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### Coursera machine learning week 5 assignment answers

function g = sigmoidGradient(z) %SIGMOIDGRADIENT returns the gradient of the sigmoid function %evaluated at z % g = SIGMOIDGRADIENT(z) computes the gradient of the sigmoid function % evaluated at z. This should work regardless if z is a matrix or a % vector. In particular, if z is a vector or matrix, you should return % the gradient for each element. g = zeros(size(z)); % ====================== YOUR CODE HERE ====================== % Instructions: Compute the gradient of the sigmoid function evaluated at % each value of z (z can be a matrix, vector or scalar). g = sigmoid(z).*(1-sigmoid(z)); % ============================================================= end

function W = randInitializeWeights(L_in, L_out) %RANDINITIALIZEWEIGHTS Randomly initialize the weights of a layer with L_in %incoming connections and L_out outgoing connections % W = RANDINITIALIZEWEIGHTS(L_in, L_out) randomly initializes the weights % of a layer with L_in incoming connections and L_out outgoing % connections. % % Note that W should be set to a matrix of size(L_out, 1 + L_in) as % the first column of W handles the "bias" terms % % You need to return the following variables correctly W = zeros(L_out, 1 + L_in); % ====================== YOUR CODE HERE ====================== % Instructions: Initialize W randomly so that we break the symmetry while % training the neural network. % % Note: The first column of W corresponds to the parameters for the bias unit % % epsilon_init = 0.12; epsilon_init = sqrt(6)/(sqrt(L_in)+sqrt(L_out)); W = - epsilon_init + rand(L_out, 1 + L_in) * 2 * epsilon_init ; % ========================================================================= end

function [J, grad] = nnCostFunction(nn_params, ... input_layer_size, ... hidden_layer_size, ... num_labels, ... X, y, lambda) %NNCOSTFUNCTION Implements the neural network cost function for a two layer %neural network which performs classification % [J grad] = NNCOSTFUNCTON(nn_params, hidden_layer_size, num_labels, ... % X, y, lambda) computes the cost and gradient of the neural network. The % parameters for the neural network are "unrolled" into the vector % nn_params and need to be converted back into the weight matrices. % % The returned parameter grad should be a "unrolled" vector of the % partial derivatives of the neural network. % % Reshape nn_params back into the parameters Theta1 and Theta2, the weight matrices % for our 2 layer neural network % DIMENSIONS: % Theta1 = 25 x 401 % Theta2 = 10 x 26 Theta1 = reshape(nn_params(1:hidden_layer_size * (input_layer_size + 1)), ... hidden_layer_size, (input_layer_size + 1)); Theta2 = reshape(nn_params((1 + (hidden_layer_size * (input_layer_size + 1))):end), ... num_labels, (hidden_layer_size + 1)); % Setup some useful variables m = size(X, 1); % You need to return the following variables correctly J = 0; Theta1_grad = zeros(size(Theta1)); %25 x401 Theta2_grad = zeros(size(Theta2)); %10 x 26 % ====================== YOUR CODE HERE ====================== % Instructions: You should complete the code by working through the % following parts. % % Part 1: Feedforward the neural network and return the cost in the % variable J. After implementing Part 1, you can verify that your % cost function computation is correct by verifying the cost % computed in ex4.m % % Part 2: Implement the backpropagation algorithm to compute the gradients % Theta1_grad and Theta2_grad. You should return the partial derivatives of % the cost function with respect to Theta1 and Theta2 in Theta1_grad and % Theta2_grad, respectively. After implementing Part 2, you can check % that your implementation is correct by running checkNNGradients % % Note: The vector y passed into the function is a vector of labels % containing values from 1..K. You need to map this vector into a % binary vector of 1's and 0's to be used with the neural network % cost function. % % Hint: We recommend implementing backpropagation using a for-loop % over the training examples if you are implementing it for the % first time. % % Part 3: Implement regularization with the cost function and gradients. % % Hint: You can implement this around the code for % backpropagation. That is, you can compute the gradients for % the regularization separately and then add them to Theta1_grad % and Theta2_grad from Part 2. % %%%%%%%%%%% Part 1: Calculating J w/o Regularization %%%%%%%%%%%%%%% X = [ones(m,1), X]; % Adding 1 as first column in X a1 = X; % 5000 x 401 z2 = a1 * Theta1'; % m x hidden_layer_size == 5000 x 25 a2 = sigmoid(z2); % m x hidden_layer_size == 5000 x 25 a2 = [ones(size(a2,1),1), a2]; % Adding 1 as first column in z = (Adding bias unit) % m x (hidden_layer_size + 1) == 5000 x 26 z3 = a2 * Theta2'; % m x num_labels == 5000 x 10 a3 = sigmoid(z3); % m x num_labels == 5000 x 10 h_x = a3; % m x num_labels == 5000 x 10 %Converting y into vector of 0's and 1's for multi-class classification %%%%% WORKING %%%%% % y_Vec = zeros(m,num_labels); % for i = 1:m % y_Vec(i,y(i)) = 1; % end %%%%%%%%%%%%%%%%%%% y_Vec = (1:num_labels)==y; % m x num_labels == 5000 x 10 %Costfunction Without regularization J = (1/m) * sum(sum((-y_Vec.*log(h_x))-((1-y_Vec).*log(1-h_x)))); %scalar %%%%%%%%%%% Part 2: Implementing Backpropogation for Theta_gra w/o Regularization %%%%%%%%%%%%% %%%%%%% WORKING: Backpropogation using for loop %%%%%%% % for t=1:m % % Here X is including 1 column at begining % % % for layer-1 % a1 = X(t,:)'; % (n+1) x 1 == 401 x 1 % % % for layer-2 % z2 = Theta1 * a1; % hidden_layer_size x 1 == 25 x 1 % a2 = [1; sigmoid(z2)]; % (hidden_layer_size+1) x 1 == 26 x 1 % % % for layer-3 % z3 = Theta2 * a2; % num_labels x 1 == 10 x 1 % a3 = sigmoid(z3); % num_labels x 1 == 10 x 1 % % yVector = (1:num_labels)'==y(t); % num_labels x 1 == 10 x 1 % % %calculating delta values % delta3 = a3 - yVector; % num_labels x 1 == 10 x 1 % % delta2 = (Theta2' * delta3) .* [1; sigmoidGradient(z2)]; % (hidden_layer_size+1) x 1 == 26 x 1 % % delta2 = delta2(2:end); % hidden_layer_size x 1 == 25 x 1 %Removing delta2 for bias node % % % delta_1 is not calculated because we do not associate error with the input % % % CAPITAL delta update % Theta1_grad = Theta1_grad + (delta2 * a1'); % 25 x 401 % Theta2_grad = Theta2_grad + (delta3 * a2'); % 10 x 26 % % end % % Theta1_grad = (1/m) * Theta1_grad; % 25 x 401 % Theta2_grad = (1/m) * Theta2_grad; % 10 x 26 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% WORKING: Backpropogation (Vectorized Implementation) %%%%%%% % Here X is including 1 column at begining A1 = X; % 5000 x 401 Z2 = A1 * Theta1'; % m x hidden_layer_size == 5000 x 25 A2 = sigmoid(Z2); % m x hidden_layer_size == 5000 x 25 A2 = [ones(size(A2,1),1), A2]; % Adding 1 as first column in z = (Adding bias unit) % m x (hidden_layer_size + 1) == 5000 x 26 Z3 = A2 * Theta2'; % m x num_labels == 5000 x 10 A3 = sigmoid(Z3); % m x num_labels == 5000 x 10 % h_x = a3; % m x num_labels == 5000 x 10 y_Vec = (1:num_labels)==y; % m x num_labels == 5000 x 10 DELTA3 = A3 - y_Vec; % 5000 x 10 DELTA2 = (DELTA3 * Theta2) .* [ones(size(Z2,1),1) sigmoidGradient(Z2)]; % 5000 x 26 DELTA2 = DELTA2(:,2:end); % 5000 x 25 %Removing delta2 for bias node Theta1_grad = (1/m) * (DELTA2' * A1); % 25 x 401 Theta2_grad = (1/m) * (DELTA3' * A2); % 10 x 26 %%%%%%%%%%%% WORKING: DIRECT CALCULATION OF THETA GRADIENT WITH REGULARISATION %%%%%%%%%%% % %Regularization term is later added in Part 3 % Theta1_grad = (1/m) * Theta1_grad + (lambda/m) * [zeros(size(Theta1, 1), 1) Theta1(:,2:end)]; % 25 x 401 % Theta2_grad = (1/m) * Theta2_grad + (lambda/m) * [zeros(size(Theta2, 1), 1) Theta2(:,2:end)]; % 10 x 26 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%% Part 3: Adding Regularisation term in J and Theta_grad %%%%%%%%%%%%% reg_term = (lambda/(2*m)) * (sum(sum(Theta1(:,2:end).^2)) + sum(sum(Theta2(:,2:end).^2))); %scalar %Costfunction With regularization J = J + reg_term; %scalar %Calculating gradients for the regularization Theta1_grad_reg_term = (lambda/m) * [zeros(size(Theta1, 1), 1) Theta1(:,2:end)]; % 25 x 401 Theta2_grad_reg_term = (lambda/m) * [zeros(size(Theta2, 1), 1) Theta2(:,2:end)]; % 10 x 26 %Adding regularization term to earlier calculated Theta_grad Theta1_grad = Theta1_grad + Theta1_grad_reg_term; Theta2_grad = Theta2_grad + Theta2_grad_reg_term; % ------------------------------------------------------------- % ========================================================================= % Unroll gradients grad = [Theta1_grad(:) ; Theta2_grad(:)]; end

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