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, 1 + L_in) as
%   the column row of W handles the "bias" terms
%W = zeros(L_out, 1 + 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(6) / (sqrt(L_out+L_in));
W = rand(L_out, 1 + L_in) * 2 * 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 vectorperturb(p) = e;% Compute Numerical Gradientnumgrad(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 Propagationa1 = [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yyJ = 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';
endJ = 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 nnTheta1_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