%%
%                            Example6_7.m
%
%
% Construct synthetic data for the 2 parameter spring model.  This data is 
% subsequent used in Example 11.21 and Example 12.11.
%

  clear all

%%
% Specify parameter and measurement variance values.
%

  K = 22.1;
  C = 1.3;
  sigma = .1;
  var = sigma^2;
  
%%
% Construct and save the synthetic data.
%

  t = 0:.01:5;
  error = sigma*randn(size(t));
  n = length(t);

  factor = sqrt(K - C^2/4);
  denom = sqrt(4*K - C^2);
  f = 2*exp(-C*t/2).*cos(factor*t);

  obs = f + error;
  obs(1) = 2;                                 % Enforce the initial condition
  Res = obs - f;
  
  %spring_data = [t' obs'];
  %save('spring_data.txt','spring_data','-ascii')



%%
% Plot the model response, synthetic data, and residuals. One notes in the residual
% plot that the 2 sigma interval contains approximately 95% of the residuals.
%

  figure(1)
  plot(t,f,'k','linewidth',2)
  hold on
  plot(t,obs,'bx','linewidth',1)
  plot(t,0*f,'k','linewidth',2)
  hold off
  set(gca,'Fontsize',[20]);
  legend('Model','Synthetic Data','Location','NorthEast')
  xlabel('Time (s)')
  ylabel('Displacement')
  %print -depsc fig3_6a

  figure(2)
  plot(t,Res,'bx',t,0*f,'k','linewidth',2)
  hold on
  plot(t,2*sigma*ones(size(t)),'--r',t,-2*sigma*ones(size(t)),'--r','linewidth',3)
  hold off
  set(gca,'Fontsize',[20]);
  xlabel('Time (s)')
  ylabel('Residuals')
  %print -depsc fig3_6b