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Opgave: EM algoritmen og Gaussiske mikstur modeller (GMM)

Foelgende matlab scripts genererer et data saet:

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


gausssampler.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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