diff toolboxes/FullBNT-1.0.7/netlab3.3/demgmm4.m @ 0:e9a9cd732c1e tip

first hg version after svn

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+%DEMGMM4 Demonstrate density modelling with a Gaussian mixture model.
+%
+%	Description
+%	 The problem consists of modelling data generated by a mixture of
+%	three Gaussians in 2 dimensions with a mixture model using full
+%	covariance matrices.  The priors are 0.3, 0.5 and 0.2; the centres
+%	are (2, 3.5), (0, 0) and (0,2); the variances are (0.16, 0.64) axis
+%	aligned, (0.25, 1) rotated by 30 degrees and the identity matrix. The
+%	first figure contains a scatter plot of the data.
+%
+%	A Gaussian mixture model with three components is trained using EM.
+%	The parameter vector is printed before training and after training.
+%	The user should press any key to continue at these points.  The
+%	parameter vector consists of priors (the column), and centres (given
+%	as (x, y) pairs as the next two columns).  The covariance matrices
+%	are printed separately.
+%
+%	The second figure is a 3 dimensional view of the density function,
+%	while the third shows the axes of the 1-standard deviation ellipses
+%	for the three components of the mixture model.
+%
+%	See also
+%	GMM, GMMINIT, GMMEM, GMMPROB, GMMUNPAK
+%
+
+%	Copyright (c) Ian T Nabney (1996-2001)
+
+
+% Generate the data
+
+ndata = 500;
+
+% Fix the seeds for reproducible results
+randn('state', 42);
+rand('state', 42);
+data = randn(ndata, 2);
+prior = [0.3 0.5 0.2];
+% Mixture model swaps clusters 1 and 3
+datap = [0.2 0.5 0.3];
+datac = [0 2; 0 0; 2 3.5];
+datacov = repmat(eye(2), [1 1 3]);
+data1 = data(1:prior(1)*ndata,:);
+data2 = data(prior(1)*ndata+1:(prior(2)+prior(1))*ndata, :);
+data3 = data((prior(1)+prior(2))*ndata +1:ndata, :);
+
+% First cluster has axis aligned variance and centre (2, 3.5)
+data1(:, 1) = data1(:, 1)*0.4 + 2.0;
+data1(:, 2) = data1(:, 2)*0.8 + 3.5;
+datacov(:, :, 3) = [0.4*0.4 0; 0 0.8*0.8];
+
+% Second cluster has variance axes rotated by 30 degrees and centre (0, 0)
+rotn = [cos(pi/6) -sin(pi/6); sin(pi/6) cos(pi/6)];
+data2(:,1) = data2(:, 1)*0.5;
+data2 = data2*rotn;
+datacov(:, :, 2) = rotn' * [0.25 0; 0 1] * rotn;
+
+% Third cluster is at (0,2)
+data3 = data3 + repmat([0 2], prior(3)*ndata, 1);
+
+% Put the dataset together again
+data = [data1; data2; data3];
+
+clc
+disp('This demonstration illustrates the use of a Gaussian mixture model')
+disp('with full covariance matrices to approximate the unconditional ')
+disp('probability density of data in a two-dimensional space.')
+disp('We begin by generating the data from a mixture of three Gaussians and')
+disp('plotting it.')
+disp(' ')
+disp('The first cluster has axis aligned variance and centre (0, 2).')
+disp('The second cluster has variance axes rotated by 30 degrees')
+disp('and centre (0, 0).  The third cluster has unit variance and centre')
+disp('(2, 3.5).')
+disp(' ')
+disp('Press any key to continue.')
+pause
+
+fh1 = figure;
+plot(data(:, 1), data(:, 2), 'o')
+set(gca, 'Box', 'on')
+
+% Set up mixture model
+ncentres = 3;
+input_dim = 2;
+mix = gmm(input_dim, ncentres, 'full');
+
+% Initialise the model parameters from the data
+options = foptions;
+options(14) = 5;	% Just use 5 iterations of k-means in initialisation
+mix = gmminit(mix, data, options);
+
+% Print out model
+clc
+disp('The mixture model has three components and full covariance')
+disp('matrices.  The model parameters after initialisation using the')
+disp('k-means algorithm are as follows')
+disp('    Priors        Centres')
+disp([mix.priors' mix.centres])
+disp('Covariance matrices are')
+disp(mix.covars)
+disp('Press any key to continue.')
+pause
+
+% Set up vector of options for EM trainer
+options = zeros(1, 18);
+options(1)  = 1;		% Prints out error values.
+options(14) = 50;		% Number of iterations.
+
+disp('We now train the model using the EM algorithm for 50 iterations.')
+disp(' ')
+disp('Press any key to continue.')
+pause
+[mix, options, errlog] = gmmem(mix, data, options);
+
+% Print out model
+disp(' ')
+disp('The trained model has priors and centres:')
+disp('    Priors        Centres')
+disp([mix.priors' mix.centres])
+disp('The data generator has priors and centres')
+disp('    Priors        Centres')
+disp([datap' datac])
+disp('Model covariance matrices are')
+disp(mix.covars(:, :, 1))
+disp(mix.covars(:, :, 2))
+disp(mix.covars(:, :, 3))
+disp('Data generator covariance matrices are')
+disp(datacov(:, :, 1))
+disp(datacov(:, :, 2))
+disp(datacov(:, :, 3))
+disp('Note the close correspondence between these parameters and those')
+disp('of the distribution used to generate the data.  The match for')
+disp('covariance matrices is not that close, but would be improved with')
+disp('more iterations of the training algorithm.')
+disp(' ')
+disp('Press any key to continue.')
+pause
+
+clc
+disp('We now plot the density given by the mixture model as a surface plot.')
+disp(' ')
+disp('Press any key to continue.')
+pause
+
+% Plot the result
+x = -4.0:0.2:5.0;
+y = -4.0:0.2:5.0;
+[X, Y] = meshgrid(x,y);
+X = X(:);
+Y = Y(:);
+grid = [X Y];
+Z = gmmprob(mix, grid);
+Z = reshape(Z, length(x), length(y));
+c = mesh(x, y, Z);
+hold on
+title('Surface plot of probability density')
+hold off
+drawnow
+
+clc
+disp('The final plot shows the centres and widths, given by one standard')
+disp('deviation, of the three components of the mixture model.  The axes')
+disp('of the ellipses of constant density are shown.')
+disp(' ')
+disp('Press any key to continue.')
+pause
+
+% Try to calculate a sensible position for the second figure, below the first
+fig1_pos = get(fh1, 'Position');
+fig2_pos = fig1_pos;
+fig2_pos(2) = fig2_pos(2) - fig1_pos(4) - 30;
+fh2 = figure('Position', fig2_pos);
+
+h3 = plot(data(:, 1), data(:, 2), 'bo');
+axis equal;
+hold on
+title('Plot of data and covariances')
+for i = 1:ncentres
+  [v,d] = eig(mix.covars(:,:,i));
+  for j = 1:2
+    % Ensure that eigenvector has unit length
+    v(:,j) = v(:,j)/norm(v(:,j));
+    start=mix.centres(i,:)-sqrt(d(j,j))*(v(:,j)');
+    endpt=mix.centres(i,:)+sqrt(d(j,j))*(v(:,j)');
+    linex = [start(1) endpt(1)];
+    liney = [start(2) endpt(2)];
+    line(linex, liney, 'Color', 'k', 'LineWidth', 3)
+  end
+  % Plot ellipses of one standard deviation
+  theta = 0:0.02:2*pi;
+  x = sqrt(d(1,1))*cos(theta);
+  y = sqrt(d(2,2))*sin(theta);
+  % Rotate ellipse axes
+  ellipse = (v*([x; y]))';
+  % Adjust centre
+  ellipse = ellipse + ones(length(theta), 1)*mix.centres(i,:);
+  plot(ellipse(:,1), ellipse(:,2), 'r-');
+end
+hold off
+
+disp('Note how the data cluster positions and widths are captured by')
+disp('the mixture model.')
+disp(' ')
+disp('Press any key to end.')
+pause
+
+close(fh1);
+close(fh2);
+clear all; 
+