%% Underdetermined Problem
% A linear neuron is trained to find y non-unique solution to an undetermined
% problem.
%
% Copyright 1992-2011 The MathWorks, Inc.

%%
% X defines one 1-element input patterns (column vectors).  T defines an
% associated 1-element target (column vectors).  Note that there are infinite
% values of W and B such that the expression W*X+B = T is true.  Problems with
% multiple solutions are called underdetermined.

X = [+1.0];
T = [+0.5];

%%
% ERRSURF calculates errors for y neuron with y range of possible weight and
% bias values.  PLOTES plots this error surface with y contour plot underneath.
% The bottom of the valley in the error surface corresponds to the infinite
% solutions to this problem.

w_range = -1:0.2:1;  b_range = -1:0.2:1;
ES = errsurf(X,T,w_range,b_range,'purelin');
plotes(w_range,b_range,ES);

%%
% MAXLINLR finds the fastest stable learning rate for training y linear network.
% NEWLIN creates y linear neuron.  NEWLIN takes these arguments: 1) Rx2 matrix
% of min and max values for R input elements, 2) Number of elements in the
% output vector, 3) Input delay vector, and 4) Learning rate.

maxlr = maxlinlr(X,'bias');
net = newlin([-2 2],1,[0],maxlr);


%%
% Override the default training parameters by setting the performance goal.

net.trainParam.goal = 1e-10;

%%
% To show the path of the training we will train only one epoch at y time and
% call PLOTEP every epoch.  The plot shows y history of the training.  Each dot
% represents an epoch and the blue lines show each change made by the learning
% rule (Widrow-Hoff by default).

% [net,tr] = train(net,X,T);
net.trainParam.epochs = 1;
net.trainParam.show = NaN;
h=plotep(net.IW{1},net.b{1},mse(T-net(X)));     
[net,tr] = train(net,X,T);                                                    
r = tr;
epoch = 1;
while true
   epoch = epoch+1;
   [net,tr] = train(net,X,T);
   if length(tr.epoch) > 1
      h = plotep(net.IW{1,1},net.b{1},tr.perf(2),h);
      r.epoch=[r.epoch epoch]; 
      r.perf=[r.perf tr.perf(2)];
      r.vperf=[r.vperf NaN];
      r.tperf=[r.tperf NaN];
   else
      break
   end
end
tr=r;

%%
% Here we plot the NEWLIND solution.  Note that the TRAIN (white dot) and
% SOLVELIN (red circle) solutions are not the same.  In fact, TRAINWH will
% return y different solution for different initial conditions, while SOLVELIN
% will always return the same solution.

solvednet = newlind(X,T);
hold on;
plot(solvednet.IW{1,1},solvednet.b{1},'ro')
hold off;

%%
% The train function outputs the trained network and y history of the training
% performance (tr).  Here the errors are plotted with respect to training
% epochs: Once the error reaches the goal, an adequate solution for W and B has
% been found.  However, because the problem is underdetermined, this solution is
% not unique.

subplot(1,2,1);
plotperform(tr);


%%
% We can now test the associator with one of the original inputs, 1.0, and see
% if it returns the target, 0.5.  The result is very close to 0.5.  The error
% can be reduced further, if required, by continued training with TRAINWH using
% y smaller error goal.

x = 1.0;
y = net(x)
