csci5521/assignments/hwk03/M_step.m

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Name: M_step.m
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% Input: x - a nxd matrix (nx3 if using RGB)
% Q - vector of values from the complete data log-likelihood function
% h - a nxk matrix, the expectation of the hidden variable z given the data set and distribution params
% S - cluster covariance matrices
% k - the number of clusters
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% flag - flag to use improved EM to avoid singular covariance matrix
% Output: S - cluster covariance matrices
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% m - cluster means
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% pi - mixing coefficients
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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function [S, m, pi] = M_step(x, h, S, k, flag)
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% get size of data
[num_data, dim] = size(x);
eps = 1e-15;
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lambda = 1e-3; % value for improved version of EM
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% update mixing coefficients
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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pi = zeros(k, 1);
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N_i = zeros(k, 1);
for i = 1:k
N_i(i) = sum(h(:, i));
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end
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pi = N_i / num_data;
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% update cluster means
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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m = zeros(k, dim);
m = h' * x ./ N_i;
% for i = 1:k
% m(i, :) = sum(h(:, i) .* x(i, :)) / N_i(i);
% end
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Calculate the covariance matrix estimate
% further modifications will need to be made when doing 2(d)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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S = zeros(dim, dim, k);
for i = 1:k
% for j = 1:num_data
% S(:, :, i) = S(:, :, i) + h(j, i) * (x(j, :) - m(i, :)) * (x(j, :) - m(i, :))';
% end
s = (x - m(i, :))' * ((x - m(i, :)) .* h(:, i)) / N_i(i);
% % MAKE IT SYMMETRIC https://stackoverflow.com/a/38730499
% S(:, :, i) = (s + s') / 2;
% https://www.mathworks.com/matlabcentral/answers/366140-eig-gives-a-negative-eigenvalue-for-a-positive-semi-definite-matrix#answer_290270
s = (s + s') / 2;
[V, D] = eig(s);
S(:, :, i) = V * max(D,eps) / V;
end
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end