My solutions to Week 4 assignments:

**Part 1: Regularied Logistic Regression**function [J, grad] = lrCostFunction(theta, X, y, lambda) %LRCOSTFUNCTION Compute cost and gradient for logistic regression with %regularization % J = LRCOSTFUNCTION(theta, X, y, lambda) computes the cost of using % theta as the parameter for regularized logistic regression and the % gradient of the cost w.r.t. to the parameters. % Initialize some useful values m = length(y); % number of training examples % You need to return the following variables correctly J = 0; grad = zeros(size(theta)); % ====================== YOUR CODE HERE ====================== % Instructions: Compute the cost of a particular choice of theta. % You should set J to the cost. % Compute the partial derivatives and set grad to the partial % derivatives of the cost w.r.t. each parameter in theta % % Hint: The computation of the cost function and gradients can be % efficiently vectorized. For example, consider the computation % % sigmoid(X * theta) % % Each row of the resulting matrix will contain the value of the % prediction for that example. You can make use of this to vectorize % the cost function and gradient computations. % % Hint: When computing the gradient of the regularized cost function, % there're many possible vectorized solutions, but one solution % looks like: % grad = (unregularized gradient for logistic regression) % temp = theta; % temp(1) = 0; % because we don't add anything for j = 0 % grad = grad + YOUR_CODE_HERE (using the temp variable) % thetaTx = (theta'*X')'; h = sigmoid(thetaTx); leftJ = -(1/m)*(sum(y'*log(h)+(1-y)'*log(1-h))); rightJ = (lambda/(2*m))*sum((theta.^2)(2:end,1)); J = leftJ+rightJ; error = h-y; grad = ((error'*X)' + (lambda*theta))/m; grad(1) = (error'*X(:,1))/m ; % ============================================================= grad = grad(:); end

**Part 2: One-vs-all classifier training**function [all_theta] = oneVsAll(X, y, num_labels, lambda) %ONEVSALL trains multiple logistic regression classifiers and returns all %the classifiers in a matrix all_theta, where the i-th row of all_theta %corresponds to the classifier for label i % [all_theta] = ONEVSALL(X, y, num_labels, lambda) trains num_labels % logisitc regression classifiers and returns each of these classifiers % in a matrix all_theta, where the i-th row of all_theta corresponds % to the classifier for label i % Some useful variables m = size(X, 1); n = size(X, 2); % You need to return the following variables correctly all_theta = zeros(num_labels, n + 1); % Add ones to the X data matrix X = [ones(m, 1) X]; % ====================== YOUR CODE HERE ====================== % Instructions: You should complete the following code to train num_labels % logistic regression classifiers with regularization % parameter lambda. % % Hint: theta(:) will return a column vector. % % Hint: You can use y == c to obtain a vector of 1's and 0's that tell use % whether the ground truth is true/false for this class. % % Note: For this assignment, we recommend using fmincg to optimize the cost % function. It is okay to use a for-loop (for c = 1:num_labels) to % loop over the different classes. % % fmincg works similarly to fminunc, but is more efficient when we % are dealing with large number of parameters. % % Example Code for fmincg: % % % Set Initial theta % initial_theta = zeros(n + 1, 1); % % % Set options for fminunc % options = optimset('GradObj', 'on', 'MaxIter', 50); % % % Run fmincg to obtain the optimal theta % % This function will return theta and the cost % [theta] = ... % fmincg (@(t)(lrCostFunction(t, X, (y == c), lambda)), ... % initial_theta, options); % options = optimset('GradObj', 'on', 'MaxIter', 50); for c = 1:num_labels initial_theta = zeros(n+1, 1); [theta] = fmincg(@(t)(lrCostFunction(t, X, (y==c), lambda)), initial_theta, options); all_theta(c,:) = theta; end % ========================================================================= end

**Part 3: One-vs-all classifier prediction**function p = predictOneVsAll(all_theta, X) %PREDICT Predict the label for a trained one-vs-all classifier. The labels %are in the range 1..K, where K = size(all_theta, 1). % p = PREDICTONEVSALL(all_theta, X) will return a vector of predictions % for each example in the matrix X. Note that X contains the examples in % rows. all_theta is a matrix where the i-th row is a trained logistic % regression theta vector for the i-th class. You should set p to a vector % of values from 1..K (e.g., p = [1; 3; 1; 2] predicts classes 1, 3, 1, 2 % for 4 examples) m = size(X, 1); num_labels = size(all_theta, 1); % You need to return the following variables correctly p = zeros(size(X, 1), 1); % Add ones to the X data matrix X = [ones(m, 1) X]; % ====================== YOUR CODE HERE ====================== % Instructions: Complete the following code to make predictions using % your learned logistic regression parameters (one-vs-all). % You should set p to a vector of predictions (from 1 to % num_labels). % % Hint: This code can be done all vectorized using the max function. % In particular, the max function can also return the index of the % max element, for more information see 'help max'. If your examples % are in rows, then, you can use max(A, [], 2) to obtain the max % for each row. % z = X*all_theta'; for i = 1:size(z, 1) for j=1:size(z, 2) z(i, j) = pinv(1+pinv(e^z(i, j))); end end pSigmoid = z; [maxProbabilities indices] = max(pSigmoid, [], 2); p = indices; % ========================================================================= end

**Part 4: Neural Network Prediction Function**function p = predict(Theta1, Theta2, X) %PREDICT Predict the label of an input given a trained neural network % p = PREDICT(Theta1, Theta2, X) outputs the predicted label of X given the % trained weights of a neural network (Theta1, Theta2) % Useful values m = size(X, 1); num_labels = size(Theta2, 1); % You need to return the following variables correctly p = zeros(size(X, 1), 1); % ====================== YOUR CODE HERE ====================== % Instructions: Complete the following code to make predictions using % your learned neural network. You should set p to a % vector containing labels between 1 to num_labels. % % Hint: The max function might come in useful. In particular, the max % function can also return the index of the max element, for more % information see 'help max'. If your examples are in rows, then, you % can use max(A, [], 2) to obtain the max for each row. % X = [ones(m, 1) X]; z = X*Theta1'; for i = 1:size(z, 1) for j=1:size(z, 2) z(i, j) = pinv(1+pinv(e^z(i, j))); end end pSigmoid = z; pSigmoid = [ones(size(pSigmoid, 1), 1) pSigmoid]; z1 = pSigmoid*Theta2'; [maxProbabilities indices] = max(z1, [], 2); p = indices; % ========================================================================= end

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