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06 Nov 2002 (Updated )

Pattern analysis toolbox.

demgmm5.m
%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 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)*0.1 + 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 again
data = [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 models
ncentres = 3;
ppca_dim = 1;

clc
disp('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.')
pause

fh1 = figure;
plot(data(:, 1), data(:, 2), 'o')
set(gca, 'Box', 'on')
axis equal
hold on

mix = 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 data
mix = 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')
pause

options(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 result
for 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-')
end

disp(' ')
disp('Press any key to exit')
pause
close (fh1);
clear all;

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