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06 Nov 2002 (Updated )

Pattern analysis toolbox.

demmlp2.m
%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 logistic outputs trained with a quasi-Newton
%	optimisation algorithm is compared with the optimal Bayesian decision
%	rule.
%
%	See also
%	MLP, MLPFWD, NETERR, QUASINEW
%

%	Copyright (c) Ian T Nabney (1996-2001)


% Set up some figure parameters
AxisShift = 0.05;
ClassSymbol1 = 'r.';
ClassSymbol2 = 'y.';
PointSize = 12;
titleSize = 10;

% Fix the seeds
rand('state', 423);
randn('state', 423);

clc
disp('This demonstration shows how an MLP with logistic outputs and')
disp('and cross entropy error 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(Chandles, 'Bayes', ...
  'Network', 3);

axis([x0 x1 y0 y1])
set(gca,'Box','on','XTick',[],'YTick',[])

title('Training Data','FontSize',titleSize);
hold off

subplot(1, 2, 2)
imagesc(X(:),Y(:),yg);
colormap hot
colorbar
axis(axis)
set(gca,'YDir','normal','XTick',[],'YTick',[])
title('Network Output','FontSize',titleSize)

clc
disp('This figure shows the training data with the decision boundary')
disp('produced by the trained network and the network''s prediction of')
disp('the posterior probability of the red class.')
disp(' ')
disp('Press any key to continue.')
pause

% 
% Now generate and classify a test data set
% 
[testdata testlabel] = gmmsamp(mix, n);
testlab=[testlabel==1 testlabel>1];

% This is the Bayesian classification
tpx_j = gmmpost(mix, testdata);
Bpost = [tpx_j(:,1), tpx_j(:,2)+tpx_j(:,3)];
[Bcon Brate]=confmat(Bpost, [testlabel==1 testlabel>1]);

% Compute network classification
yt = mlpfwd(net, testdata);
% Convert single output to posteriors for both classes
testpost = [yt 1-yt];
[C trate]=confmat(testpost,[testlabel==1 testlabel>1]);

fh4 = figure;
set(fh4, 'Name', 'Decision Boundaries');
whitebg(fh4, 'k');
hold on
plot(testdata((testlabel==1),1),testdata((testlabel==1),2),...
  ClassSymbol1, 'MarkerSize', PointSize)
plot(testdata((testlabel>1),1),testdata((testlabel>1),2),...
  ClassSymbol2, 'MarkerSize', PointSize)
% Bayesian decision boundary
[cB, hB] = contour(xrange,yrange,p1_x,[0.5 0.5],'b-');
set(hB, 'LineWidth', 2);
% Network decision boundary
[cN, hN] = contour(xrange,yrange,yg,[0.5 0.5],'r-');
set(hN, 'LineWidth', 2);
Chandles = [hB(1) hN(1)];
legend(Chandles, 'Bayes decision boundary', ...
  'Network decision boundary', -1);
axis([x0 x1 y0 y1])
title('Test Data')
set(gca,'Box','On','Xtick',[],'YTick',[])

clc
disp('This figure shows the test data with the decision boundary')
disp('produced by the trained network and the optimal Bayes rule.')
disp(' ')
disp('Press any key to continue.')
pause

fh5 = figure;
set(fh5, 'Name', 'Test Set Performance');
whitebg(fh5, 'w');
% Bayes rule performance
subplot(1,2,1)
plotmat(Bcon,'b','k',12)
set(gca,'XTick',[0.5 1.5])
set(gca,'YTick',[0.5 1.5])
grid('off')
set(gca,'XTickLabel',['Red   ' ; 'Yellow'])
set(gca,'YTickLabel',['Yellow' ; 'Red   '])
ylabel('True')
xlabel('Predicted')
title(['Bayes Confusion Matrix (' num2str(Brate(1)) '%)'])

% Network performance
subplot(1,2, 2)
plotmat(C,'b','k',12)
set(gca,'XTick',[0.5 1.5])
set(gca,'YTick',[0.5 1.5])
grid('off')
set(gca,'XTickLabel',['Red   ' ; 'Yellow'])
set(gca,'YTickLabel',['Yellow' ; 'Red   '])
ylabel('True')
xlabel('Predicted')
title(['Network Confusion Matrix (' num2str(trate(1)) '%)'])

disp('The final figure shows the confusion matrices for the')
disp('two rules on the test set.')
disp(' ')
disp('Press any key to exit.')
pause
whitebg(fh1, 'w');
whitebg(fh2, 'w');
whitebg(fh3, 'w');
whitebg(fh4, 'w');
whitebg(fh5, 'w');
close(fh1); close(fh2); close(fh3);
close(fh4); close(fh5);
clear all;

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