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;

% ====================== 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
%         the cost function with respect to Theta1 and Theta2 in Theta1_grad and
%         Theta2_grad, respectively. After implementing Part 2, you can check
%
%         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
%               and Theta2_grad from Part 2.
%
%数据集X:5000*400
%y:1*m的向量
yNum = zeros(m,num_labels);
for i=1:m
yNum(i,y(i))=1;
end
%开始训练
a1=X
%5000*25
%第一层开始训练了
%Theta1 25*401
z2=(Theta1 * [ones(m,1) a1]')';
a2=sigmoid(z2);
%5000*10
%第二层开始训练
z3 = (Theta2 * [ones(m, 1) a2]')';
a3 = sigmoid(z3);
%计算代价函数
for i = 1:m
J = J + (-yNum(i, :) * log(a3(i, :))' - (1.- yNum(i, :)) * log(1.- a3(i, :))');
end
%取第i行与a3(算出来的最后结果)的第i列做点积即可
J = J / m
%正则化
t1 = Theta1(:, 2:end);
t2 = Theta2(:, 2:end);
regularization = lambda / (2 * m) * (sum(sum(t1 .^ 2)) + sum(sum(t2 .^ 2)));
J = J + regularization;
%取非1的元素,平方求和
% 5000 x 10
d3 = a3-yNum;
% 5000 x 25
%不包含第1列:全部为1
d2 = (d3 * Theta2(:, 2:end)) .* sigmoidGradient(z2);
%这里是求最后的梯度下降值
%偏差乘以原来的元素就是梯度下降值
a2_with_a0 = [ones(m, 1) a2];
D2 = d3' * a2_with_a0;
%正则化,不对第一行处理,因为是bias量
regularization = lambda / m * [zeros(size(Theta2, 1), 1) Theta2(:, 2:end)];
a1_with_a0 = [ones(m, 1) a1];
D1 = d2' * a1_with_a0;
regularization = lambda / m * [zeros(size(Theta1, 1), 1) Theta1(:, 2:end)];

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

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

end

D是$\Delta$,d是$\delta$,

%evaluated at z
%   evaluated at z. This should work regardless if z is a matrix or a
%   vector. In particular, if z is a vector or matrix, you should return
%   the gradient for each element.

g = zeros(size(z));

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the gradient of the sigmoid function evaluated at
%               each value of z (z can be a matrix, vector or scalar).
%直接计算导函数,每一个元素的导函数值都存进去,所以说之间加了个.
g = sigmoid(z) .* (1.- sigmoid(z));

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

end