function [S_n,et,MSE] = simple_exp_smoothing(Z,w,S_0)
% SIMPLE_EXP_SMOOTHING - Performs simple exponential smoothing
% 
% Inputs: 
%   Z  : the original sequence
%   w  : the relaxation factor 
%  S_0 : the initial smooth value 
% 
% Outputs: 
%   S_n : the smoothed sequence
%   et  : the one step-ahead residuals 
%  MSE  : the mean square error   
%
% Written by:
% -- 
% John L. Weatherwax                2009-04-21
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

if( nargin<3 ) S_0 = median(Z); end

n   = length(Z); 
S_n = zeros(n,1); 
et  = zeros(n,1); 

alpha = 1 - w; 

et(1)  = Z(1) - S_0; 
S_n(1) = alpha * Z(1) + w * S_0;              % do the first point by hand ... 

for ii=2:length(Z),                           % do the rest ... 
  et(ii)  = Z(ii) - S_n(ii-1);
  S_n(ii) = alpha * Z(ii) + w * S_n(ii-1);
end

MSE = mean( et.^2 );