function W = randInitializeWeights(L_in, L_out)

%RANDINITIALIZEWEIGHTS Randomly initialize the weights of a layer with L_in

%incoming connections and L_out outgoing connections

%   W = RANDINITIALIZEWEIGHTS(L_in, L_out) randomly initializes the weights

%   of a layer with L_in incoming connections and L_out outgoing

%   connections.

%

%   Note that W should be set to a matrix of size(L_out,  + L_in) as

%   the column row of W handles the "bias" terms

%

W = zeros(L_out,  + L_in);

% Instructions: Initialize W randomly so that we break the symmetry while

%               training the neural network.

%

% Note: The first row of W corresponds to the parameters for the bias units

%

epsilon_init = sqrt() / (sqrt(L_out+L_in));

W = rand(L_out,  + L_in) *  * epsilon_init - epsilon_init;

end

function numgrad = computeNumericalGradient(J, theta)

%COMPUTENUMERICALGRADIENT Computes the gradient using "finite differences"

%and gives us a numerical estimate of the gradient.

%   numgrad = COMPUTENUMERICALGRADIENT(J, theta) computes the numerical

%   gradient of the function J around theta. Calling y = J(theta) should

%   return the function value at theta.

% Notes: The following code implements numerical gradient checking, and

%        returns the numerical gradient.It sets numgrad(i) to (a numerical

%        approximation of) the partial derivative of J with respect to the

%        i-th input argument, evaluated at theta. (i.e., numgrad(i) should

%        be the (approximately) the partial derivative of J with respect

%        to theta(i).)

%                

numgrad = zeros(size(theta));

perturb = zeros(size(theta));

e = 1e-4;

for p = 1:numel(theta)

    % Set perturbation vector

    perturb(p) = e;

    % Compute Numerical Gradient

    numgrad(p) = ( J(theta + perturb) - J(theta - perturb)) / (2*e);

    perturb(p) = 0;

end

end

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: 1->2; Theta2: 2->3

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);

J = 0;

Theta1_grad = zeros(size(Theta1));

Theta2_grad = zeros(size(Theta2));

%         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.

for i = 1:m

    % compute activation by Forward Propagation

    a1 = [1; X(i,:)'];

    z2 = Theta1 * a1;

    a2 = [1; sigmoid(z2)];

    z3 = Theta2 * a2;

    h = sigmoid(z3);

    yy = zeros(num_labels,1);

    yy(y(i)) = 1;              % 训练集的真实值yy

    J = J + sum(-yy .* log(h) - (1-yy) .* log(1-h));

    % Back Propagation

    delta3 = h - yy;

    delta2 = (Theta2(:,2:end)' * delta3) .* sigmoidGradient(z2); %注意要除去偏移单元的连接权重

    Theta2_grad = Theta2_grad + delta3 * a2';

    Theta1_grad = Theta1_grad + delta2 * a1';

end

J = J / m + lambda * (sum(sum(Theta1(:,2:end) .^ 2)) + sum(sum(Theta2(:,2:end) .^ 2))) / (2*m);

Theta2_grad = Theta2_grad / m;

Theta2_grad(:,2:end) = Theta2_grad(:,2:end) + lambda * Theta2(:,2:end) / m; % regularized nn

Theta1_grad = Theta1_grad / m;

Theta1_grad(:,2:end) = Theta1_grad(:,2:end) + lambda * Theta1(:,2:end) / m; % regularized nn

% Unroll gradients

grad = [Theta1_grad(:) ; Theta2_grad(:)];

end