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

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.

Abstract: EMD 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. 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.

Figures

Fig. 1. Comparative study of 3A-EMO versus Huang's (10) and Rilling's (20) reference (Ref) methods.

Full text

HAL Id: hal-00910730
https://hal.archives-ouvertes.fr/hal-00910730
Submitted on 6 Dec 2013
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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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
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
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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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
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






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









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



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

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


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
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





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




















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

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

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
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


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

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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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_
              
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         
 3A-EMD trivariate decomposition of the signal  Dashed dark line: real IMF Continuous red line: estimated IMF.
          
          
          
 
        
 
       

       
   
        
 

 
     
       
    
       


      
 

         
      
        

      
 
 
       

 
          
    


 

   

     

     
     

        
     
   
      
    

    
  
   

       
 
 
  
       
      
   
    
 

 
  

       
 

Frequently asked questions

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. 

Q2. What future works have the authors mentioned in the paper "3a-emd: a generalized approach for monovariate and multivariate emd" ?

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