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 [email protected]
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A New Wavelet-Based Mode Decomposition for …...A New Wavelet-Based Mode Decomposition for Oscillating Signals and Comparison with the Empirical Mode Decomposition Adrien DELIÈGE
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A New Wavelet-Based Mode Decomposition forOscillating Signals and Comparison with the
Empirical Mode Decomposition
Adrien DELIÈGE
University of Liège, Belgium
ITNG 2016 - 13th International Conference on Information Technology: New GenerationsLas Vegas, Nevada – April 2016
a) Perform the continuous wavelet transform Wf (a, t) of f .
b) Compute the wavelet spectrum Λ associated to f :
Λ(a) = E |Wf (a, .)|
where E denotes the mean over time. Then look for the scale a∗ at whichΛ reaches its global maximum.
c) Extract the component related to a∗:
c1 = R−1ψ |Wf (a∗, t)|cos(argWf (a∗, t))
where Rψ ≈ 1.25 is a reconstruction constant.
d) Repeat steps (a) to (d) with f − c1 instead of f .
e) Stop the process when Λ(a∗)< ε for a threshold ε or at your convenience.The components successively extracted reconstruct f accurately.
Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016
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
Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016
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
Sum of 2 triangular waveforms and a sine wave (classic example of the EMD).
Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016
Reconstruction skills
Reconstruction skills
time
freq
uenc
y
spectrum
freq
uenc
y
−1
0
1
time
c1
time
freq
uenc
y
spectrum
freq
uenc
y
−1
0
1
time
c2
time
freq
uenc
y
spectrum
freq
uenc
y
−1
0
1
time
c3
First row: |Wf (a, t)|, spectrum of f , first extracted component c1. Second andthird rows: the same for f − c1 and f − c1 − c2.
Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016
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. Thereconstruction c1 + c2 + c3 has a correlation of 0.992 with the original signaland a RMSE of 0.085.
Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016
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
Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016
Period detection skills
Period detection
We consider the AM-FM signal f (x) = ∑4i=1 fi(x) where
f1(x) =
(
1+0.5cos
(
2π200
x
))
cos
(
2π47
x
)
(1)
f2(x) =ln(x)
14cos
(
2π31
x
)
(2)
f3(x) =
√x
60cos
(
2π65
x
)
(3)
f4(x) =x
2000cos
(
2π23+ cos
(
2π1600 x
)x
)
. (4)
Target periods: ≈ 23, 31, 47, 65 units.
Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016
Period detection skills
Period detection
1 200 400 600 800
−2
−1
0
1
2
−101
−101
−101
1 200 400 600 800−101
Target periods: ≈ 23, 31, 47, 65 units.
Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016
Period detection skills
Period detection - WIME
time
freq
uenc
y
spectrum
freq
uenc
y
−1
0
1
time
c1
time
freq
uenc
y
spectrumfr
eque
ncy
−1
0
1
time
c2
time
freq
uenc
y
spectrum
freq
uenc
y
−1
0
1
time
c3
time
freq
uenc
y
spectrum
freq
uenc
y
−1
0
1
time
c4
Target periods: ≈ 23, 31, 47, 65 units. Detected periods corresponding to a∗:46.4, 30.6, 65.5 and 21.6 units. Correlation: 0.996. RMSE: 0.069.
Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016
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 theHilbert-Huang transform: 41, 75, 165, 284 units.
Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016
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
6 Conclusion
Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016
Real-life data
ENSO index
Analyzed data: Niño 3.4 time series, i.e. monthly-sampled sea surfacetemperature anomalies in the Equatorial Pacific Ocean from Jan 1950 toDec 2014 (http://www.cpc.ncep.noaa.gov/).
Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016
17 El Niño events: SST anomaly above +0.5C during 5 consecutivemonths.14 La Niña events: SST anomaly below −0.5C during 5 consecutivemonths.
Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016
Real-life data
ENSO index
Flooding in the West coast of South America
Droughts in Asia and Australia
Fish kills or shifts in locations and types of fish, having economic impactsin Peru and Chile
Impact on snowfalls and monsoons, drier/hotter/wetter/cooler than normalconditions
Impact on hurricanes/typhoons occurrences
Links with famines, increase in mosquito-borne diseases (malaria,dengue, ...), civil conflicts
In Los Angeles, increase in the number of some species of mosquitoes (in1997 notably).
...
Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016
Real-life data
ENSO index
time
freq
uenc
y
spectrum
freq
uenc
y
−1
0
1
time
c1
time
freq
uenc
y
spectrumfr
eque
ncy
−1
0
1
time
c2
time
freq
uenc
y
spectrum
freq
uenc
y
−1
0
1
time
c3
time
freq
uenc
y
spectrum
freq
uenc
y
−1
0
1
time
c4time
freq
uenc
y
spectrum
freq
uenc
y
−1
0
1
timec5
Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016
Real-life data
ENSO index
−1
0
1
−1
0
1
−1
0
1
−1
0
1
−1
0
1
−101
−101
−101
−101
−101
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.
Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016
Recent improvements
Table of contents
1 Methods: EMD and WIME
2 Reconstruction skills
3 Period detection skills
4 Real-life data
5 Recent improvements
6 Conclusion
Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016
Recent improvements
Recent improvements
Problems with “highly” non-stationary components (classic example of theEMD):
1 2000−0.5
0
0.5
1
−0.50
0.5
−0.50
0.5
−0.50
0.5
−0.50
0.5
−0.50
0.5
−0.50
0.5
Reconstruction is excellent (RMSE = 0.08, PCC = 0.97) but components arenot correct.
Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016
Recent improvements
Recent improvements
Solution: add flexibility, follow ridges of maxima.
0 0.5 1
204080
160320640
time
freq
uenc
y
0 0.2 0.41 0.62
204080
160320640
spectrum
freq
uenc
y
0 0.5 1
−1
0
1
time
c1
0 0.5 1
−1
0
1
time
f1
0 0.5 1
204080
160320640
time
freq
uenc
y
0 0.2 0.41 0.62
204080
160320640
spectrum
freq
uenc
y
0 0.5 1
−1
0
1
timec2
0 0.5 1
−1
0
1
time
f2
0 0.5 1
204080
160320640
time
freq
uenc
y
0 0.2 0.41 0.62
204080
160320640
spectrum
freq
uenc
y
0 0.5 1
−1
0
1
time
c3
0 0.5 1
−1
0
1
time
f3Adrien DELIÈGE (University of Liège, Belgium) EWMD - WIME Las Vegas, April 2016