eemd.m

%function allmode=eemd(Y,Nstd,NE)
%
% This is an EMD/EEMD program
%
% INPUT:
%       Y: Inputted data;1-d data only
%       Nstd: ratio of the standard deviation of the added noise and that of Y;
%       NE: Ensemble number for the EEMD
% OUTPUT:
%       A matrix of N*(m+1) matrix, where N is the length of the input
%       data Y, and m=fix(log2(N))-1. Column 1 is the original data, columns 2, 3, ...
%       m are the IMFs from high to low frequency, and comlumn (m+1) is the
%       residual (over all trend).
%
% NOTE:
%       It should be noted that when Nstd is set to zero and NE is set to 1, the
%       program degenerates to a EMD program.(for EMD Nstd=0,NE=1)
%       This code limited sift number=10 ,the stoppage criteria can't change. 
%
% References:   
%  Wu, Z., and N. E Huang (2008), 
%  Ensemble Empirical Mode Decomposition: a noise-assisted data analysis method. 
%   Advances in Adaptive Data Analysis. Vol.1, No.1. 1-41.  
%
% code writer: Zhaohua Wu. 
% footnote:S.C.Su 2009/03/04
%
% There are three loops in this code coupled together.
%  1.read data, find out standard deviation ,devide all data by std
%  2.evaluate TNM as total IMF number--eq1. 
%    TNM2=TNM+2,original data and residual included in TNM2
%    assign 0 to TNM2 matrix
%  3.Do EEMD NE times-------------------------------------------------------------loop EEMD start
%     4.add noise
%     5.give initial values before sift 
%     6.start to find an IMF------------------------------------------------IMF loop start
%     7.sift 10 times to get IMF--------------------------sift loop  start  and end
%     8.after 10 times sift --we got IMF
%     9.subtract IMF from data ,and let the residual to find next IMF by loop
%     6.after having all the IMFs---------------------------------------------IMF loop end
%     9.after TNM IMFs ,the residual xend is over all trend
%  3.Sum up NE decomposition result-------------------------------------------------loop EEMD end
% 10.Devide EEMD summation by NE,std be multiply back to data
%
% Association: no
% this function ususally used for doing 1-D EEMD with fixed 
% stoppage criteria independently.
%
% Concerned function: extrema.m 
%                     above mentioned m file must be put together

function allmode=eemd(Y,Nstd,NE)

%part1.read data, find out standard deviation ,devide all data by std
xsize=length(Y);
dd=1:1:xsize;
Ystd=std(Y);
Y=Y/Ystd;

%part2.evaluate TNM as total IMF number,ssign 0 to TNM2 matrix
TNM=fix(log2(xsize))-1;
TNM2=TNM+2;
for kk=1:1:TNM2, 
    for ii=1:1:xsize,
        allmode(ii,kk)=0.0;
    end
end

%part3 Do EEMD  -----EEMD loop start
for iii=1:1:NE,   %EEMD loop -NE times EMD sum together
  
    %part4 --Add noise to original data,we have X1
    for i=1:xsize,
        temp=randn(1,1)*Nstd;
        X1(i)=Y(i)+temp;
    end

    %part4 --assign original data in the first column  
    for jj=1:1:xsize,
        mode(jj,1) = Y(jj);
    end
    
    %part5--give initial 0 to xorigin and xend
    xorigin = X1;
    xend = xorigin;
    
    %part6--start to find an IMF-----IMF loop start
    nmode = 1;
    while nmode <= TNM,
        xstart = xend; %last loop value assign to new iteration loop 
                       %xstart -loop start data
        iter = 1;      %loop index initial value
 
        %part7--sift 10 times to get IMF---sift loop  start 
        while iter<=10,
            [spmax, spmin, flag]=extrema(xstart);  %call function extrema 
            %the usage of  spline ,please see part11.  
            upper= spline(spmax(:,1),spmax(:,2),dd); %upper spline bound of this sift 
            lower= spline(spmin(:,1),spmin(:,2),dd); %lower spline bound of this sift 
            mean_ul = (upper + lower)/2;%spline mean of upper and lower  
            xstart = xstart - mean_ul;%extract spline mean from Xstart
            iter = iter +1;
        end
        %part7--sift 10 times to get IMF---sift loop  end      
        
        %part8--subtract IMF from data ,then let the residual xend to start to find next IMF 
         xend = xend - xstart;
   
   	     nmode=nmode+1;
        
        %part9--after sift 10 times,that xstart is this time IMF 
        for jj=1:1:xsize,
            mode(jj,nmode) = xstart(jj);
        end

    end
    %part6--start to find an IMF-----IMF loop end

    %part 10--after gotten  all(TNM) IMFs ,the residual xend is over all trend
    %                        put them in the last column  
    for jj=1:1:xsize,
        mode(jj,nmode+1)=xend(jj);
    end
    %after part 10 ,original +TNM-IMF+overall trend  ---those are all in mode    
     allmode=allmode+mode;
    
end
%part3 Do EEMD  -----EEMD loop end

%part10--devide EEMD summation by NE,std be multiply back to data
allmode=allmode/NE;
allmode=allmode*Ystd;

%part11--the syntax of the matlab function spline
%yy= spline(x,y,xx); this means
%x and y are matrixs of n1 points ,use n1 set (x,y) to form the cubic spline
%xx and yy are matrixs of n2 points,we want know the spline value yy(y-axis) in the xx (x-axis)position
%after the spline is formed by n1 points ,find coordinate value on the spline for [xx,yy] --n2 position.