function [w_opt,S_n,et,MSE,omega_grid,MSE_omega] = simple_exp_smoothing_optimum(Z,S_0,omega_grid)
% SIMPLE_EXP_SMOOTHING_OPTIUM - Find the optimum relaxation coefficient to use with simple exponential smoothing
% 
% Inputs: 
%   Z        : the original sequence
%  S_0       : the initial smooth value (optional)
% omega_grid : the grid of possible omega values to search over (optional)
% 
% Outputs: 
% w_opt      : the optimal value of omega the relaxation coefficient
%   S_n      : the optimal smoothed sequence of z_n
%   et       : the optimal one step-ahead residuals 
%  MSE       : the optimal mean square error   
% omega_grid : the grid of omegas searched over (will be the same if passed in)
% MSE_omega  : the in-sample mean square error for each value of omega in omega_grid 
%
% 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 )         % a reasonable default for the grid of omegas to search over
  omega_grid = linspace(0.1,0.99,100); 
end
if( nargin<2 )         % a reasonable default for the initial value of the smooth 
  S_0 = median(Z); 
end

MSE_omega  = zeros(1,length(omega_grid));
for ii=1:length(omega_grid),
  [ Ys, et, MSE_omega(ii) ] = simple_exp_smoothing( Z, omega_grid(ii), S_0 ); 
end

[min_value,min_indx] = min( MSE_omega ); w_opt = omega_grid(min_indx);

[ S_n, et, MSE ] = simple_exp_smoothing( Z, w_opt, S_0 );