nnCostFunction1.m

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
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));

%theta1 = uint8(theta1);
%theta2 = uint8(theta2);				 
				 
% 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.
%
X = [ones(m, 1) X];

for i=1:m
    % activations for each layer
    a1 = X(i,:)';
    z2 = theta1 * a1;
    a2 = [1; sigmoid(z2)];
    z3 = theta2 * a2;
    a3 = sigmoid(z3);
    % final layer activation is output vector
    h = a3;

    % create a boolean vector from a numeric label
    yVec = (1:num_labels) == Y(i);
    J = J + sum(-yVec .* log(h) - (1 - yVec) .* log(1 - h));

    % backpropagation
    delta3 = a3 - yVec;
    delta2 = theta2' * delta3 .* (a2 .* (1 - a2));
    theta2_grad = theta2_grad + delta3 * a2';
    theta1_grad = theta1_grad + delta2(2:end) * a1';
end;

% scaling cost function and gradients
J = J / m;
theta1_grad = theta1_grad / m;
theta2_grad = theta2_grad / m;

% regularization
new = (lambda/(2*m)) * (sum(sum(theta1(:,2:end).^2)) + sum(sum(theta2(:,2:end).^2)));
%new = (lambda / (2 * m)) * (sumsq(theta1(:, 2:end)(:)) + sumsq(theta2(:, 2:end)(:)));
J = J + new;
theta1_grad = theta1_grad + (lambda / m) * [zeros(size(theta1, 1), 1) theta1(:,2:end)];
theta2_grad = theta2_grad + (lambda / m) * [zeros(size(theta2, 1), 1) theta2(:,2:end)];

% =========================================================================

% Unroll gradients
grad = [theta1_grad(:) ; theta2_grad(:)];


end