%  Please read the full homework problem 3 (from homework 2) for a
%  description of what we're doing here.  

%  Start by loading data 

cd ~/matlab/Data
load HW2_prob3.mat

%  Now, let's divide the data (x and y), using the first six years
%  to train a multiple regression equation, and use the last five as an
%  independent verification:

train_set = 1:7;
x_train = x(train_set,:);
y_train = y(train_set,:);

ver_set = 8:11;
x_ind = x(ver_set,:);
y_ind = y(ver_set,:);

%  Start by predicting y using only the first column of x.  Do this using
%  the normal equations:

cols = 1;  %  Use only the first column.

X = [ones(7,1) x_train(:,cols)];
a = (X'*X) \ (X'*y_train);  %  The \ operator acts similarly to the
                            %  inverse function:
                            %    a = inv(X'*X) * (X'*y_train);

%  Generate dependent and independent predictions of y:

y_pred_dep = X * a;
y_pred_ind = [ones(4,1) x_ind(:,cols)] * a;

%  Now, evaluate the explained and unexplained variance of y, and the
%  root mean square error of the forecast based on the dependent and
%  independent predictions of y:

corr(y_pred_dep, y_train)^2    %  Explained variance of y, 
                               %  based on dependent data
1-corr(y_pred_dep, y_train)^2  %  Unexplained variance of y, 
                               %  based on dependent data
sqrt(1-corr(y_pred_dep, y_train)^2)  %  RMSE from dependent data

corr(y_pred_ind, y_ind)^2      %  Explained var., independ. data
1-corr(y_pred_ind, y_ind)^2    %  Unxplained var., independ. data
sqrt(1-corr(y_pred_ind, y_ind)^2)    %  RMSE, independ. data


%  Now, repeat this, adding one predictor time series each time.  To save
%  room, let's just do this in a 'do loop'.  We'll just remember the
%  correlations

%  Initialize outputs (this makes MATLAB happy)
corr_dependent = zeros(6,1);
corr_independent = zeros(6,1);

for i = 1:6;
  
  cols = 1:i;
  X = [ones(7,1) x_train(:,cols)];
  a = (X'*X) \ (X'*y_train);  
  
  y_pred_dep = X * a;
  y_pred_ind = [ones(4,1) x_ind(:,cols)] * a;

  corr_dependent(i) = corr(y_pred_dep, y_train);
  corr_independent(i) = corr(y_pred_ind, y_ind);
  
end

%  Finally, evaluate the explained and unexplained variance, and the RMSE
%  for the dependent and independent data:

var_expl_dependent = corr_dependent.^2;
var_unexpl_dependent = 1-var_expl_dependent;
rmse_dependent = sqrt(var_unexpl_dependent);

var_expl_independent = corr_independent.^2;
var_unexpl_independent = 1-var_expl_independent;
rmse_independent = sqrt(var_unexpl_independent);