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Proceedings ArticleDOI

3A-EMD: A generalized approach for monovariate and multivariate EMD

10 May 2010-pp 300-303

TL;DR: A novel EMD approach called 3A-EMD is proposed, essentially based on the redefinition of the mean envelope operator and allows, under certain conditions, a straightforward decomposition of monovariate and multivariate signals without any change in the core of the algorithm.

AbstractEMD is an emerging topic in signal processing research and is applied in various practical fields. Its recent extension to multivariate signals, motivated by the need to jointly analyze multi-channel signals, is an active topic of research. However, all the existing extensions specifically hold either mono-, bi- or tri-variate signals or require multiple projections that complexify the original process. In this communication, a novel EMD approach called 3A-EMD is proposed. It is essentially based on the redefinition of the mean envelope operator and allows, under certain conditions, a straightforward decomposition of monovariate and multivariate signals without any change in the core of the algorithm. A comparative study with classical monovariate and bivariate methods is presented and shows the competitiveness of 3A-EMD. A trivariate decomposition is also given to illustrate the extension of the proposed algorithm to any signal dimension, D>2.

Summary (1 min read)

1. INTRODUCTION

  • Empirical Mode Decomposition (EMD) was originally in troduced in the late 1990s to study water surface wave evolution [1].
  • The latter components, referred to as the Intrinsic Mode Functions (lMFs), are estimated using an iterative procedure called sifting process.

2. A NEW EMD APPROACH: 3A-EMD

  • The proposed 3A-EMD method aims at providing a sim ple algorithm working for multivariate signals without any modification.
  • For D = 1, the reader could check that the computed extrema using the previous property in clude the signal scalar extrema (of Huang [1]) but also the saddle points.
  • A com plete oscillation is defined between P1 and P2 and the as sociated oscillation barycenter, Mp1--+P2, is obtained by: (5) The mean trend M( {s(t)}) is then computed by interpo lating between the barycenters.
  • (A3) The energy of the different IMF derivatives of the signal should ideally decrease or, at least, the energy of the nth IMF dn should not be "too much" bigger than the en ergy of the following IMFs dn+1.
  • For any given EMD algorithm, let's call N the number of extracted IMF, Kn the number of sifting iterations performed to extract the nth IMF, and dn,k the signal considered at the kth iterations of the sifting pro cess.

3. RESULTS

  • The objective of this section is twofold: to compare the performance of3A-EMD algorithm with classical approa ches and to illustrate its extensibility to multivariate sig nals (D > 2).
  • Let A second criterion evaluates the ability of the algorithm to minimize border effects.
  • Finally, the computational complexity analysis suggests that, even if no significant differences are visible on monovariate signals, 3A-EMD generally requires less sifting iterations and less computa tional operations than the Rilling's method.
  • The trivariate (D > 2) signal 831 projected on the three main axis (from left to right).
  • The three IMFs and the residue are displayed from the top to the bottom of the figure.

4. CONCLUSION AND PERSPECTIVES

  • The obtained results show that, under certain as sumptions on the signals, this alternative definition en ables to decompose both monovariate and multivariate sig nals without any modification in the core of the algorithm.
  • This last point is the one main difference with regard to the existing approaches of the literature.
  • The comparative study also suggests that 3A-EMD seems to offer compet itive core and border performances as well as some in teresting performances in terms of computational com plexity.
  • A restricted scope of application mainly due to the use of the first derivative in the algorithm remains the most important limitation of their approach.
  • As pointed before, those restrictions should not be too much restrictive in practical fields.

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3A-EMD: A Generalized Approach for Monovariate and
Multivariate EMD.
Julien Fleureau, Amar Kachenoura, Jean-Claude Nunes, Laurent Albera, Lot
Senhadji
To cite this version:
Julien Fleureau, Amar Kachenoura, Jean-Claude Nunes, Laurent Albera, Lot Senhadji. 3A-
EMD: A Generalized Approach for Monovariate and Multivariate EMD.. Information Sciences,
Signal Processing and their Applications, May 2010, Kuala Lumpur, Malaysia. pp.300 - 303,
�10.1109/ISSPA.2010.5605465�. �hal-00910730�

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=


 


 


= 










      

 
           


 
       




 
  
           
       

 

 
    
  
      
 
     

  
    
 
       
      

  
 
 
  


       
        
  


    
    

     

 


 

 






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     
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
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
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       
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 Computational complexity of some reference algorithms and
of 3A-EMO method.
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
    
 
   
  















































































 

         


 


























A




 
 
         



 








 

  
 
 







  Comparative study of 3A-EMO versus Huang's (10) and
Rilling's (20) reference (Re methods.


       

       

  
      
   
     

  
  
 
  
  
     
      
 

 
       
  


      
         

 
 
 

              
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



              
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
_
              
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         
 3A-EMD trivariate decomposition of the signal  Dashed dark line: real IMF Continuous red line: estimated IMF.
          
          
          
 
        
 
       

       
   
        
 

 
     
       
    
       


      
 

         
      
        

      
 
 
       

 
          
    


 

   

     

     
     

        
     
   
      
    

    
  
   

       
 
 
  
       
      
   
    
 

 
  

       
 
Citations
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Journal ArticleDOI
TL;DR: A novel EMD approach, which allows for a straightforward decomposition of mono- and multivariate signals without any change in the core of the algorithm, is proposed, and Qualitative results illustrate the good behavior of the proposed algorithm whatever the signal dimension is.
Abstract: Empirical Mode Decomposition (EMD) is an emerging topic in signal processing research, applied in various practical fields due in particular to its data-driven filter bank properties. In this paper, a novel EMD approach called X-EMD (eXtended-EMD) is proposed, which allows for a straightforward decomposition of mono- and multivariate signals without any change in the core of the algorithm. Qualitative results illustrate the good behavior of the proposed algorithm whatever the signal dimension is. Moreover, a comparative study of X-EMD with classical mono- and multivariate methods is presented and shows its competitiveness. Besides, we show that X-EMD extends the filter bank properties enjoyed by monovariate EMD to the case of multivariate EMD. Finally, a practical application on multichannel sleep recording is presented.

69 citations


Cites background or methods from "3A-EMD: A generalized approach for ..."

  • ...Recall that, in [19, 20], an elementary oscillation of a given function s with values in RD (D≥ 1) is considered as a piece of s defined between two consecutive oscillation extrema of s....

    [...]

  • ...In recent works, [19, 20], we proposed an EMD algorithm, called 2T-EMD (Turning Tangent EMD), ables to process mono- and multivariate signals whatever the signal dimension D is, without any change in the algorithm....

    [...]

  • ...Method D Numerical complexity F (dn,k+1) Huang [1] 1 18L+ 15MH(dn,k) Rilling [16] 2 L(11P + 2) + 15 ∑P/2 p=1MR(dn,k, p) Rehman [18] N∗ LP (2D + 18) + 15∑Pp=1MRM(dn,k, p) 2T-EMD [19, 20] N∗ D(19L+ 16M2T(dn,k)) +M2T(dn,k) X-EMD N∗ D(19L+ 15MX(dn,k))...

    [...]

  • ...To deal with this problem, we propose [19, 20] a new and general way to compute extrema in multidimensional output space RD....

    [...]

  • ...As far as the cases of multivariate signals are concerned, two methods, namely 2T-EMD algorithm [19, 20] and Rehman’s method [18], are considered....

    [...]


Journal ArticleDOI
TL;DR: A novel empirical mode decomposition (EMD) algorithm, called 2T-EMD, for both mono- and multivariate signals is proposed in this correspondence, essentially based on a redefinition of the signal mean envelope, computed thanks to new characteristic points, which offers the possibility to decomposeMultivariate signals without any projection.
Abstract: A novel empirical mode decomposition (EMD) algorithm, called 2T-EMD, for both mono- and multivariate signals is proposed in this correspondence. It differs from the other approaches by its computational lightness and its algorithmic simplicity. The method is essentially based on a redefinition of the signal mean envelope, computed thanks to new characteristic points, which offers the possibility to decompose multivariate signals without any projection. The scope of application of the novel algorithm is specified, and a comparison of the 2T-EMD technique with classical methods is performed on various simulated mono- and multivariate signals. The monovariate behaviour of the proposed method on noisy signals is then validated by decomposing a fractional Gaussian noise and an application to real life EEG data is finally presented.

33 citations


Cites background from "3A-EMD: A generalized approach for ..."

  • ...In order to get a unified framework for mono- and multivariate EMD [ 12 ], the signal mean trend and consequently the mean operator have to be redefined....

    [...]

  • ...[ 12 ] J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. New...

    [...]


DissertationDOI
01 Nov 2011

1 citations


References
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Journal ArticleDOI
Abstract: A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the empirical mode decomposition method with which any complicated data set can be dec...

16,171 citations


"3A-EMD: A generalized approach for ..." refers background or methods in this paper

  • ...Comparative study This section compares 3A-EMD per­ formances with those obtained by Huang's algorithm [1] in monovariate case and Rilling's method [7] in bivariate context....

    [...]

  • ...Empirical Mode Decomposition (EMD) was originally in­ troduced in the late 1990s to study water surface wave evolution [1]....

    [...]

  • ...In Huang [1], the mean envelope M( {dn,k (t)}) is cal­ culated as the half sum of the upper and the lower en-...

    [...]

  • ...For D = 1, the reader could check that the computed extrema using the previous property in­ clude the signal scalar extrema (of Huang [1]) but also the saddle points....

    [...]

  • ...Let's also call MHkua ng the number of extrema den, tected in dn,k by the monovariate Huang's procedure [1], MR'k' ll ing the number of extrema detected in the pth pron, ,p jection of dn,k by the second geometrical bivariate Rilling's procedure [7] with P projection planes, and M�1 the num­ ber of barycenters detected in dn,k by 3A-EMD method....

    [...]


01 Jun 2003
TL;DR: Empirical Mode Decomposition is presented, and issues related to its effective implementation are discussed, and an interpretation of the method in terms of adaptive constant-Q filter banks is supported.
Abstract: Huang’s data-driven technique of Empirical Mode Decomposition (EMD) is presented, and issues related to its effective implementation are discussed. A number of algorithmic variations, including new stopping criteria and an on-line version of the algorithm, are proposed. Numerical simulations are used for empirically assessing performance elements related to tone identification and separation. The obtained results support an interpretation of the method in terms of adaptive constant-Q filter banks.

1,380 citations


"3A-EMD: A generalized approach for ..." refers background or methods in this paper

  • ...Several studies dealing with EMD have been reported in the last decade: those addressed to improve the Huang's algorithm [4, 5], and those proposing an extension of EMD to bi- or trivariate signals [6, 7, 8] and even recently to multivariate signals [9]....

    [...]

  • ...Sifting process stop criterion The sifting process termi­ nation is handled by means of a classical modified Cauchy­ like criterion [4]....

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Journal ArticleDOI
TL;DR: The proposed algorithm to use real-valued projections along multiple directions on hyperspheres in order to calculate the envelopes and the local mean of multivariate signals, leading to multivariate extension of EMD.
Abstract: Despite empirical mode decomposition (EMD) becoming a de facto standard for time-frequency analysis of nonlinear and non-stationary signals, its multivariate extensions are only emerging; yet, they are a prerequisite for direct multichannel data analysis. An important step in this direction is the computation of the local mean, as the concept of local extrema is not well defined for multivariate signals. To this end, we propose to use real-valued projections along multiple directions on hyperspheres ( n -spheres) in order to calculate the envelopes and the local mean of multivariate signals, leading to multivariate extension of EMD. To generate a suitable set of direction vectors, unit hyperspheres ( n -spheres) are sampled based on both uniform angular sampling methods and quasi-Monte Carlo-based low-discrepancy sequences. The potential of the proposed algorithm to find common oscillatory modes within multivariate data is demonstrated by simulations performed on both hexavariate synthetic and real-world human motion signals.

676 citations


"3A-EMD: A generalized approach for ..." refers background in this paper

  • ...Several studies dealing with EMD have been reported in the last decade: those addressed to improve the Huang's algorithm [4, 5], and those proposing an extension of EMD to bi- or trivariate signals [6, 7, 8] and even recently to multivariate signals [9]....

    [...]


Journal ArticleDOI
TL;DR: The empirical mode decomposition is extended to bivariate time series that generalizes the rationale underlying the EMD to the bivariate framework and is designed to extract zero-mean rotating components.
Abstract: The empirical mode decomposition (EMD) has been introduced quite recently to adaptively decompose nonstationary and/or nonlinear time series. The method being initially limited to real-valued time series, we propose here an extension to bivariate (or complex-valued) time series that generalizes the rationale underlying the EMD to the bivariate framework. Where the EMD extracts zero-mean oscillating components, the proposed bivariate extension is designed to extract zero-mean rotating components. The method is illustrated on a real-world signal, and properties of the output components are discussed. Free Matlab/C codes are available at http://perso.ens-lyon.fr/patrick.flandrin.

452 citations


"3A-EMD: A generalized approach for ..." refers background or methods in this paper

  • ...Comparative study This section compares 3A-EMD per­ formances with those obtained by Huang's algorithm [1] in monovariate case and Rilling's method [7] in bivariate context....

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  • ...Let's also call MHkua ng the number of extrema den, tected in dn,k by the monovariate Huang's procedure [1], MR'k' ll ing the number of extrema detected in the pth pron, ,p jection of dn,k by the second geometrical bivariate Rilling's procedure [7] with P projection planes, and M�1 the num­ ber of barycenters detected in dn,k by 3A-EMD method....

    [...]

  • ...Several studies dealing with EMD have been reported in the last decade: those addressed to improve the Huang's algorithm [4, 5], and those proposing an extension of EMD to bi- or trivariate signals [6, 7, 8] and even recently to multivariate signals [9]....

    [...]

  • ...More precisely, the second geometric approach proposed in [7] is used as reference for bivariate decompo­ sitions with 8 projection planes....

    [...]


Journal ArticleDOI
TL;DR: A method for the empirical mode decomposition (EMD) of complex-valued data is proposed based on the filter bank interpretation of the EMD mapping and by making use of the relationship between the positive and negative frequency component of the Fourier spectrum.
Abstract: A method for the empirical mode decomposition (EMD) of complex-valued data is proposed. This is achieved based on the filter bank interpretation of the EMD mapping and by making use of the relationship between the positive and negative frequency component of the Fourier spectrum. The so-generated intrinsic mode functions (IMFs) are complex-valued, which facilitates the extension of the standard EMD to the complex domain. The analysis is supported by simulations on both synthetic and real-world complex-valued signals

235 citations


"3A-EMD: A generalized approach for ..." refers background in this paper

  • ...Several studies dealing with EMD have been reported in the last decade: those addressed to improve the Huang's algorithm [4, 5], and those proposing an extension of EMD to bi- or trivariate signals [6, 7, 8] and even recently to multivariate signals [9]....

    [...]


Frequently Asked Questions (2)
Q1. What have the authors contributed in "3a-emd: a generalized approach for monovariate and multivariate emd" ?

However, all the existing extensions specifically hold either mono-, bior tri-variate signals or require mul­ tiple projections that complexify the original process. A comparative study with classical monovariate and bivariate methods is presented and shows the competitiveness of 3A-EMD. 

More simulated and real data decompositions should be performed in future works to verify the 3A-EMD interest in practical context.