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6 changed files with 54 additions and 21 deletions
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assignments/hwk03/2a.png
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assignments/hwk03/2a.png
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@ -65,13 +65,8 @@ function [h, m, Q] = EMG(x, k, epochs, flag)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% TODO: Store the value of the complete log-likelihood function
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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L = 0;
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% for i = 1:num_data
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% for j = 1:k
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% prior = mvnpdf(x, m(j, :), S(:, :, j));
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% L = L + h(i, j) * (log(pi(i)) + log(prior(i)));
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% end
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% end
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Q(2*n - 1) = Q_step(x, m, S, k, pi, h);
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%%%%%%%%%%%%%%%%
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% M-step
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@ -82,6 +77,7 @@ function [h, m, Q] = EMG(x, k, epochs, flag)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% TODO: Store the value of the complete log-likelihood function
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Q(2*n) = Q_step(x, m, S, k, pi, h);
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end
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@ -20,18 +20,25 @@ function [S, m, pi] = M_step(x, h, S, k, flag)
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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);
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m = zeros(k, dim);
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for i = 1:k
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N_i(i) = sum(h(:, i));
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for j = 1:num_data
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m(i, :) = m(i, :) + h(j, i) * x(j, :);
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end
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end
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pi = N_i / num_data;
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for i = 1:k
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m(i, :) = m(i, :) ./ N_i(i);
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end
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% update cluster means
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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m = zeros(k, dim);
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m = h' * x ./ N_i;
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% m = zeros(k, dim);
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% m = h' * x ./ N_i;
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% for i = 1:k
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% m(i, :) = sum(h(:, i) .* x(i, :)) / N_i(i);
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% end
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@ -40,19 +47,24 @@ function [S, m, pi] = M_step(x, h, S, k, flag)
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% Calculate the covariance matrix estimate
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% further modifications will need to be made when doing 2(d)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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S = zeros(dim, dim, k);
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S = zeros(dim, dim, k) + eps;
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for i = 1:k
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% for j = 1:num_data
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% S(:, :, i) = S(:, :, i) + h(j, i) * (x(j, :) - m(i, :)) * (x(j, :) - m(i, :))';
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% end
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s = (x - m(i, :))' * ((x - m(i, :)) .* h(:, i)) / N_i(i);
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s = zeros(dim, dim);
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for j = 1:num_data
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s = s + h(j, i) * (x(j, :) - m(i, :))' * (x(j, :) - m(i, :));
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end
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s = s / N_i(i);
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% s = (x - m(i, :))' * ((x - m(i, :)) .* h(:, i)) / N_i(i);
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% % MAKE IT SYMMETRIC https://stackoverflow.com/a/38730499
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% S(:, :, i) = (s + s') / 2;
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% https://www.mathworks.com/matlabcentral/answers/366140-eig-gives-a-negative-eigenvalue-for-a-positive-semi-definite-matrix#answer_290270
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s = (s + s') / 2;
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% https://www.mathworks.com/matlabcentral/answers/57411-matlab-sometimes-produce-a-covariance-matrix-error-with-non-postive-semidefinite#answer_69524
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[V, D] = eig(s);
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S(:, :, i) = V * max(D,eps) / V;
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s = V * max(D, eps) / V;
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S(:, :, i) = s;
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end
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end
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@ -26,25 +26,26 @@ function [] = Problem2()
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figure();
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for k = 4:4:12
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fprintf("k=%d\n", k);
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% call EM on data
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[h, m, Q] = EMG(stadium_x, k, epochs, false);
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% get compressed version of image
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[~,class_index] = max(h,[],2);
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compress = m(class_index,:);
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% 2(a), plot compressed image
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subplot(3,2,index)
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imagesc(permute(reshape(compress, [width, height, depth]),[2 1 3]))
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index = index + 1;
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% 2(b), plot complete data likelihood curves
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subplot(3,2,index)
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x = 1:size(Q);
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c = repmat([1 0 0; 0 1 0], length(x)/2, 1);
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scatter(x,Q,20,c);
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index = index + 1;
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pause;
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end
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shg
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@ -69,6 +70,7 @@ function [] = Problem2()
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% TODO: plot goldy image after using clusters from k-means
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% begin code here
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[~, ~, ~, D] = kmeans(goldy_x, k);
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% end code here
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shg
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11
assignments/hwk03/Q_step.m
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11
assignments/hwk03/Q_step.m
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@ -0,0 +1,11 @@
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function [LL] = Q_step(x, m, S, k, pi, h)
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[num_data, ~] = size(x);
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LL = 0;
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for i = 1:k
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N = mvnpdf(x, m(i, :), S(:, :, i));
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for j = 1:num_data
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LL = LL + h(j, i) * (log(pi(i) + eps) + log(N(j) + eps));
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end
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end
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end
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@ -66,4 +66,16 @@ Updates:
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&= - sum_t (r^t-y^t) (v_1 frac(diff z^t_1, diff w_j) + v_2 frac(diff z^t_2, diff w_j)) \
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&= - sum_t (r^t-y^t) (x^t_j v_1 cases(0 "if" ww dot xx < 0, 1 "otherwise") + x^t_j v_2 (1 - tanh^2 (ww dot xx))) \
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&= - sum_t (r^t-y^t) x^t_j (v_1 cases(0 "if" ww dot xx < 0, 1 "otherwise") + v_2 (1 - tanh^2 (ww dot xx))) \
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$
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$
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#pagebreak()
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= Problem 2a + 2b
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#image("2a.png")
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= Problem 2c
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= Problem 2d
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MLE of $Sigma_i$
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