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));

% 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.
%


%
%	Activations ....
%
a1 = [ones(m, 1) X];
a2 = [ones(m,1) sigmoid(a1*Theta1')];
a3 = sigmoid(a2*Theta2');
h = a3;

%
%	Transform y to .....
%
Y = zeros(m,num_labels);
for i=1:m,
	Y(i,y(i))=1;
end

%
%	This is the J without ....
%
J = sum(sum( Y .* log(h) .+ (1-Y) .* log(1-h) )) / (-m);
J = J + (lambda/(2*m))*sum(sum(Theta1(:,2:size(Theta1,2)).^2));
J = J + (lambda/(2*m))*sum(sum(Theta2(:,2:size(Theta2,2)).^2));

%
%	Gradient ....
%
d3 = a3 .- Y;
d2 = d3 * Theta2 .* (a2.*(1-a2));

dlt1 = zeros(size(Theta1));
dlt2 = zeros(size(Theta2));
for i=1:m,
	dlt2 = dlt2 + d3(i,:)'*a2(i,:);
	t = d2(i,:)'*a1(i,:);
	t = t(2:size(t,1),:);
	dlt1 = dlt1 + t;
end

Theta1_grad = dlt1/m;
Theta2_grad = dlt2/m;

%
%	Regularization ....
%
r1 = lambda*Theta1/m;
r2 = lambda*Theta2/m;

t1s = size(Theta1);
r1 = [ zeros(t1s(1),1) r1(:,2:t1s(2))];

t2s = size(Theta2);
r2 = [ zeros(t2s(1),1) r2(:,2:t2s(2))];

Theta1_grad = Theta1_grad + r1;
Theta2_grad = Theta2_grad + r2;


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

% Unroll gradients
grad = [Theta1_grad(:) ; Theta2_grad(:)];


end