its globalmaximum.

c) Extractthe component related to a∗:

c1

=

Rψ−1

|

Wf

(

a∗

,

t

)|

cos(arg Wf

(

a∗

,

t

))

where R_ψ

_≈

1.25 is a reconstruction constant.

d) Repeatsteps (a) to (d) with f

₋

c1instead of f .

e) Stopthe process when

Λ(

a∗

) <

ε

for a threshold

ε

or at your convenience.

Reconstruction skills

Table of contents

1 Methods: EMD and WIME

2 Reconstruction skills

3 _{Period detection skills}

4 _{Real-life data}

5 _{Recent improvements}

6 Conclusion

Reconstruction skills Reconstruction skills First example: 1 1024 −1.5 0 1.5 −1 0 1 −1 0 1 −1 0 1 −1 0 1

Reconstruction skills Reconstruction skills time frequency spectrum frequency −1 0 1 time c1 time frequency spectrum frequency −1 0 1 time c2 time frequency spectrum frequency −1 0 1 time c3

First row:

_|

Wf

(

a

,

t

)

_|

, spectrum of f , first extracted component c1. Second and third rows: the same for f

₋

c1and f

−

c1

−

c2.

Reconstruction skills

Reconstruction skills

WIME Expected EMD

−1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1

First column: components extracted with WIME. Second column: the real (expected) components. Third column: the IMFs extracted with the EMD. The reconstruction c1

+

c2

+

c3has a correlation of 0.992 with the original signal and a RMSE of 0.085.

Period detection skills

Table of contents

1 Methods: EMD and WIME

2 Reconstruction skills

3 _{Period detection skills}

4 _{Real-life data}

5 _{Recent improvements}

6 Conclusion

Period detection skills

Period detection

We consider the AM-FM signal f

(

x

) =

∑

4

i=1fi

(

x

)

where f1

(

x

) =



1

+

0.5 cos



2

π

200x



cos



2

π

47x



(1) f2

(

x

) =

ln(x

)

14 cos



2

π

31x



(2) f3

(

x

) =

√

x 60 cos



2

π

65x



(3) f4

(

x

) =

x 2000cos 2

π

23

+

cos ₁₆₀₀2π x

x

!

.

(4)

Period detection skills Period detection 1 200 400 600 800 −2 −1 0 1 2 −1 0 1 −1 0 1 −1 0 1 1 200 400 600 800 −1 0 1

Target periods:

_≈

23, 31, 47, 65 units.

Period detection skills

Period detection - WIME

time frequency spectrum frequency −1 0 1 time c1 time frequency spectrum frequency −1 0 1 time c2 time frequency spectrum frequency −1 0 1 time c3 time frequency spectrum frequency −1 0 1 time c4

Target periods:

_≈

23, 31, 47, 65 units. Detected periods corresponding to a∗:

Period detection skills

Period detection - EMD

−1 0 1 −1 0 1 1 200 400 600 800 −1 0 1 1 200 400 600 800 −1 0 1

Target periods:

_≈

23, 31, 47, 65 units. Periods extracted from the

Hilbert-Huang transform: 41, 75, 165, 284 units.

Real-life data

Table of contents

1 Methods: EMD and WIME

2 Reconstruction skills

3 _{Period detection skills}

4 _{Real-life data}

5 _{Recent improvements}

Real-life data

ENSO index

Analyzed data: Niño 3.4 time series, i.e. monthly-sampled sea surface temperature anomalies in the Equatorial Pacific Ocean from Jan 1950 to Dec 2014 (http://www.cpc.ncep.noaa.gov/).

Real-life data ENSO index Niño 3.4 index: 1950 1960 1970 1980 1990 2000 2010 2020 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 time (years) Nino 3.4 index

17 El Niño events: SST anomaly above

+0.5

◦_{C during 5 consecutive}

months.

14 La Niña events: SST anomaly below

₋

0.5◦_{C during 5 consecutive}

Real-life data

ENSO index

Floodingin the West coast of South America

Droughtsin Asia and Australia

Fish killsor shifts in locations and types of fish, havingeconomic impacts

in Peru and Chile

Impact on snowfalls andmonsoons, drier/hotter/wetter/cooler than normal

conditions

Impact onhurricanes/typhoonsoccurrences

Links with famines, increase inmosquito-borne diseases(malaria,

dengue, ...), civil conflicts

In Los Angeles, increase in the number of some species of mosquitoes (in 1997 notably).

...

Real-life data ENSO index time frequency spectrum frequency ₋₁ 0 1 time c1 time frequency spectrum frequency ₋₁ 0 1 time c2 time frequency spectrum frequency ₋₁ 0 1 time c3 time frequency spectrum frequency ₋₁ 0 1 time c4 time frequency spectrum frequency ₋₁ 0 1 time c5

Real-life data ENSO index −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1 −1 0 1

Components extracted with WIME and the IMFs from the EMD. Periods: 44.8, 28.6, 17, 65.6, 140.6 months (WIME) and 9.8, 21, 38.6, 75.9, 138.4 (EMD). Those from WIME are more in agreement with some previous works.

Recent improvements

Table of contents

1 Methods: EMD and WIME

2 Reconstruction skills

3 _{Period detection skills}

4 _{Real-life data}

5 _{Recent improvements}

Recent improvements

Recent improvements

Problems with “highly” non-stationary components (classic example of the EMD): 1 2000 −0.5 0 0.5 1 −0.5 0 0.5 −0.5 0 0.5 −0.5 0 0.5 −0.5 0 0.5 −0.5 0 0.5 −0.5 0 0.5

Reconstruction is excellent (RMSE

=

0.08, PCC

=

0.97) but components are

not correct.

Recent improvements

Recent improvements

Solution: add flexibility, follow ridges of maxima.

0 0.5 1 20 40 80 160 320 640 time frequency 0 0.2 0.41 0.62 20 40 80 160 320 640 spectrum frequency 0 0.5 1 −1 0 1 time c1 0 0.5 1 −1 0 1 time f1 0 0.5 1 20 40 80 160 320 640 time frequency 0 0.2 0.41 0.62 20 40 80 160 320 640 spectrum frequency 0 0.5 1 −1 0 1 time c2 0 0.5 1 −1 0 1 time f2 0 0.5 1 20 40 80 160 320 640 time frequency 0 0.2 0.41 0.62 20 40 80 160 320 640 spectrum frequency 0 0.5 1 −1 0 1 time c3 0 0.5 1 −1 0 1 time f3

Recent improvements

Recent improvements

Crossings in the TF plane:

f1

(

t

) =

1.25 cos



−(

t

₋

3)3

+

180t



(5) f2

(

t

) =

cos 1.8t

+

20t

(6) f3

(

t

) =



0.75 cos

₍₋

1.6

π

t2

+

20

π

t

)

(t

<

5) 0.75 cos(4

π

t

)

(t

_≥

5) (7) 0 10 20 −3 −2 −1 0 1 2 3 time f

Recent improvements Recent improvements WIME 0 10 20 1.25 2.5 5 10 20 40 80 time frequency 0 0.39 0.78 1.17 1.25 2.5 5 10 20 40 80 spectrum frequency 0 10 20 −2 −1 0 1 2 time c1 0 10 20 −2 −1 0 1 2 time f1 0 10 20 1.25 2.5 5 10 20 40 80 time frequency 0 0.39 0.78 1.17 1.25 2.5 5 10 20 40 80 spectrum frequency 0 10 20 −2 −1 0 1 2 time c2 0 10 20 −2 −1 0 1 2 time f2 0 10 20 1.25 2.5 5 10 20 40 80 time frequency 0 0.39 0.78 1.17 1.25 2.5 5 10 20 40 80 spectrum frequency 0 10 20 −2 −1 0 1 2 time c3 0 10 20 −2 −1 0 1 2 time f3

Recent improvements Recent improvements EMD 0 10 20 1.25 2.5 5 10 20 40 80 time frequency 0 10 20 −2 −1 0 1 2 time IMF1 0 10 20 1.25 2.5 5 10 20 40 80 time frequency 0 10 20 −2 −1 0 1 2 time IMF2 0 10 20 1.25 2.5 5 10 20 40 80 time frequency 0 10 20 −2 −1 0 1 2 time IMF3

Conclusion

Table of contents

1 Methods: EMD and WIME

2 Reconstruction skills

3 _{Period detection skills}

4 _{Real-life data}

5 _{Recent improvements}

Conclusion

Conclusion

WIME

is a wavelet-baseddecompositionmethod

hasreconstruction skillscomparable to the EMD

hasperiod detection skillsbetter than the EMD in mode-mixing problems

handlescrossing patternsin the TF plane

gives accurate results withreal-lifesignals

Future works: study the mathematical properties of the method, test its tolerance to noise, limit edge effects,...

Thanks for your attention.

Please refer to the paper for relevant references.