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    Metadata details:
    filename: demmlp2.m
    path of file: ./repos/bnt/netlab3.3
    
    the code of the file until where you have to start completion: %DEMMLP2 Demonstrate simple classification using a multi-layer perceptron
%
%	Description
%	The problem consists of input data in two dimensions drawn from a
%	mixture of three Gaussians: two of which are assigned to a single
%	class.  An MLP with logisti
    
function can be trained to model the')
disp('posterior class probabilities in a classification problem.')
disp('The results are compared with the optimal Bayes rule classifier,')
disp('which can be computed exactly as we know the form of the generating')
disp('distribution.')
disp(' ')
disp('Press any key to continue.')
pause

fh1 = figure;
set(fh1, 'Name', 'True Data Distribution');
whitebg(fh1, 'k');

% 
% Generate the data
% 
n=200;

% Set up mixture model: 2d data with three centres
% Class 1 is first centre, class 2 from the other two
mix = gmm(2, 3, 'full');
mix.priors = [0.5 0.25 0.25];
mix.centres = [0 -0.1; 1 1; 1 -1];
mix.covars(:,:,1) = [0.625 -0.2165; -0.2165 0.875];
mix.covars(:,:,2) = [0.2241 -0.1368; -0.1368 0.9759];
mix.covars(:,:,3) = [0.2375 0.1516; 0.1516 0.4125];

[data, label] = gmmsamp(mix, n);

% 
% Calculate some useful axis limits
% 
x0 = min(data(:,1));
x1 = max(data(:,1));
y0 = min(data(:,2));
y1 = max(data(:,2));
dx = x1-x0;
dy = y1-y0;
expand = 5/100;			% Add on 5 percent each way
x0 = x0 - dx*expand;
x1 = x1 + dx*expand;
y0 = y0 - dy*expand;
y1 = y1 + dy*expand;
resolution = 100;
step = dx/resolution;
xrange = [x0:step:x1];
yrange = [y0:step:y1];
% 					
% Generate the grid
% 
[X Y]=meshgrid([x0:step:x1],[y0:step:y1]);
% 
% Calculate the class conditional densities, the unconditional densities and
% the posterior probabilities
% 
px_j = gmmactiv(mix, [X(:) Y(:)]);
px = reshape(px_j*(mix.priors)',size(X));
post = gmmpost(mix, [X(:) Y(:)]);
p1_x = reshape(post(:, 1), size(X));
p2_x = reshape(post(:, 2) + post(:, 3), size(X));

% 
% Generate some pretty pictures !!
% 
colormap(hot)
colorbar
subplot(1,2,1)
hold on
plot(data((label==1),1),data(label==1,2),ClassSymbol1, 'MarkerSize', PointSize)
plot(data((label>1),1),data(label>1,2),ClassSymbol2, 'MarkerSize', PointSize)
contour(xrange,yrange,p1_x,[0.5 0.5],'w-');
axis([x0 x1 y0 y1])
set(gca,'Box','On')
title('The Sampled Data');
rect=get(gca,'Position');
rect(1)=rect(1)-AxisShift;
rect(3)=rect(3)+AxisShift;
set(gca,'Position',rect)
hold off

subplot(1,2,2)
imagesc(X(:),Y(:),px);
hold on
[cB, hB] = contour(xrange,yrange,p1_x,[0.5 0.5],'w:');
set(hB,'LineWidth', 2);
axis([x0 x1 y0 y1])
set(gca,'YDir','normal')
title('Probability Density p(x)')
hold off

drawnow;
clc;
disp('The first figure shows the data sampled from a mixture of three')
disp('Gaussians, the first of which (whose centre is near the origin) is')
disp('labelled red and the other two are labelled yellow.  The second plot')
disp('shows the unconditional density of the data with the optimal Bayesian')
disp('decision boundary superimposed.')
disp(' ')
disp('Press any key to continue.')
pause
fh2 = figure;
set(fh2, 'Name', 'Class-conditional Densities and Posterior Probabilities');
whitebg(fh2, 'w');

subplot(2,2,1)
p1=reshape(px_j(:,1),size(X));
imagesc(X(:),Y(:),p1);
colormap hot
colorbar
axis(axis)
set(gca,'YDir','normal')
hold on
plot(mix.centres(:,1),mix.centres(:,2),'b+','MarkerSize',8,'LineWidth',2)
title('Density p(x|red)')
hold off

subplot(2,2,2)
p2=reshape((px_j(:,2)+px_j(:,3)),size(X));
imagesc(X(:),Y(:),p2);
colorbar
set(gca,'YDir','normal')
hold on
plot(mix.centres(:,1),mix.centres(:,2),'b+','MarkerSize',8,'LineWidth',2)
title('Density p(x|yellow)')
hold off

subplot(2,2,3)
imagesc(X(:),Y(:),p1_x);
set(gca,'YDir','normal')
colorbar
title('Posterior Probability p(red|x)')
hold on
plot(mix.centres(:,1),mix.centres(:,2),'b+','MarkerSize',8,'LineWidth',2)
hold off

subplot(2,2,4)
imagesc(X(:),Y(:),p2_x);
set(gca,'YDir','normal')
colorbar
title('Posterior Probability p(yellow|x)')
hold on
plot(mix.centres(:,1),mix.centres(:,2),'b+','MarkerSize',8,'LineWidth',2)
hold off

% Now set up and train the MLP
nhidden=6;
nout=1;
alpha = 0.2;	% Weight decay
ncycles = 60;	% Number of training cycles. 
% Set up MLP network
net = mlp(2, nhidden, nout, 'logistic', alpha);
options = zeros(1,18);
options(1) = 1;                 % Print out error values
options(14) = ncycles;

mlpstring = ['We now set up an MLP with ', num2str(nhidden), ...
    ' hidden units, logistic output and cross'];
trainstring = ['entropy error function, and train it for ', ...
    num2str(ncycles), ' cycles using the'];
wdstring = ['quasi-Newton optimisation algorithm with weight decay of ', ...
    num2str(alpha), '.'];

% Force out the figure before training the MLP
drawnow;
disp(' ')
disp('The second figure shows the class conditional densities and posterior')
disp('probabilities for each class. The blue crosses mark the centres of')
disp('the three Gaussians.')
disp(' ')
disp(mlpstring)
disp(trainstring)
disp(wdstring)
disp(' ')
disp('Press any key to continue.')
pause

% Convert targets to 0-1 encoding
target=[label==1];

% Train using quasi-Newton.
[net] = netopt(net, options, data, target, 'quasinew');
y = mlpfwd(net, data);
yg = mlpfwd(net, [X(:) Y(:)]);
yg = reshape(yg(:,1),size(X));

fh3 = figure;
set(fh3, 'Name', 'Network Output');
whitebg(fh3, 'k')
subplot(1, 2, 1)
hold on
plot(data((label==1),1),data(label==1,2),'r.', 'MarkerSize', PointSize)
plot(data((label>1),1),data(label>1,2),'y.', 'MarkerSize', PointSize)
% Bayesian decision boundary
[cB, hB] = contour(xrange,yrange,p1_x,[0.5 0.5],'b-');
[cN, hN] = contour(xrange,yrange,yg,[0.5 0.5],'r-');
set(hB, 'LineWidth', 2);
set(hN, 'LineWidth', 2);
Chandles = [hB(1) hN(1)];
legend