nnCostFunction.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));

% 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.
%

%warning("error", "all");

%predict output 
x = [ones(size(X, 1), 1) X];
z2 = x*Theta1';
a = sigmoid(z2);
a = [ones(size(a, 1), 1) a];
pred = sigmoid(a*Theta2');

Y_M = [];


%go thought all samples in the data
%predict 
for im=1:m;
    Y = zeros(num_labels, 1);
    Y(y(im)) = 1;

    %mount a global y vector
    Y_M = [Y_M; Y'];

    %calculate the J(theta)
    precalc_1 = -Y'*log(pred(im,:)');
    precalc_2 = (1.-Y)'*log(1-pred(im,:)');

    J += precalc_1 - precalc_2;
end

error_a3 = pred - Y_M;
gradient_a3 = error_a3'*a;
cov_error_a3 = cov(error_a3);

% hidden layer error
error_a2 = (Theta2'*error_a3');
error_a2 = error_a2(2:size(error_a2, 1), :)';
error_a2 = error_a2 .* sigmoidGradient(z2);
gradient_a2 = error_a2'*x;
cov_error_a2 = cov(error_a2);

regularization = sum(sum((Theta2(:, 2:size(Theta2, 2)).^2)'*cov_error_a3));
regularization += sum(sum((Theta1(:, 2:size(Theta1, 2)).^2)'*cov_error_a2));
regularization *= (lambda/2);

J = (J+regularization) / m;

% Calculate the theta regularized
Theta1_r = Theta1;
Theta2_r = Theta2;

for l=1:size(Theta1_r, 1)
    Theta1_r(l, 1) = 0;
end
for l=1:size(Theta2_r, 1)
    Theta2_r(l, 1) = 0;
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%generate a persample gradient matrix, to compute gradient
all_gradient_a3 = zeros(size(gradient_a3, 1), size(gradient_a3, 2), m);
cov_gradient_a3 = zeros(size(gradient_a3, 1), size(gradient_a3, 2), size(gradient_a3, 2));

for it1=1:size(all_gradient_a3, 1)
    for it2=1:size(all_gradient_a3, 2)
       all_gradient_a3(it1, it2, :) = error_a3(:,it1) .* a(:,it2);
    end 
end

for it1=1:size(all_gradient_a3, 1)
     m1 = (reshape(all_gradient_a3(it1, :, :), size(all_gradient_a3, 2), size(all_gradient_a3, 3)))';
     cov_gradient_a3(it1, :, :) = cov( m1 );
end

all_gradient_a2 = zeros(size(gradient_a2, 1), size(gradient_a2, 2), m);
cov_gradient_a2 = zeros(size(gradient_a2, 1), size(gradient_a2, 2), size(gradient_a2, 2));

for it1=1:size(all_gradient_a2, 1)
    for it2=1:size(all_gradient_a2, 2)
        all_gradient_a2(it1, it2, :) = error_a2(:,it1) .* x(:,it2);
    end
end

for it1=1:size(all_gradient_a2, 1)
    m1 = reshape(all_gradient_a2(it1, :, :), size(all_gradient_a2, 2), size(all_gradient_a2, 3))';
    cov_gradient_a2(it1, :, :) = cov( m1 );
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
for it1=1:size(cov_gradient_a3, 1)
    m1 = reshape(cov_gradient_a3(it1, :, :), size(cov_gradient_a3, 2), size(cov_gradient_a3, 3));
    Theta2_r(it1, :) = Theta2_r(it1, :)*m1;
end

for it1=1:size(cov_gradient_a2, 1)
    m1 = reshape(cov_gradient_a2(it1, :, :), size(cov_gradient_a2, 2), size(cov_gradient_a2, 3));
    Theta1_r(it1, :) = Theta1_r(it1, :)*m1;
end
%$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

% calculate the delta error gradient
Theta1_grad = ((gradient_a2) + (Theta1_r*lambda) )/m;
Theta2_grad = ((gradient_a3) + (Theta2_r*lambda) )/m;

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

end