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
%

X = [ones(m, 1), X];  % add a first colum of ones (bias term)
A_2 = sigmoid(X*Theta1');
A_2 = [ones(m, 1), A_2];  % (bias term)
A_3 = sigmoid(A_2*Theta2');
h_0 = A_3;
%disp(round(h_0));

% y is 1x5000 and holds the labels as numbers, turn it into 5000x10,
% each row holding the label as vectors, e.g.  [0 1 0 0 0 ... ] for 2.
y = eye(num_labels)(y,:);  % y is used as an index, it gets a row,
                           % e.g. [0 0 0 1 ... 0 0]
assert(size(y) == [m num_labels]);


J = 1/m * sum(sum(-y.*log(h_0) - (1-y).*log(1-h_0)));
assert(size(J) == [1 1]);

% 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.
%

D_1 = zeros(size(Theta1));
D_2 = zeros(size(Theta2));

for t = 1:m
  % feed forward this training sample
  % ---------------------------------------------------------------------------
  a_1 = X(t,:);
  % (X already has 1-column)
  assert(size(a_1) == [1, input_layer_size+1]);

  z_2 = a_1*Theta1';
  a_2 = sigmoid(z_2);
  a_2 = [ones(size(a_2, 1)), a_2];  % (bias term)
  assert(size(a_2) == [1, hidden_layer_size+1]);

  z_3 = a_2*Theta2';
  a_3 = sigmoid(z_3);
  h_0 = a_3;
  assert(size(h_0) == [1, num_labels]);

  % back propagate / error
  % ---------------------------------------------------------------------------
  assert(size(y) == [m num_labels]);
  d_3 = a_3 - y(t,:);
  assert(size(d_3) == [1, num_labels]);

  d_2 = d_3*Theta2 .* [1, sigmoidGradient(z_2)];
  d_2 = d_2(2:end);
  assert(size(d_2) == [1, hidden_layer_size]);

  % accumulate over all m training examples
  D_2 = D_2 + d_3'*a_2;
  D_1 = D_1 + d_2'*a_1;
end

% average
D_2 /= m;
D_1 /= m;
Theta2_grad = D_2;
Theta1_grad = D_1;

% 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.
%

% Note: Theta1/2 are matrixes here, we want all their rows, but skip their
% first column (not regularizing the bias term).
J_regularization_term = lambda/(2*m) * ...
                      (sum(sum(Theta1(:,2:end).^2)) ...
                       + sum(sum(Theta2(:,2:end).^2)));
assert(size(J_regularization_term) == [1 1]);
J += J_regularization_term;

Theta2_grad_regularization_term = lambda/m * [zeros(size(Theta2, 1), 1) Theta2(:,2:end)];
Theta1_grad_regularization_term = lambda/m * [zeros(size(Theta1, 1), 1) Theta1(:,2:end)];
Theta2_grad += Theta2_grad_regularization_term;
Theta1_grad += Theta1_grad_regularization_term;

% -------------------------------------------------------------

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

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


end