%   
% 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.
% 
%-----

close all; drawnow; clc; clear; clear functions; 

n = 24; 

% Part (a): Holt's method:
% 
beta_0_n = 30; 
beta_1_n = 2;

alpha_1 = 0.2; 
alpha_2 = 0.1; 
omega_1 = 1 - alpha_1;
omega_2 = 1 - alpha_2; 

% the values of the new measurements: 
Zs = [ 28, 31, 36 ]; 

% implement the first Hold update by hand: 
mu_0   = beta_0_n;
beta_0 = beta_1_n;

mus    = zeros(3,1); 
betas  = zeros(3,1); 
z_pred = zeros(3,2);

mus(1)      = ( 1 - omega_1 ) * Zs(1) + omega_1 * ( mu_0 + beta_0 ); 
betas(1)    = ( 1 - omega_2 ) * ( mus(1) - mu_0 ) + omega_2 * beta_0;
z_pred(1,1) = mus(1) + betas(1) * 1;
z_pred(1,2) = mus(1) + betas(1) * 2; 

for ni=2:3
  mus(ni)   = ( 1 - omega_1 ) * Zs(ni) + omega_1 * ( mus(ni-1) + betas(ni-1) );
  betas(ni) = ( 1 - omega_2 ) * ( mus(ni) - mus(ni-1) ) + omega_2 * betas(ni-1);
  z_pred(ni,1) = mus(ni) + betas(ni) * 1;
  z_pred(ni,2) = mus(ni) + betas(ni) * 2; 
end

% Part (b): Double exponential smoothing:
% 

% first convert the initial values of beta_1_n and beta_0_n ... into initial values of our smooth (Equation 3.43 in the book): 
%
alpha = 0.1; w = 1 - alpha; 
S_0_1 = beta_0_n - (     w / ( 1 - w ) ) * beta_1_n;           % these are S_0^{[1]}, and S_0^{[2]} 
S_0_2 = beta_0_n - ( 2 * w / ( 1 - w ) ) * beta_1_n; 

[xHats, S_1, S_2, et, MSE ] = double_exp_smoothing( Zs, 1-alpha, S_0_1, S_0_2 );

% compute the two step look ahead values from S_1 and S_2: 
% 
xHats_2 = ( 2 + (alpha/w) * 2 ) * S_1 - ( 1 + (alpha/w) * 2 ) * S_2; 

[ xHats, xHats_2 ]