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Opgave: EM algoritmen og Gaussiske mikstur modeller (GMM)

Foelgende matlab scripts genererer et datasæt:

gmmdataset.m:

% Makes a 2 dimensional data set for which you can fit a 3 component
% Gaussian Mixture Model (GMM) to.
clear all;


prior = [0.2, 0.5, 0.3];

m = [0.1, 0.4;
     0.6, 0.6;
     1.0, 0.7]';

R45 = [cos(pi/4) -sin(pi/4);
       sin(pi/4) cos(pi/4)]

S = cell(1,3);
S{1} = R45' * diag([0.01, 0.25]);
S{2} = R45 * diag([0.005, 0.25]);
S{3} = R45' * diag([0.01, 0.25]);

N = 500;
X = zeros(2,N);
for i=1:N
    % Sample component k from prior
    k = randpmf([1,2,3], prior, 1);

    % Sample from component k Gaussian
    X(:, i) = gausssamples(1, m(:, k), S{k});

end


figure(1)
plot(m(1,:), m(2,:), 'o')
hold on;

plot(X(1,:),X(2,:), '.')
hold off


gausssamples.m:

function X=gausssamples(N, m, S)
% X=gausssamples(N, m, S)
%
% Take N from a multivariate Gaussian distribution with mean m and
% covariance matrix S.

D=length(m);
[V,E]=eig(S);
X = sqrt(E) * V * randn(D,N) + repmat(m, 1, N);


randpmf.m:

function r = randpmf(x, p, dim)
%  function r = randpmf(x, p, dim)
%   x   - Labels to sample from
%   p   - Distribution of labels  
%   dim - Dimension of the output matrix
%   r   - A matrix with random samples of p(x)
%
% Sample elements from the vector x following the discrete probability
% mass function p(x). 
%
%  By Kim S. Pedersen, ITU, 2004  
  
[n,m]=size(p);

if n==0 & m==0
  error('p must be a vector');
end
 
if n>1 && m>1
  error('p must be a vector');
end
 
% Force p to be a 1xN vector
if n>1
  p=p';
end
   
cp = cumsum(p);
r = rand(dim);

for i=1:prod(dim)
  idx = find(r(i) >= [0 cp(1:end-1)]);
  % r < cp(2:end)
  % idx2 = find(r < cp(2:end))+1
 
  r(i) = x(idx(end));
end


Modeller dette data saet med en 3 komponent Gaussisk mikstur model. Implementer EM algoritmen for GMM og anvend den til at estimere parametrene i modellen.

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