% varpro_example.m
% Sample problem illustrating the use of varpro.m
%
% Observations y(t) are given at 10 values of t.
%
% The model for this function y(t) is 
%
%   eta(t) = c1 exp(-alpha2 t)*cos(alpha3 t) 
%          + c2 exp(-alpha1 t)*cos(alpha2 t)
%
% The linear parameters are c1 and c2, and the 
%  nonlinear parameters are alpha1, alpha2, and alpha3.
%
% The two nonlinear functions in the model are
%
%   Phi1(alpha,t) = exp(-alpha2 t)*cos(alpha3 t),
%   Phi2(alpha,t) = exp(-alpha1 t)*cos(alpha2 t).
%
% Dianne P. O'Leary and Bert W. Rust, September 2010

disp('****************************************')
disp('Sample problem illustrating the use of varpro.m')

% Data observations y(t) were taken at these times:

t = [0;.1;.22;.31;.46;.50;.63;.78;.85;.97];

y = [ 6.9842;  5.1851;  2.8907;  1.4199; -0.2473; 
     -0.5243; -1.0156; -1.0260; -0.9165; -0.6805];

% The weights for the least squares fit are stored in w.

w = [ 1.0; 1.0; 1.0; 0.5; 0.5; 1.0; 0.5; 1.0; 0.5; 0.5];       

% Set varpro options to display the progress of the iteration
% and to check the correctness of the derivative formulas.

options = optimset('Display','iter','DerivativeCheck','on');

% Set the initial guess for alpha and call varpro to estimate
% alpha and c.

alphainit = [0.5; 2; 3];  % initial guess

tic
[alpha,c,wresid,resid_norm,y_est,Regression] = ...
     varpro(y,w,alphainit,2,@(alpha)adaex(alpha,t),[],[],options);
toc

% The data y(t) were generated using the parameters 
%         alphatrue = [1.0; 2.5; 4.0], ctrue = [6; 1].  
% Noise was added to the data, so these parameters do not provide
% the best fit.  The computed solution is:
%    Linear Parameters:
%      5.8416357e+00    1.1436854e+00 
%    Nonlinear Parameters:
%      1.0132255e+00    2.4968675e+00    4.0625148e+00 
% Note that there are many sets of parameters that fit the data well.

% Display some regression diagnostics.

Regression
disp('Correlation matrix')
disp(Regression.CorMx)

disp('End of sample problem.')
disp('****************************************')