wolffd@0: %DEMGMM3 Demonstrate density modelling with a Gaussian mixture model.
wolffd@0: %
wolffd@0: %	Description
wolffd@0: %	 The problem consists of modelling data generated by a mixture of
wolffd@0: %	three Gaussians in 2 dimensions with a mixture model using diagonal
wolffd@0: %	covariance matrices.  The priors are 0.3, 0.5 and 0.2; the centres
wolffd@0: %	are (2, 3.5), (0, 0) and (0,2); the covariances are all axis aligned
wolffd@0: %	(0.16, 0.64), (0.25, 1) and the identity matrix. The first figure
wolffd@0: %	contains a scatter plot of the data.
wolffd@0: %
wolffd@0: %	A Gaussian mixture model with three components is trained using EM.
wolffd@0: %	The parameter vector is printed before training and after training.
wolffd@0: %	The user should press any key to continue at these points.  The
wolffd@0: %	parameter vector consists of priors (the column), and centres (given
wolffd@0: %	as (x, y) pairs as the next two columns).  The diagonal entries of
wolffd@0: %	the covariance matrices are printed separately.
wolffd@0: %
wolffd@0: %	The second figure is a 3 dimensional view of the density function,
wolffd@0: %	while the third shows the axes of the 1-standard deviation circles
wolffd@0: %	for the three components of the mixture model.
wolffd@0: %
wolffd@0: %	See also
wolffd@0: %	GMM, GMMINIT, GMMEM, GMMPROB, GMMUNPAK
wolffd@0: %
wolffd@0: 
wolffd@0: %	Copyright (c) Ian T Nabney (1996-2001)
wolffd@0: 
wolffd@0: % Generate the data
wolffd@0: ndata = 500;
wolffd@0: 
wolffd@0: % Fix the seeds for reproducible results
wolffd@0: randn('state', 42);
wolffd@0: rand('state', 42);
wolffd@0: data = randn(ndata, 2);
wolffd@0: prior = [0.3 0.5 0.2];
wolffd@0: % Mixture model swaps clusters 1 and 3
wolffd@0: datap = [0.2 0.5 0.3];
wolffd@0: datac = [0 2; 0 0; 2 3.5];
wolffd@0: datacov = [1 1;1 0.25; 0.4*0.4 0.8*0.8];
wolffd@0: data1 = data(1:prior(1)*ndata,:);
wolffd@0: data2 = data(prior(1)*ndata+1:(prior(2)+prior(1))*ndata, :);
wolffd@0: data3 = data((prior(1)+prior(2))*ndata +1:ndata, :);
wolffd@0: 
wolffd@0: % First cluster has axis aligned variance and centre (2, 3.5)
wolffd@0: data1(:, 1) = data1(:, 1)*0.4 + 2.0;
wolffd@0: data1(:, 2) = data1(:, 2)*0.8 + 3.5;
wolffd@0: 
wolffd@0: % Second cluster has axis aligned variance and centre (0, 0)
wolffd@0: data2(:,2) = data2(:, 2)*0.5;
wolffd@0: 
wolffd@0: % Third cluster is at (0,2) with identity matrix for covariance
wolffd@0: data3 = data3 + repmat([0 2], prior(3)*ndata, 1);
wolffd@0: 
wolffd@0: % Put the dataset together again
wolffd@0: data = [data1; data2; data3];
wolffd@0: 
wolffd@0: clc
wolffd@0: disp('This demonstration illustrates the use of a Gaussian mixture model')
wolffd@0: disp('with diagonal covariance matrices to approximate the unconditional')
wolffd@0: disp('probability density of data in a two-dimensional space.')
wolffd@0: disp('We begin by generating the data from a mixture of three Gaussians')
wolffd@0: disp('with axis aligned covariance structure and plotting it.')
wolffd@0: disp(' ')
wolffd@0: disp('The first cluster has centre (0, 2).')
wolffd@0: disp('The second cluster has centre (0, 0).')
wolffd@0: disp('The third cluster has centre (2, 3.5).')
wolffd@0: disp(' ')
wolffd@0: disp('Press any key to continue')
wolffd@0: pause
wolffd@0: 
wolffd@0: fh1 = figure;
wolffd@0: plot(data(:, 1), data(:, 2), 'o')
wolffd@0: set(gca, 'Box', 'on')
wolffd@0: 
wolffd@0: % Set up mixture model
wolffd@0: ncentres = 3;
wolffd@0: input_dim = 2;
wolffd@0: mix = gmm(input_dim, ncentres, 'diag');
wolffd@0: 
wolffd@0: options = foptions;
wolffd@0: options(14) = 5;	% Just use 5 iterations of k-means in initialisation
wolffd@0: % Initialise the model parameters from the data
wolffd@0: mix = gmminit(mix, data, options);
wolffd@0: 
wolffd@0: % Print out model
wolffd@0: disp('The mixture model has three components and diagonal covariance')
wolffd@0: disp('matrices.  The model parameters after initialisation using the')
wolffd@0: disp('k-means algorithm are as follows')
wolffd@0: disp('    Priors        Centres')
wolffd@0: disp([mix.priors' mix.centres])
wolffd@0: disp('Covariance diagonals are')
wolffd@0: disp(mix.covars)
wolffd@0: disp('Press any key to continue.')
wolffd@0: pause
wolffd@0: 
wolffd@0: % Set up vector of options for EM trainer
wolffd@0: options = zeros(1, 18);
wolffd@0: options(1)  = 1;		% Prints out error values.
wolffd@0: options(14) = 20;		% Number of iterations.
wolffd@0: 
wolffd@0: disp('We now train the model using the EM algorithm for 20 iterations.')
wolffd@0: disp(' ')
wolffd@0: disp('Press any key to continue.')
wolffd@0: pause
wolffd@0: 
wolffd@0: [mix, options, errlog] = gmmem(mix, data, options);
wolffd@0: 
wolffd@0: % Print out model
wolffd@0: disp(' ')
wolffd@0: disp('The trained model has priors and centres:')
wolffd@0: disp('    Priors        Centres')
wolffd@0: disp([mix.priors' mix.centres])
wolffd@0: disp('The data generator has priors and centres')
wolffd@0: disp('    Priors        Centres')
wolffd@0: disp([datap' datac])
wolffd@0: disp('Model covariance diagonals are')
wolffd@0: disp(mix.covars)
wolffd@0: disp('Data generator covariance diagonals are')
wolffd@0: disp(datacov)
wolffd@0: disp('Note the close correspondence between these parameters and those')
wolffd@0: disp('of the distribution used to generate the data.')
wolffd@0: disp(' ')
wolffd@0: disp('Press any key to continue.')
wolffd@0: pause
wolffd@0: 
wolffd@0: clc
wolffd@0: disp('We now plot the density given by the mixture model as a surface plot.')
wolffd@0: disp(' ')
wolffd@0: disp('Press any key to continue.')
wolffd@0: pause
wolffd@0: 
wolffd@0: % Plot the result
wolffd@0: x = -4.0:0.2:5.0;
wolffd@0: y = -4.0:0.2:5.0;
wolffd@0: [X, Y] = meshgrid(x,y);
wolffd@0: X = X(:);
wolffd@0: Y = Y(:);
wolffd@0: grid = [X Y];
wolffd@0: Z = gmmprob(mix, grid);
wolffd@0: Z = reshape(Z, length(x), length(y));
wolffd@0: c = mesh(x, y, Z);
wolffd@0: hold on
wolffd@0: title('Surface plot of probability density')
wolffd@0: hold off
wolffd@0: drawnow
wolffd@0: 
wolffd@0: clc
wolffd@0: disp('The final plot shows the centres and widths, given by one standard')
wolffd@0: disp('deviation, of the three components of the mixture model.  The axes')
wolffd@0: disp('of the ellipses of constant density are shown.')
wolffd@0: disp(' ')
wolffd@0: disp('Press any key to continue.')
wolffd@0: pause
wolffd@0: 
wolffd@0: % Try to calculate a sensible position for the second figure, below the first
wolffd@0: fig1_pos = get(fh1, 'Position');
wolffd@0: fig2_pos = fig1_pos;
wolffd@0: fig2_pos(2) = fig2_pos(2) - fig1_pos(4);
wolffd@0: fh2 = figure('Position', fig2_pos);
wolffd@0: 
wolffd@0: h = plot(data(:, 1), data(:, 2), 'bo');
wolffd@0: hold on
wolffd@0: axis('equal');
wolffd@0: title('Plot of data and covariances')
wolffd@0: for i = 1:ncentres
wolffd@0:   v = [1 0];
wolffd@0:   for j = 1:2
wolffd@0:     start=mix.centres(i,:)-sqrt(mix.covars(i,:).*v);
wolffd@0:     endpt=mix.centres(i,:)+sqrt(mix.covars(i,:).*v);
wolffd@0:     linex = [start(1) endpt(1)];
wolffd@0:     liney = [start(2) endpt(2)];
wolffd@0:     line(linex, liney, 'Color', 'k', 'LineWidth', 3)
wolffd@0:     v = [0 1];
wolffd@0:   end
wolffd@0:   % Plot ellipses of one standard deviation
wolffd@0:   theta = 0:0.02:2*pi;
wolffd@0:   x = sqrt(mix.covars(i,1))*cos(theta) + mix.centres(i,1);
wolffd@0:   y = sqrt(mix.covars(i,2))*sin(theta) + mix.centres(i,2);
wolffd@0:   plot(x, y, 'r-');
wolffd@0: end
wolffd@0: hold off
wolffd@0: 
wolffd@0: disp('Note how the data cluster positions and widths are captured by')
wolffd@0: disp('the mixture model.')
wolffd@0: disp(' ')
wolffd@0: disp('Press any key to end.')
wolffd@0: pause
wolffd@0: 
wolffd@0: close(fh1);
wolffd@0: close(fh2);
wolffd@0: clear all;
wolffd@0: