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19th International Conference on Computational Statistics Paris - France, August 22-27 Empirical Mode Decomposition for Trend Extraction. Application to Electrical Data Jean-Michel Poggi Laboratoire de Mathmatique, Universit dOrsay,

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SLIDE 1

Empirical Mode Decomposition for Trend Extraction. Application to Electrical Data

Jean-Michel Poggi

Laboratoire de Mathématique, Université d’Orsay, France

Joint work with Farouk Mhamdi, Mériem Jaïdane-Saïdane

U2S, ENIT, Tunis, Tunisia

19th International Conference

  • n Computational Statistics

Paris - France, August 22-27

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SLIDE 2

Outline

  • Motivation: Blind Trend Extraction
  • Empirical Mode Decomposition (EMD)
  • EMD trend vs. Hodrick Prescott (HP) trend
  • Simulated seasonal series
  • Tunisian daily peak load 2000-2006
  • EMD vs. HP, wavelet trends

This work was supported by 2005/2009 VRR research project between ENIT and Tunisian Society of Electricity and Gas (STEG)

2 EMD for trend extraction

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SLIDE 3

Trend?

  • Trend = some “smooth additive component that contain

information about global change” Alexandrov et al. (2009)

  • The problem: to extract trends from seasonal time series

without strong modeling and global estimation

  • Ex: long term electricity load or airline traffic forecasting

3 EMD for trend extraction

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SLIDE 4

Seasonal time series intrinsic trend

  • Additive time series model:

Y(t) = T(t) + SC(t) + e(t) i.e. trend + seasonal-cycles + error

  • Several methods are used for time series components extraction

including

– local or global regressions – MA filtering, X11, X12 – Hodrick Prescott filter

see Alexandrov et al. (2009) for a recent review

  • The idea: to directly extract trend without identifying the other

components of the observed signal

EMD for trend extraction 4

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SLIDE 5

EMD? EMD and trend?

EMD transformation is nonlinear and suitable for non- stationary signals Huang et al. (2004)

Signal is decomposed as a superposition of local sums of

  • scillatory components called Intrinsic Mode Functions (IMF)
  • f different time scales intrinsic to the signal

Y(t) =  IMFk(t) + r(t) with IMF = function with zero mean and having as many zero crossings as maxima or minima and r = a monotone function

IMFs are fully data-driven and local in time

After the IMF extraction process (sifting) , it remains r(t) a monotone function candidate to be an estimate of T(t) since it is a trend free of oscillatory components

5 EMD for trend extraction

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SLIDE 6

EMD extraction algorithm

EMD for trend extraction 6

Residue = s(t), I1(t) = Residue i = 1, k = 1 while Residue not equal zero or not monotone while Ii has non-negligible local mean U(t) = spline through local maxima of Ii L(t) = spline through local minima of Ii Av(t) = 1/2 (U(t) + L(t)) Ii(t) = Ii(t) - Av(t), i = i + 1 end IMFk(t) = Ii(t) Residue = Residue - IMFk k = k+1 end

Credit: Suz Tolwinski University of Arizona Program in Applied Mathematics Spring 2007 RTG

IMF?

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SLIDE 7

EMD acting on an example Analyzed signal = tone + chirp

tone chirp tone + chirp

Credit: Rilling and Flandrin from http://perso.ens-lyon.fr/patrick.flandrin/emd.html

7 EMD for trend extraction

+ =

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SLIDE 8

EMD acting on an example

iteration 0 - Analyzed signal = starting point

10 20 30 40 50 60 70 80 90 100 110 120

  • 2
  • 1

1 2 IMF 1; iteration 0

8 EMD for trend extraction

Residue = s(t) I1(t) = Residue i = 1 k = 1 while Residue not equal zero or not monotone while Ii has non-negligible local mean U(t) = spline through local maxima of Ii L(t) = spline through local minima of Ii Av(t) = 1/2 (U(t) + L(t)) Ii(t) = Ii(t) - Av(t) i = i + 1 end IMFk(t) = Ii(t) Residue = Residue - IMFk k = k+1 end

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SLIDE 9

EMD acting on an example Extract local maxima

10 20 30 40 50 60 70 80 90 100 110 120

  • 2
  • 1

1 2 IMF 1; iteration 0

9 EMD for trend extraction

Residue = s(t) I1(t) = Residue i = 1 k = 1 while Residue not equal zero or not monotone while Ii has non-negligible local mean U(t) = spline through local maxima of Ii L(t) = spline through local minima of Ii Av(t) = 1/2 (U(t) + L(t)) Ii(t) = Ii(t) - Av(t) i = i + 1 end IMFk(t) = Ii(t) Residue = Residue - IMFk k = k+1 end

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SLIDE 10

EMD acting on an example Maxima envelope by interpolation

10 20 30 40 50 60 70 80 90 100 110 120

  • 2
  • 1

1 2 IMF 1; iteration 0

10 EMD for trend extraction

Residue = s(t) I1(t) = Residue i = 1 k = 1 while Residue not equal zero or not monotone while Ii has non-negligible local mean U(t) = spline through local maxima of Ii L(t) = spline through local minima of Ii Av(t) = 1/2 (U(t) + L(t)) Ii(t) = Ii(t) - Av(t) i = i + 1 end IMFk(t) = Ii(t) Residue = Residue - IMFk k = k+1 end

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SLIDE 11

EMD acting on an example Extract local minima

10 20 30 40 50 60 70 80 90 100 110 120

  • 2
  • 1

1 2 IMF 1; iteration 0

11 EMD for trend extraction

Residue = s(t) I1(t) = Residue i = 1 k = 1 while Residue not equal zero or not monotone while Ii has non-negligible local mean U(t) = spline through local maxima of Ii L(t) = spline through local minima of Ii Av(t) = 1/2 (U(t) + L(t)) Ii(t) = Ii(t) - Av(t) i = i + 1 end IMFk(t) = Ii(t) Residue = Residue - IMFk k = k+1 end

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SLIDE 12

EMD acting on an example Minima envelope by interpolation

10 20 30 40 50 60 70 80 90 100 110 120

  • 2
  • 1

1 2 IMF 1; iteration 0

12 EMD for trend extraction

Residue = s(t) I1(t) = Residue i = 1 k = 1 while Residue not equal zero or not monotone while Ii has non-negligible local mean U(t) = spline through local maxima of Ii L(t) = spline through local minima of Ii Av(t) = 1/2 (U(t) + L(t)) Ii(t) = Ii(t) - Av(t) i = i + 1 end IMFk(t) = Ii(t) Residue = Residue - IMFk k = k+1 end

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SLIDE 13

EMD acting on an example Mean of maxima and minima envelopes

10 20 30 40 50 60 70 80 90 100 110 120

  • 2
  • 1

1 2 IMF 1; iteration 0

13 EMD for trend extraction

Residue = s(t) I1(t) = Residue i = 1 k = 1 while Residue not equal zero or not monotone while Ii has non-negligible local mean U(t) = spline through local maxima of Ii L(t) = spline through local minima of Ii Av(t) = 1/2 (U(t) + L(t)) Ii(t) = Ii(t) - Av(t) i = i + 1 end IMFk(t) = Ii(t) Residue = Residue - IMFk k = k+1 end

Local low frequency component

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SLIDE 14

EMD acting on an example Subtract mean envelope from signal

10 20 30 40 50 60 70 80 90 100 110 120

  • 2
  • 1

1 2 IMF 1; iteration 0 10 20 30 40 50 60 70 80 90 100 110 120

  • 1.5
  • 1
  • 0.5

0.5 1 1.5 residue

14 EMD for trend extraction

Residue = s(t) I1(t) = Residue i = 1 k = 1 while Residue not equal zero or not monotone while Ii has non-negligible local mean U(t) = spline through local maxima of Ii L(t) = spline through local minima of Ii Av(t) = 1/2 (U(t) + L(t)) Ii(t) = Ii(t) - Av(t) (“residue”-->) i = i + 1 end IMFk(t) = Ii(t) Residue = Residue - IMFk k = k+1 end

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SLIDE 15

EMD acting on an example Is the residue a IMF? No

10 20 30 40 50 60 70 80 90 100 110 120

  • 2
  • 1

1 2 IMF 1; iteration 0 10 20 30 40 50 60 70 80 90 100 110 120

  • 1.5
  • 1
  • 0.5

0.5 1 1.5 residue

15 EMD for trend extraction

Residue = s(t) I1(t) = Residue i = 1 k = 1 while Residue not equal zero or not monotone while Ii has non-negligible local mean U(t) = spline through local maxima of Ii L(t) = spline through local minima of Ii Av(t) = 1/2 (U(t) + L(t)) Ii(t) = Ii(t) - Av(t) (“residue”-->) i = i + 1 end IMFk(t) = Ii(t) Residue = Residue - IMFk k = k+1 end

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SLIDE 16

EMD acting on an example No, so iterate the loop (sifting)

10 20 30 40 50 60 70 80 90 100 110 120

  • 1.5
  • 1
  • 0.5

0.5 1 1.5 IMF 1; iteration 1 10 20 30 40 50 60 70 80 90 100 110 120

  • 1.5
  • 1
  • 0.5

0.5 1 1.5 residue

16 EMD for trend extraction

Residue = s(t) I1(t) = Residue i = 1 k = 1 while Residue not equal zero or not monotone while Ii has non-negligible local mean U(t) = spline through local maxima of Ii L(t) = spline through local minima of Ii Av(t) = 1/2 (U(t) + L(t)) Ii(t) = Ii(t) - Av(t) i = i + 1 end IMFk(t) = Ii(t) Residue = Residue - IMFk k = k+1 end

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SLIDE 17

EMD acting on an example IMF1 - iteration 1 - maxima

10 20 30 40 50 60 70 80 90 100 110 120

  • 1.5
  • 1
  • 0.5

0.5 1 1.5 IMF 1; iteration 1 10 20 30 40 50 60 70 80 90 100 110 120

  • 1.5
  • 1
  • 0.5

0.5 1 1.5 residue

17 EMD for trend extraction

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SLIDE 18

EMD acting on an example IMF1 - iteration 1 - minima

10 20 30 40 50 60 70 80 90 100 110 120

  • 1.5
  • 1
  • 0.5

0.5 1 1.5 IMF 1; iteration 1 10 20 30 40 50 60 70 80 90 100 110 120

  • 1.5
  • 1
  • 0.5

0.5 1 1.5 residue

18 EMD for trend extraction

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SLIDE 19

EMD acting on an example IMF1 - iteration 2 – mean

10 20 30 40 50 60 70 80 90 100 110 120

  • 1.5
  • 1
  • 0.5

0.5 1 1.5 IMF 1; iteration 2 10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 residue

19 EMD for trend extraction

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SLIDE 20

EMD acting on an example IMF1 - iteration 3

10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 IMF 1; iteration 3 10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 residue

20 EMD for trend extraction

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SLIDE 21

EMD acting on an example IMF1 - iteration 4

10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 IMF 1; iteration 4 10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 residue

21 EMD for trend extraction

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SLIDE 22

EMD acting on an example IMF1 - iteration 5

10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 IMF 1; iteration 5 10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 residue

22 EMD for trend extraction

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SLIDE 23

EMD acting on an example IMF1 - iteration 6

10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 IMF 1; iteration 6 10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 residue

23 EMD for trend extraction

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SLIDE 24

EMD acting on an example IMF1 - iteration 7

10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 IMF 1; iteration 7 10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 residue

24 EMD for trend extraction

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SLIDE 25

EMD acting on an example IMF1 - iteration 8

10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 IMF 1; iteration 8 10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 residue

IMF1

25 EMD for trend extraction

Residue = s(t) I1(t) = Residue i = 1 k = 1 while Residue not equal zero or not monotone while Ii has non-negligible local mean U(t) = spline through local maxima of Ii L(t) = spline through local minima of Ii Av(t) = 1/2 (U(t) + L(t)) Ii(t) = Ii(t) - Av(t) i = i + 1 end IMFk(t) = Ii(t) Residue = Residue - IMFk k = k+1 end

negligible local mean

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SLIDE 26

EMD acting on an example IMF2 - iteration 0

10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 IMF 2; iteration 0 10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 residue

26 EMD for trend extraction

Residue = s(t) I1(t) = Residue i = 1 k = 1 while Residue not equal zero or not monotone while Ii has non-negligible local mean U(t) = spline through local maxima of Ii L(t) = spline through local minima of Ii Av(t) = 1/2 (U(t) + L(t)) Ii(t) = Ii(t) - Av(t) i = i + 1 end IMFk(t) = Ii(t) Residue = Residue - IMFk k = k+1 end

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SLIDE 27

EMD acting on an example IMF2 – iteration 1

10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 IMF 2; iteration 1 10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 residue

27 EMD for trend extraction

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SLIDE 28

EMD acting on an example IMF2 – iteration 2

10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 IMF 2; iteration 2 10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 residue

28 EMD for trend extraction

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SLIDE 29

EMD acting on an example IMF2 – iteration 3

10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 IMF 2; iteration 3 10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 residue

29 EMD for trend extraction

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SLIDE 30

EMD acting on an example IMF2 – iteration 4

10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 IMF 2; iteration 4 10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 residue

30 EMD for trend extraction

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SLIDE 31

EMD acting on an example IMF2 – iteration 5

10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 IMF 2; iteration 5 10 20 30 40 50 60 70 80 90 100 110 120

  • 1
  • 0.5

0.5 1 residue

IMF2

31 EMD for trend extraction

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SLIDE 32

Trend definitions and EMD

  • No consensus about what is a trend, various definitions

related to data peculiarities and fields of application

  • EMD already used to extract trends :

– Suling et al. 2009, local trends for time scales – Flandrin et al. 2004, sum of nonzero mean IMFs – Zhou et al. 2008, power-system oscillation data

  • This paper: long-term trend for seasonal time series and

comparison with Hodrick Prescott (HP) and a remark about wavelets

  • Three weeks ago:

Moghtaderi, Borgnat, Flandrin (april 2010) in ”Trend Filtering: Empirical Mode Decompositions Versus l1 and Hodrick–Prescott” introduce the energy-ratio approach to select IMFs The idea: select low frequency IMFs from coarse to fine and when it does not differ from some noise reference

32 EMD for trend extraction

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SLIDE 33

Hodrick-Prescott filtering

  • Comparison with the nonparametric trend extraction method based
  • n HP filtering
  • Widely used by economists , Pollock (2003)

Trend = argmin { (y-T)² +  ²(T) }

  • Penalized least squares estimation
  • Usual values for  in economic time series:

33 EMD for trend extraction

Quartely UK Gas data  = 1600 Monthly Airline data  = 14400

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SLIDE 34

HP filter / simulated seasonal time series

  • To select  automatically, see Schlicht (2005)
  • Here, we prefer to use Empirical tuning based on simulated

load curve for   [10²,1015], a bootstrap-like scheme

34 EMD for trend extraction

  • ne week simulated daily load
  • ne year simulated daily peak load
  • and additive trend (linear or exp.)
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SLIDE 35

HP vs EMD and suitable 

  • Calibration of HP parameter  for different kinds of

artificial trends for daily data

  • Comparison of HP and EMD trends for different linear,

quasi linear and exponential trends

  • EMD-trend very close to the optimal HP one

 EMD is an effective alternative for trend extraction

  • Then we can use these values to compare EMD-trend and

“optimal” HP one on real electrical daily data

35 EMD for trend extraction

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SLIDE 36

Tunisian daily peak load 2000-2006

Daily peak load IMFs + final trend

EMD for trend extraction 36

EMD

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SLIDE 37

IMFs / seasonal load components

Sum of IMFs 1-2 Short term week component Sum of IMFs 6-7-8 Long term annual component

12 weeks 4 years

37 EMD for trend extraction

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SLIDE 38

Tunisian daily peak load HP and EMD long-term trends

38 EMD for trend extraction

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SLIDE 39

Tunisian daily peak load HP and EMD trends: end effects

Dealing with EMD end effects:

  • Windowing the signal, Ren et al. (2006)
  • Extrapolate end maxima and minima, Zhaohua et al. (2009)

End effects

39 EMD for trend extraction

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SLIDE 40

Tunisian monthly electrical data HP and EMD trends (1980-2006)

40 EMD for trend extraction

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SLIDE 41

Wavelet trends: A7, A8, A9?

41 EMD for trend extraction

Approximations (/trends?) Details (/IMFs?) Choosing Good Level?

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SLIDE 42

Wavelet trends: daub5, sym8?

42 EMD for trend extraction

Choosing Good Wavelet?

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SLIDE 43

Conclusion

  • EMD eligible method for trend extraction from seasonal

time series

  • Comparison with

– HP Filter: widely used method in economics (?) – Wavelets: another time-scale/frequency method (wavelet?)

  • EMD trend extraction does not require any tuning

parameter thanks to its adaptive nature

  • Perspective: extensions to other trends and comparisons

with proposals from Moghtaderi et al. (2010)

43 EMD for trend extraction

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SLIDE 44

References

  • Alexandrov, Bianconcini, Bee Dagum, Maass, Mcelroy (2009) U.S. Census

Bureau

  • Flandrin, Goncalves, Rilling (2004) EUSIPCO
  • Huang, Shen, Long, Wu, Shih, Zheng, Yen, Tung, Liu (1998) Royal Society

London

  • Ould Mohamed Mahmoud, Mhamdi, Jaidane-Saidane (2009) IEEE PowerTech
  • Moghtaderi, Borgnat, Flandrin (april 2010) Submitted CSDA and AADA
  • Pollock (2003) Journal of Statistical Planning and Inference
  • Ren, Yang, Wu, Yan (2006) International Technology and Innovation Conf.
  • Schlicht (2005) Journal of Japan Statistic Society
  • Suling, Yanqin, Qiang, Jian (2009) World Congress on Engineering and

Computer Science

  • Wu, Huang, Long, Peng (2007) PNAS
  • Zhou, Trudnowski, Pierre, Sarawgi, Bhatt (2008) IEEE Power Elec. Syst.

44 EMD for trend extraction