How to extract the oscillating components of a signal? A wavelet-based approach compared to the Empirical Mode Decomposition

Transcription

1 How to extract the oscillating components of a signal? A wavelet-based approach compared to the Empirical Mode Decomposition Adrien DELIÈGE University of Liège, Belgium Louvain-La-Neuve, February 7, 27 adrien.deliege@ulg.ac.be

2 Introduction Decomposing series into several modes has become more and more popular and useful in signal analysis. Methods such as EMD, SSA, STFT, EWT, wavelets,... have been successfully applied in medicine, finance, climatology,... Old but gold: Fourier transform allows to decompose a signal as f(t) J k= c k cos(ω k t+ φ k ). Problem: often too many components in the decomposition. Idea: Considering the amplitudes and frequencies as functions of t to decrease the number of terms: f(t)= K k= a k (t)cos(φ k (t)) with K J (AM-FM signals). We will focus on the EMD and a wavelet-based method.

3 EMD Description of the method Illustration 2 WIME Description of the method Illustration 3 EMD vs WIME Crossings in the TF plane Mode-mixing problem Resistance to noise Real-life example: ECG Some conclusions 4 Edge effects The problem A possible solution 5 Wavelets and forecasting? ENSO index Analysis Model and skills Some conclusions

4 EMD EMD Description of the method Illustration 2 WIME Description of the method Illustration 3 EMD vs WIME Crossings in the TF plane Mode-mixing problem Resistance to noise Real-life example: ECG Some conclusions 4 Edge effects The problem A possible solution 5 Wavelets and forecasting? ENSO index Analysis Model and skills Some conclusions

5 EMD Description of the method EMD Empirical Mode Decomposition Empirical = no strong theoretical background Decomposes a signal into IMFs (Intrinsic Mode Functions) Is often used with the Hilbert-Huang transform to represent the IMFs in the TF plane (not shown here).

6 EMD Description of the method EMD ) For a signal X(t), let m, (t)= u,(t)+l, (t) 2 be the mean of its upper and lower envelopes u(t) and l(t) as determined from a cubic-spline interpolation of local maxima and minima. 2) Compute h, (t) as: h, (t)=x(t) m, (t). 3) Now h, (t) is treated as the data, m, (t) is the mean of its upper and lower envelopes, and the process is iterated ( sifting process ): h, (t)=h, (t) m, (t). 4) The sifting process is repeated k s, i.e. until a stopping criterion is satisfied. h,k (t)=h,k (t) m,k (t),

7 EMD Description of the method EMD 5) Then h,k (t) is considered as the component c (t) of the signal and the whole process is repeated with the rest r (t)=x(t) c (t) instead of X(t). Get c 2 (t) then repeat with r 2 (t)=r (t) c 2 (t),... By construction, the number of extrema is decreased when going from r i to r i+, and the whole decomposition is guaranteed to be completed with a finite number of modes. Stopping criterion for the sifting process: When computing m i,j (t), also compute a i,j (t)= u i,j(t) l i,j (t) 2 σ i,j (t)= m i,j (t) a i,j (t) The sifting is iterated until σ(t)<.5 for 95% of the total length of X(t) and σ(t)<.5 for the remaining 5%..

8 EMD Illustration EMD - Illustration Show!

9 WIME EMD Description of the method Illustration 2 WIME Description of the method Illustration 3 EMD vs WIME Crossings in the TF plane Mode-mixing problem Resistance to noise Real-life example: ECG Some conclusions 4 Edge effects The problem A possible solution 5 Wavelets and forecasting? ENSO index Analysis Model and skills Some conclusions

10 WIME Description of the method Continuous wavelet transform Given a wavelet ψ and a function f, the wavelet transform of f at t and at scale a> is defined as ( ) x t dx W f (t,a)= f(x) ψ a a where ψ is the complex conjugate of ψ. We use the wavelet ψ defined by its Fourier transform as ˆψ(ν)=sin ( πν )e (ν Ω)2 2 2Ω with Ω=π 2/ln2, which is similar to the Morlet wavelet but with exactly one vanishing moment.

11 WIME Description of the method CWT Real and Imaginary parts of ψ

12 WIME Description of the method CWT Real and Imaginary parts of ψ compared to a cosine R(W f (,a)), I(W f (,a)) thus W f (,a).

13 WIME Description of the method CWT Real and Imaginary parts of ψ compared to a cosine R(W f (,a)), I(W f (,a)) thus W f (,a).

14 WIME Description of the method CWT Real and Imaginary parts of ψ compared to a cosine R(W f (,a)), I(W f (,a)) thus W f (,a).

15 WIME Description of the method CWT Real and Imaginary parts of ψ compared to a cosine shifted R(W f (,a)) 2/2, I(W f (,a)) 2/2 thus W f (,a).

16 WIME Description of the method CWT One has ˆψ(ν) < 5 if ν thus ψ can be considered as a progressive wavelet (i.e. ˆψ(ν)= if ν ). Property: If f(x)=acos(ωx+ ϕ), then W f (t,a)= A 2 ei(tω+ϕ)ˆψ(aω). Consequence: Given t, if a is the scale at which reaches its maximum, then a W f (t,a) a ω=ω. The value of ω can be obtained (if unknown) and f is recovered as f(x)=2r(w f (x,a ))=2 W f (x,a (x)) cos(argw f (x,a (x))).

17 WIME Description of the method WIME - Wavelet-Induced Mode Extraction ) Perform the CWT of the signal f : W f (t,a). 2) Compute the wavelet spectrum Λ associated with f : a Λ(a)=E t W f (t,a) where E t denotes the mean over. 3) Segment the spectrum to isolate the scale a at which Λ reaches its highest local maximum between the scales a and a 2 at whichλdisplays the left and right local minima that are the closest to a. We set A=[a,a 2 ]. 4) Choose a starting point(t,a(t )) with a(t ) A, e.g. (t,a(t ))=argmax W f (t,a). t, a A

18 WIME Description of the method WIME 5) Compute the ridge t (t,a(t)) forward and backward that stems from (t,a(t )): a) Compute b and b 2 such that b 2 b = a 2 a and a(t )=(b + b 2 )/2, i.e. center a(t ) in a frequency band of the same length as the initial one. b) Among the scales at which the function a [b,b 2 ] W f (t +,a) reaches a local maximum, define a(t + ) as the closest one to a(t ). If there is no local maximum, then a(t + )=a(t ). c) Repeat step 5) with (t +,a(t + )) instead of (t,a(t )) until the end of the signal. d) Proceed in the same way backward from (t,a(t )) until the beginning of the signal. 6) Extract the component associated with the ridge: t 2R(W f (t,a(t)))=2 W f (t,a(t)) cos(argw f (t,a(t))).

19 WIME Description of the method WIME 7) That component is c. The whole process is repeated with the rest r = f c instead of f. Get c 2 then repeat with r 2 (t)=r (t) c 2 (t),... 8) Stop the process when the extracted components are not relevant anymore, e.g. at c n if f n i= c i <.5 f. Alternative stopping criterion : EMD-like method. Very useful:(t,a) W f (t,a) can be seen as a TF representation of f.

20 WIME Illustration WIME - Illustration Show again!

21 f WIME Illustration WIME - Illustration We consider the function f = f + f 2 defined on [,] by { cos(6πt) if t.5 f (t)= cos(8πt 5π) if t >.5 f 2 (t)=cos(πt+ πt 2 )

22 WIME Illustration WIME - Illustration frequency Time-frequency representation of f : (t,a) W f (t,a).

23 WIME Illustration WIME - Illustration frequency frequency spectrum Time-frequency representation of f and spectrum a Λ(a)=E t W f (t,a).

24 WIME Illustration WIME - Illustration frequency First ridge.

25 WIME Illustration WIME - Illustration c f.5.5 First component c extracted and expected component f.

26 WIME Illustration WIME - Illustration frequency Time-frequency representation of r = f c.

27 WIME Illustration WIME - Illustration frequency frequency spectrum Time-frequency representation of r = f c and spectrum.

28 WIME Illustration WIME - Illustration frequency Second ridge.

29 WIME Illustration WIME - Illustration c2 f2.5.5 Second component c 2 extracted from r = f c and expected component f 2.

30 f WIME Illustration WIME - Illustration Original and reconstructed signal.

31 WIME Illustration WIME - Illustration With an AM-FM signal f (t)=(2+sin(5πt))cos((t.5) 3 + t) { (.5+t)cos(.2e f 2 (t)= t + 35t) if t.5 t cos( 3t 2 + t) if t >.5 f

32 WIME Illustration WIME - Illustration frequency frequency c c

33 EMD vs WIME EMD Description of the method Illustration 2 WIME Description of the method Illustration 3 EMD vs WIME Crossings in the TF plane Mode-mixing problem Resistance to noise Real-life example: ECG Some conclusions 4 Edge effects The problem A possible solution 5 Wavelets and forecasting? ENSO index Analysis Model and skills Some conclusions

34 EMD vs WIME Crossings in the TF plane EMD vs WIME Round Crossings in the TF plane

35 EMD vs WIME Crossings in the TF plane EMD vs WIME: Crossings in the TF plane We consider f = f + f 2 + f 3 (on [,]) made of three FM-components with constant amplitudes of.25,,.75: f f (t)=.25cos((t 7) 3 8t) f 2 (t)=cos(36(.5) t 2t) {.75cos(25t+ cos(3t)) if t.5 f 3 (t)=.75cos( 5t t) if t >

36 EMD vs WIME Crossings in the TF plane EMD vs WIME: Crossings in the TF plane - WIME frequency frequency frequency c c2 c

37 EMD vs WIME Crossings in the TF plane EMD vs WIME: Crossings in the TF plane - EMD IMF IMF2 IMF frequency frequency frequency

38 EMD vs WIME Crossings in the TF plane EMD vs WIME: Crossings in the TF plane - WIME-EMD

39 EMD vs WIME Crossings in the TF plane EMD vs WIME: Crossings in the TF plane The influence of the crossings between the patterns in the TF plane remains limited for WIME. The energy-based hierarchy among the components is respected for WIME The EMD follows an upper ridge first scheme and can t proceed otherwise.

40 EMD vs WIME Mode-mixing problem EMD vs WIME Round 2 Mode-mixing problem

41 EMD vs WIME Mode-mixing problem EMD vs WIME: Mode-mixing problem We consider a signal made of AM-FM components that are not well-separated with respect to their frequency nor with their amplitudes. Objective: recover the original frequencies used to build the signal. We consider f = i f i with ( ( )) ( ) 2π 2π f (t)= +.5cos 2 t cos 47 t f 2 (t)= ln(t) ( ) 2π 4 cos 3 t ( ) t 2π f 3 (t)= 6 cos 65 t ( ) f 4 (t)= t 2 cos 2π ( 2π 23+cos 6 t)t. Target frequencies: /47, /3, /65, and /23 Hz. Note that t takes integer values from to 8.

42 f EMD vs WIME Mode-mixing problem EMD vs WIME: Mode-mixing problem Target frequencies: /47, /3, /65, and /23 Hz.

43 EMD vs WIME Mode-mixing problem EMD vs WIME: Mode-mixing problem - WIME /2 /4 /8 /6 /32 /64 /28 / frequency /2 /4 /8 /6 /32 /64 /28 / frequency /2 /4 Target:/47 /8 Detect:/46.3 /6 /32 /64 /28 / spectrum /2 /4 Target:/3 /8 Detect:/3.6 /6 /32 /64 /28 / spectrum

44 EMD vs WIME Mode-mixing problem EMD vs WIME: Mode-mixing problem - WIME /2 /4 /8 /6 /32 /64 /28 / frequency /2 /4 /8 /6 /32 /64 /28 / frequency /2 /4 Target:/65 /8 Detect:/65.5 /6 /32 /64 /28 / spectrum /2 /4 Target:/23 /8 Detect:/2.6 /6 /32 /64 /28 / spectrum

45 EMD vs WIME Mode-mixing problem EMD vs WIME: Mode-mixing problem - WIME c c f f

46 EMD vs WIME Mode-mixing problem EMD vs WIME: Mode-mixing problem - WIME c3 c f3 f

47 EMD vs WIME Mode-mixing problem EMD vs WIME: Mode-mixing problem - EMD IMF IMF IMF2 IMF

48 EMD vs WIME Mode-mixing problem EMD vs WIME: Mode-mixing problem - WIME - EMD Target WIME EMD /23 /2.6 - /3 /3.6 - /47 /46.3 /4 /65 /65.5 / / /284 IMF is almost the signal itself - correlation of.93. EMD cannot resolve the mode-mixing problem. WIME provides accurate information.

49 EMD vs WIME Resistance to noise EMD vs WIME Round 3 Resistance to noise

50 EMD vs WIME Resistance to noise Resistance to noise We consider the chirp f defined on [,] by f(t)=cos(7t+ 3t 2 ) and a Gaussian white noise X of zero mean and variance and we run WIME on f, f + X, f + 2X and f + 3X.

51 EMD vs WIME Resistance to noise Resistance to noise: WIME f.5.5 f WIME with f and f + X. frequency frequency

52 EMD vs WIME Resistance to noise Resistance to noise: WIME f f WIME with f + 2X and f + 3X. frequency frequency

53 EMD vs WIME Resistance to noise Resistance to noise: EMD Not performed since: It is known (and obvious) that EMD is not noise-resistant. It first gives many noisy IMFS. It is not fair to compare EMD with WIME; improved versions of the EMD should be used instead, e.g. Ensemble Empirical Mode Decomposition (EEMD) and Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN). Improvements of EMD are made to the detriment of computational costs. WIME is naturally resistant and the scales to use for the reconstruction can be selected.

54 EMD vs WIME Real-life example: ECG Real-life example: ECG Real-life example Electrocardiogram

55 EMD vs WIME Real-life example: ECG Real-life example: ECG f P frequency c.4.2.2

56 EMD vs WIME Real-life example: ECG Real-life example: ECG.6 frequency c frequency c

57 EMD vs WIME Real-life example: ECG Real-life example: ECG The Dirac-like impulses make it an approximation of an AM-FM signal. WIME extracts valuable information. It decomposes the signal into simpler components easy to analyze. It could be useful to compare hundreds of patients. EMD provides 4 IMFS, many of them are noisy.

58 EMD vs WIME Some conclusions EMD vs WIME: Some conclusions EMD is fully data-driven but sensitive to noise and not flexible (black box). EMD extract components before visualizing them. EMD follows upper ridge first principle, thus have problems with intersecting frequencies and mode mixing. EMD has codes available on the internet.... WIME is flexible but works in the frequency domain. Visualization prior to the analysis allows more freedom. WIME respects the hierarchical structure imposed by the energy of the components thus have better skills when EMD is in trouble. WIME is naturally tolerant to noise. WIME can provide a finer analysis of the data....

59 Edge effects EMD Description of the method Illustration 2 WIME Description of the method Illustration 3 EMD vs WIME Crossings in the TF plane Mode-mixing problem Resistance to noise Real-life example: ECG Some conclusions 4 Edge effects The problem A possible solution 5 Wavelets and forecasting? ENSO index Analysis Model and skills Some conclusions

60 Edge effects The problem Edge effects What you often see in practice 5 C4 25 Period age

61 Edge effects The problem Edge effects In practice: the signal has to be padded at its edges to obtain the CWT. Possibilities: zero-padding constant padding orthogonal symmetry (mirroring) central symmetry (inverse mirroring) periodization If possible, the padding needs to have the same properties as the signal. Zero-padding: universality, independent of the signal.

62 Edge effects The problem Zero-padding Expected modulus for f(t)=cos(ωt+ ϕ): W f (t,ω/ω) =.5. What happens with a simple cosine: Ridge : straight line. The amplitude decreases at the borders. The instantaneous frequency increases or decreases depending on ϕ (not shown). This confirms intuition.

63 Edge effects The problem Zero-padding Expected: W f (t,ω/ω)= 2 eitω thus W f (t,ω/ω) =.5. What happens with a simple cosine f(x) = cos(2π/x): Ridge : straight line. The amplitude decreases at the borders. The instantaneous frequency increases or decreases depending on ϕ (not shown). This confirms intuition.

64 Edge effects The problem Zero-padding Expected: W f (t,ω/ω)= 2 eitω thus W f (t,ω/ω) =.5. What happens with a simple cosine f(x) = cos(2π/x): Ridge : straight line. The amplitude decreases at the borders. The instantaneous frequency increases or decreases depending on ϕ (not shown). This confirms intuition.

65 Edge effects The problem Zero-padding: In theory This is due to the finite length of the signal. Mathematically, in this case, f(x)=cos(ωx)χ ],] (x) and thus for a=ω/ω, W f (t,ω/ω)= 2 eitω z(t) with and R(z(t))= 2 2 π I(z(t))= 2 π ˆψ(Ωx)(x 2 2x ) (x 2 sin(tω( x))dx )(3 x) ˆψ(Ωx) (x+ )(3 x) cos(tω( x))dx.

66 Edge effects The problem Zero-padding: In theory W f (t,ω/ω)= 2 eitω z(t), study of z(t): Amplitude and argument of z as function of t. These confirm intuition and experiments.

67 Edge effects A possible solution A possible solution? Iterations! The theoretical result is difficult to use in practice. All the energy has not be drained from the TF plane, there is still some energy left at the borders. Iterate the extraction process along the same ridge to sharpen the component before getting interested in another ridge. Stop iterations when the component extracted is not significant anymore, e.g. at iteration J if the extracted component at iteration J has less than 95% of the energy of the extracted component at the first extraction.

68 Edge effects A possible solution A possible solution? Iterations! Iteration

69 Edge effects A possible solution A possible solution? Iterations! Iteration

70 Edge effects A possible solution A possible solution? Iterations! Iteration

71 Edge effects A possible solution A possible solution? Iterations! Iteration

72 Edge effects A possible solution A possible solution? Iterations! Iteration

73 Edge effects A possible solution A possible solution? Iterations! Iteration 5 4.5e 4 4e 4 3.5e 4 3e 4 2.5e 4 2e 4.5e 4 e 4 5e 5 e e 4 7e 4 6e 4 5e 4 4e 4 3e 4 2e 4 e 4 e e 4 2e 4 3e 4 4e

74 Edge effects A possible solution A possible solution? Iterations! Iteration

75 Edge effects A possible solution A possible solution? Iterations! Iteration

76 Edge effects A possible solution A possible solution? Iterations! Iteration

77 Edge effects A possible solution A possible solution? Iterations! Iteration

78 Edge effects A possible solution A possible solution? Iterations! Iteration

79 Edge effects A possible solution A possible solution? Iterations! Iteration

80 Edge effects A possible solution A possible solution? Iterations! Iteration

81 Edge effects A possible solution A possible solution? Iterations! Iteration

82 Edge effects A possible solution A possible solution? Iterations! Iteration

83 Wavelets and forecasting? EMD Description of the method Illustration 2 WIME Description of the method Illustration 3 EMD vs WIME Crossings in the TF plane Mode-mixing problem Resistance to noise Real-life example: ECG Some conclusions 4 Edge effects The problem A possible solution 5 Wavelets and forecasting? ENSO index Analysis Model and skills Some conclusions

84 Wavelets and forecasting? Some ideas Perfect correction of border effects Terrific forecasts! Idea: perform the CWT, extract dominant components (with corrected border effects), extrapolate the components (smooth AM-FM signals), then add the components to reconstruct and predict the signal. Great idea. Doesn t work. Instead: build a model based on the information provided by the CWT.

85 Wavelets and forecasting? ENSO index ENSO index Analyzed data: Niño 3.4 series, i.e. monthly-sampled sea surface temperature anomalies in the Equatorial Pacific Ocean from Jan 95 to Dec 24 (

86 Wavelets and forecasting? ENSO index ENSO index Niño 3.4 index: Nino 3.4 index (years) 7 El Niño events: SST anomaly above+.5 C during 5 consecutive months. 4 La Niña events: SST anomaly below.5 C during 5 consecutive months.

87 Wavelets and forecasting? ENSO index 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 impacts in Peru and Chile Impact on snowfalls and monsoons, drier/hotter/wetter/cooler than normal conditions 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 (in 997 notably)....

88 Wavelets and forecasting? Analysis Analysis 5 Period (month)

89 Wavelets and forecasting? Analysis Analysis Amplitude Period (months)

90 Wavelets and forecasting? Analysis Analysis

91 Wavelets and forecasting? Analysis Analysis

92 Wavelets and forecasting? Analysis Analysis Periods of 7,3,43,6,4 months in agreement with previous studies. Period of 34 months can be an artifact; will be neglected. The low frequency components (corresponding to 3,43,6,4 months) capture 9% of the variability of the signal. These components appear relatively stationary thus easier to model.

93 Wavelets and forecasting? Model and skills Idea of the model Model the decadal oscillation and subtract it. Model a 6-months component phased with warm events and subtract it. Model a 3-months component phased with cold events and subtract it. Model a 43-months component phased with remaining warm and cold events. Extrapolate these modeled components and add them to obtain a forecast.

94 Wavelets and forecasting? Model and skills Model Idea: build components that mimic the low-frequency ones and that are easy to extrapolate. Let us assume we have the signal up to T (between 995 and 25).. Model the decadal oscillation. The amplitude A 4 is estimated with the WS of s as.35 and we set y 4 (t)=a 4 cos(2πt/4+2.2). 2. We now work with s = s y 4. The WS of s gives A 6 =.435. Phase y 6 with the strongest warm events of s, which occur approximately every 5 years: find the position p of the last local maximum of s such that s (p)>.5. If s (p)>.9 then we set y 6 (t)=a 6 cos(2π(t p)/6); else y 6 (t)= A 6 cos(2π(t p)/6).

95 Wavelets and forecasting? Model and skills Model 3. We now work with s 2 = s y 6. The WS of s 2 gives A 3 =.42. Phase y 3 with the cold events of s 2, which occur approximately every 2.5 years. Find the position p of the last local minimum of s 2 such that s 2 (p)<.5 and we set y 3 (t)= A 3 cos(2π(t p)/3). 4. We now work with s 3 = s 2 y 3. The WS of s 3 gives A 43 =.485. y 43 has to explain the remaining warm and cold events of s 3. Find the position p of the last local maximum of s 3 such that s 3 (p)>.5 and we set y 43(t)=A 43 cos(2π(t p)/43). Then we find the position p of the last local minimum of s 3 such that s 3 (p)<.8 and we set Finally, we define y 2 43(t)= A 43 cos(2π(t p)/43). y 43 =(y 43+ y 2 43)/2.

96 Wavelets and forecasting? Model and skills Model 5. Extend the signals(y i ) i I up to T + N for N large enough (at least the number of data to be predicted). Then y = y i i I stands for a first reconstruction (for t T ) and forecast (for t > T ) of s. 6. We set s(t)=y(t) for t > T, perform the CWT of s and extract the components ĉ j at scales j corresponding to 6, 2, 7, 3, 43, 6 and 4 months. These are considered as our final AM-FM components and ĉ = j ĉ j both reconstructs (for t T ) and forecasts (for t > T ) the initial ONI signal in a smooth and natural way.

97 Wavelets and forecasting? Model and skills Skills of the model Temperature anomaly Temperature anomaly Temperature anomaly Temperature anomaly

98 Wavelets and forecasting? Model and skills Skills of the model Temperature anomaly Temperature anomaly Temperature anomaly Temperature anomaly

99 Wavelets and forecasting? Model and skills Skills of the model 2 ONI ONI ONI

100 Wavelets and forecasting? Model and skills Skills of the model RMSE Lead (months) Correlation Lead (months) (slightly unfair) comparison with other works.

101 Wavelets and forecasting? Some conclusions Some conclusions The periods detected are in agreement with previous works The information provided by the CWT allows to build a model for long-term forecasting Early signs of major EN and LN events can be detected 2-3 years in advance The ideas could be combined with other models that are better for short-term predictions We could improve the model with seasonal and annual variations We could make the amplitudes vary through The important feature is the phase-locking of the components

102 Wavelets and forecasting? Some conclusions Some references EMD: [4, 5, 9,, 2] and CWT: [, 2, 6, 7, ] WIME: [3, 8] + coming soon

103 Wavelets and forecasting? Some conclusions Thank you Thank you for your attention

104 Wavelets and forecasting? Some conclusions I. Daubechies. Ten Lectures on Wavelets. SIAM, 992. I. Daubechies, J. Lu, and H.-T. Wu. Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool. Journal of Applied and Computational Harmonic Analysis, 3(2):243 26, 2. A. Deliège and S. Nicolay. A new wavelet-based mode decomposition for oscillating signals and comparison with the empirical mode decomposition. In Springer Proceedings of the 3th International Conference on Information Technology : New Generations, Las Vegas (NV), USA, 26. P. Flandrin, G. Rilling, and P. Goncalves. Empirical mode decomposition as a filter bank. IEEE Signal Processing Letters, (2):2 4, 24.

105 Wavelets and forecasting? Some conclusions N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, and H. H. Liu. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary series analysis. Proceedings of the Royal Society of London A, 454:93 995, 998. S. Mallat. A Wavelet Tour of Signal Processing. Academic Press, 999. Y. Meyer and D. Salinger. Wavelets and Operators, volume. Cambridge university press, 995. S. Nicolay. A wavelet-based mode decomposition. European Physical Journal B, 8: , 2.

106 Wavelets and forecasting? Some conclusions G. Rilling, P. Flandrin, and P. Goncalves. On empirical mode decomposition and its algorithms. In IEEE-EURASIP Workshop Nonlinear Signal Image Processing (NSIP), 23. C. Torrence and G. Compo. A practical guide to wavelet analysis. Bulletin of the American Meteorological Society, 79:6 78, 998. M. E. Torres, M. A. Colominas, G. Schlotthauer, and P. Flandrin. A complete ensemble empirical mode decomposition with adaptive noise. In IEEE International Conference on Acoustic, Speech and Signal Processing (ICASSP), 2. Z. Wu and N. E. Huang. Ensemble empirical mode decomposition: a noise-assisted data analysis method. Advances in Adaptative Data Analysis, : 4, 29.