My solutions to Week 9 Exercises (Anomaly Detection and Recommender Systems) -
1) Estimate Gaussian Parameters [ estimateGaussian.m ]
function [mu sigma2] = estimateGaussian(X) %ESTIMATEGAUSSIAN This function estimates the parameters of a %Gaussian distribution using the data in X % [mu sigma2] = estimateGaussian(X), % The input X is the dataset with each n-dimensional data point in one row % The output is an n-dimensional vector mu, the mean of the data set % and the variances sigma^2, an n x 1 vector % % Useful variables [m, n] = size(X); % You should return these values correctly mu = zeros(n, 1); sigma2 = zeros(n, 1); % ====================== YOUR CODE HERE ====================== % Instructions: Compute the mean of the data and the variances % In particular, mu(i) should contain the mean of % the data for the i-th feature and sigma2(i) % should contain variance of the i-th feature. % onesMatrix = ones(1, size(X, 1)); mu = (onesMatrix * X)/m; for j = 1:n sigma2(j) = sum((X(:, j)-mu(j)).^2)/m; end % other way for calculating variance % http://stackoverflow.com/questions/5967940/matlab-quickly-subtract-1xn-array-from-mxn-matrix-elements % http://stackoverflow.com/questions/2651267/why-is-sumx-1-the-sum-of-the-columns-in-matlab sigma2 = sum(bsxfun(@minus, X, mu).^2, 1)/m; % using mean and var functions % mu = mean(X); % sigma2 = var(X, 1); % ============================================================= end
2) Select Threshold [ selectThreshold.m ]
function [bestEpsilon bestF1] = selectThreshold(yval, pval) %SELECTTHRESHOLD Find the best threshold (epsilon) to use for selecting %outliers % [bestEpsilon bestF1] = SELECTTHRESHOLD(yval, pval) finds the best % threshold to use for selecting outliers based on the results from a % validation set (pval) and the ground truth (yval). % bestEpsilon = 0; bestF1 = 0; F1 = 0; stepsize = (max(pval) - min(pval)) / 1000; for epsilon = min(pval):stepsize:max(pval) % ====================== YOUR CODE HERE ====================== % Instructions: Compute the F1 score of choosing epsilon as the % threshold and place the value in F1. The code at the % end of the loop will compare the F1 score for this % choice of epsilon and set it to be the best epsilon if % it is better than the current choice of epsilon. % % Note: You can use predictions = (pval < epsilon) to get a binary vector % of 0's and 1's of the outlier predictions cvPredictions = size(size(pval, 1), 1); for i=1:size(pval, 1) if pval(i)>=epsilon cvPredictions(i) = 0; else cvPredictions(i) = 1; end end fp = sum((cvPredictions'==1) & (yval==0)); tp = sum((cvPredictions'==1) & (yval==1)); fn = sum((cvPredictions'==0) & (yval==1)); prec = tp/(tp+fp); rec = tp/(tp+fn); F1 = (2*prec*rec)/(prec+rec); % ============================================================= if F1 > bestF1 bestF1 = F1; bestEpsilon = epsilon; end end end
3, 4, 5, 6) Collaborative Filtering Cost, Collaborative Filtering Gradient, Regularized Cost, Regularized Gradient [ cofiCostFunc.m ]
function [J, grad] = cofiCostFunc(params, Y, R, num_users, num_movies, ... num_features, lambda) %COFICOSTFUNC Collaborative filtering cost function % [J, grad] = COFICOSTFUNC(params, Y, R, num_users, num_movies, ... % num_features, lambda) returns the cost and gradient for the % collaborative filtering problem. % % Unfold the U and W matrices from params X = reshape(params(1:num_movies*num_features), num_movies, num_features); Theta = reshape(params(num_movies*num_features+1:end), ... num_users, num_features); % You need to return the following values correctly J = 0; X_grad = zeros(size(X)); Theta_grad = zeros(size(Theta)); % ====================== YOUR CODE HERE ====================== % Instructions: Compute the cost function and gradient for collaborative % filtering. Concretely, you should first implement the cost % function (without regularization) and make sure it is % matches our costs. After that, you should implement the % gradient and use the checkCostFunction routine to check % that the gradient is correct. Finally, you should implement % regularization. % % Notes: X - num_movies x num_features matrix of movie features % Theta - num_users x num_features matrix of user features % Y - num_movies x num_users matrix of user ratings of movies % R - num_movies x num_users matrix, where R(i, j) = 1 if the % i-th movie was rated by the j-th user % % You should set the following variables correctly: % % X_grad - num_movies x num_features matrix, containing the % partial derivatives w.r.t. to each element of X % Theta_grad - num_users x num_features matrix, containing the % partial derivatives w.r.t. to each element of Theta % J = sum(sum(((X*Theta'-Y).^2).*R))/2; % temp = 0; % for i=1:num_movies % for j=1:num_users % if R(i, j)==1 temp = temp + ((Theta(j, :))*X(i, :)' - Y(i,j))^2; % end % end % end % % J = temp/2; % Gradient - Unvectorized %for i = 1:num_movies % for k=1:num_features % t = 0; % for j=1:num_users % if R(i, j)==1 t = t+ sum(Theta(j, :)*X(i, :)' - Y(i,j))*Theta(j, k); end % end % X_grad(i, k) = t; % end %end %for j=1:num_users % for k=1:num_features % t = 0; % for i=1:num_movies % if R(i, j)==1 t = t + sum(Theta(j, :)*X(i, :)' - Y(i, j))*X(i, k); end % end % Theta_grad(j, k) = t; % end %end %Gradient - Vectorized for i=1:num_movies idx = find(R(i, :)==1); ThetaTemp = Theta(idx, :); YTemp = Y(i, idx); X_grad(i, :) = ((X(i, :)*ThetaTemp' - YTemp)*ThetaTemp); reg = lambda * X(i, :); X_grad(i, :) = X_grad(i, :)+reg; end for j=1:num_users idx = find(R(:, j)==1); ThetaTemp = Theta(j, :); XTemp = X(idx, :); YTemp = Y(idx, j); first = (XTemp*ThetaTemp' - YTemp)'; Theta_grad(j, :) = (first*XTemp); reg = lambda * Theta(j, :); Theta_grad(j, :) = Theta_grad(j, :)+reg; end ThetaSquared = sum(sum(Theta .^ 2)); XSquared = sum(sum(X .^ 2)); reg = ((ThetaSquared + XSquared)*lambda)/2; J = J+reg; % ============================================================= grad = [X_grad(:); Theta_grad(:)]; end
