function [J grad] = nnCostFunction(nn_params, ...
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
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));
Theta2_grad = zeros(size(Theta2));
% ====================== 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.
a1 = [ones(m, 1) X];
% a1 has size m*401;Theta1 has size 25*401 ; z2 has size m*25
z2 = a1 * Theta1' ;
a2 = sigmoid(z2);
% a2 has size m*26 ; Theta2 has size 10*26 ; a3 has size m*num_labels
a2 = [ones(m,1) a2];
z3 = a2 * Theta2';
a3 = sigmoid(z3);
Y = zeros(m,num_labels);
Y(i,y(i)) = 1;
theta1 = Theta1;
theta2 = Theta2;
theta1(:,1) = 0;
theta2(:,1) = 0;
firstReg = sum(sum(theta1.*theta1,2));
secondReg = sum(sum(theta2.*theta2,2));
J = 1/m * sum(sum(-Y.*log(a3)-(1 - Y).*log(1-a3),2)) + lambda/(2*m)*(firstReg+secondReg);
Delta1 = zeros(size(Theta1));
Delta2 = zeros(size(Theta2));
for i = 1:m
delta3 = a3(i,:)- Y(i,:);
delta2 = (delta3*Theta2)(:,2:end).*sigmoidGradient(z2(i,:));
Delta2 = Delta2 + delta3'*a2(i,:);
Delta1 = Delta1 + delta2'*a1(i,:);
Theta1_grad = Delta1/m ;
Theta2_grad = Delta2/m ;
Theta2_grad(:,2:end) = Theta2_grad(:,2:end) .+ lambda * Theta2(:,2:end) / m;
Theta1_grad(:,2:end) = Theta1_grad(:,2:end) .+ lambda * Theta1(:,2:end) / m;
% Unroll gradients
grad = [Theta1_grad(:) ; Theta2_grad(:)];