%DEMGMM5 Demonstrate density modelling with a PPCA 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 mixture model with three one-dimensional PPCA components is trained%	using EM.  The parameter vector is printed before training and after%	training.  The parameter vector consists of priors (the column), and%	centres (given as (x, y) pairs as the next two columns).%%	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 together with the one%	standard deviation along the principal component of each mixture%	model component.%%	See also%	GMM, GMMINIT, GMMEM, GMMPROB, PPCA%%	Copyright (c) Ian T Nabney (1996-2001)ndata = 500;data = randn(ndata, 2);prior = [0.3 0.5 0.2];% Mixture model swaps clusters 1 and 3datap = [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)*0.1 + 2.0;data1(:, 2) = data1(:, 2)*0.8 + 3.5;datacov(:, :, 3) = [0.1*0.1 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)*0.2;data2 = data2*rotn;datacov(:, :, 2) = rotn' * [0.04 0; 0 1] * rotn;% Third cluster is at (0,2)
data3(:, 2) = data3(:, 2)*0.1;data3 = data3 + repmat([0 2], prior(3)*ndata, 1);% Put the dataset together againdata = [data1; data2; data3];ndata = 100;			% Number of data points.noise = 0.2;			% Standard deviation of noise distribution.x = [0:1/(2*(ndata - 1)):0.5]';randn('state', 1);rand('state', 1);t = sin(2*pi*x) + noise*randn(ndata, 1);% Fit three one-dimensional PPCA modelsncentres = 3;ppca_dim = 1;clcdisp('This demonstration illustrates the use of a Gaussian mixture model')disp('with a probabilistic PCA covariance structure to approximate the')disp('unconditional 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 variance parallel to the x-axis is significantly greater')disp('than that parallel to the y-axis.')disp('The second cluster has variance axes rotated by 30 degrees')disp('and centre (0, 0).  The third cluster has significant variance')disp('parallel to the y-axis and centre (2, 3.5).')disp(' ')disp('Press any key to continue.')pausefh1 = figure;plot(data(:, 1), data(:, 2), 'o')set(gca, 'Box', 'on')axis equalhold onmix = gmm(2, ncentres, 'ppca', ppca_dim);options = foptions;options(14) = 10;options(1) = -1;  % Switch off all warnings% Just use 10 iterations of k-means in initialisation% Initialise the model parameters from the datamix = gmminit(mix, data, options);disp('The mixture model has three components with 1-dimensional')disp('PPCA subspaces.  The model parameters after initialisation using')disp('the k-means algorithm are as follows')disp('    Priors        Centres')disp([mix.priors' mix.centres])disp(' ')disp('Press any key to continue')pauseoptions(1)  = 1;		% Prints out error values.options(14) = 30;		% Number of iterations.disp('We now train the model using the EM algorithm for up to 30 iterations.')disp(' ')disp('Press any key to continue.')pause[mix, options, errlog] = gmmem(mix, data, options);disp('The trained model has priors and centres:')disp('    Priors        Centres')disp([mix.priors' mix.centres])% Now plot the resultfor i = 1:ncentres  % Plot the PC vectors  v = mix.U(:,:,i);  start=mix.centres(i,:)-sqrt(mix.lambda(i))*(v');  endpt=mix.centres(i,:)+sqrt(mix.lambda(i))*(v');  linex = [start(1) endpt(1)];  liney = [start(2) endpt(2)];  line(linex, liney, 'Color', 'k', 'LineWidth', 3)
  % Plot ellipses of one standard deviation  theta = 0:0.02:2*pi;  x = sqrt(mix.lambda(i))*cos(theta);  y = sqrt(mix.covars(i))*sin(theta);  % Rotate ellipse axes  rot_matrix = [v(1) -v(2); v(2) v(1)];  ellipse = (rot_matrix*([x; y]))';  % Adjust centre  ellipse = ellipse + ones(length(theta), 1)*mix.centres(i,:);  plot(ellipse(:,1), ellipse(:,2), 'r-')enddisp(' ')disp('Press any key to exit')pauseclose (fh1);clear all;