Coursera吴恩达机器学习编程作业（ex4）包含BackPropagation算法的NeuralNetwork多层分类

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));

%传入的参数y是一个列向量，每个元素就是每个样本对应的数字
%有两个求和，依然是向量化后进行矩阵运算，最后用两次sum即可
%在神经网络的CostFunction中要求y和h用0/1矩阵来表示，所以对于每个样本，其结果要从数字改成一个0/1向量
%最后的J应是一个数
X=[ones(m,1) X];
z2=X*Theta1';
a2=[ones(m,1) sigmoid(z2)];
z3=a2*Theta2';
h=sigmoid(z3);%h的元素并非都是0/1而是代表概率的数，不需要把他们改为0/1！！！但是CostFunction公式中的y一定要改成0/1矩阵！！！

%非常骚气的操作，for循环真正作用的地方：
yk=zeros(m,num_labels);
for i=1:m
  yk(i,y(i))=1;%得到修改后用于公式中的0/1矩阵
end;
J=(1/m)*sum(sum(-1*yk.*log(h)-(1.-yk).*log(1.-h)));

%正则项,注意Theta1和Theta2里的第一列是怎么去除的
r=(lambda/(2*m))*(sum(sum(Theta1(:,2:end).^2))+sum(sum(Theta2(:,2:end).^2)));
J=J+r


%BackPropagation
for ex=1:m
  a1=X(ex,:);
  a1=a1';
  z2=Theta1*a1;
  a2=[1;sigmoid(z2)];
  z3=Theta2*a2;
  a3=sigmoid(z3);
  y=yk(ex,:);
  delta3=a3-y';
  delta2=Theta2(:,2:end)'*delta3.*sigmoidGradient(z2);
  Theta1_grad=Theta1_grad+delta2*a1';
  Theta2_grad=Theta2_grad+delta3*a2';
end;

Theta1_grad=Theta1_grad./m;
Theta2_grad=Theta2_grad./m;
%正则化
Theta1(:,1)=0;
Theta2(:,1)=0;
Theta1_grad=Theta1_grad+lambda/m*Theta1;
Theta2_grad=Theta2_grad+lambda/m*Theta2;

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


end

BP算法的流程，结合代码理解！