1.

function [J, grad] = lrCostFunction(theta, X, y, lambda)
%LRCOSTFUNCTION Compute cost and gradient for logistic regression with 
%regularization
%   J = LRCOSTFUNCTION(theta, X, y, lambda) computes the cost of using
%   theta as the parameter for regularized logistic regression and the
%   gradient of the cost w.r.t. to the parameters. 

% Initialize some useful values
m = length(y); % number of training examples

% You need to return the following variables correctly 
J = 0;
grad = zeros(size(theta));

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost of a particular choice of theta.
%               You should set J to the cost.
%               Compute the partial derivatives and set grad to the partial
%               derivatives of the cost w.r.t. each parameter in theta
%
% Hint: The computation of the cost function and gradients can be
%       efficiently vectorized. For example, consider the computation
%
%           sigmoid(X * theta)
%
%       Each row of the resulting matrix will contain the value of the
%       prediction for that example. You can make use of this to vectorize
%       the cost function and gradient computations. 
%
% Hint: When computing the gradient of the regularized cost function, 
%       there're many possible vectorized solutions, but one solution
%       looks like:
%           grad = (unregularized gradient for logistic regression)
%           temp = theta; 
%           temp(1) = 0;   % because we don't add anything for j = 0  
%           grad = grad + YOUR_CODE_HERE (using the temp variable)
%
Z=sum((theta'.*X)')';%依然是X的維度
h=sigmoid(Z);
J=sum((-y)'*log(h)-(1-y)'*log(1-h))/m;
J=J+lambda*sum(theta([2:end],:).^2)/m/2;

grad1=X(:,1)'*(h-y)/m;
grad2=X(:,[2:size(theta)])'*(h-y)/m+lambda*theta([2:size(theta)],:)/m;
grad=[grad1,grad2'];

grad = grad(:);

end
 

總結：在程式設計的時候，遇到了一個問題就是Z的求解時，應該是theta向量對應到每個資料集的對應特徵相乘，所以需要用到.*的形式。除此之外，在grad求解的時候，X不是整個X，而是每個對應有一個特徵,也就是資料集的一列。

2.

function [all_theta] = oneVsAll(X, y, num_labels, lambda)
%ONEVSALL trains multiple logistic regression classifiers and returns all
%the classifiers in a matrix all_theta, where the i-th row of all_theta 
%corresponds to the classifier for label i
%   [all_theta] = ONEVSALL(X, y, num_labels, lambda) trains num_labels
%   logistic regression classifiers and returns each of these classifiers
%   in a matrix all_theta, where the i-th row of all_theta corresponds 
%   to the classifier for label i

% Some useful variables
m = size(X, 1);
n = size(X, 2);
initial_theta = zeros(n + 1, 1);
% You need to return the following variables correctly 
all_theta = zeros(num_labels, n + 1);
options = optimset('GradObj', 'on', 'MaxIter', 50);
% Add ones to the X data matrix
X = [ones(m, 1) X];

% ====================== YOUR CODE HERE ======================
% Instructions: You should complete the following code to train num_labels
%               logistic regression classifiers with regularization
%               parameter lambda. 
%
% Hint: theta(:) will return a column vector.
%
% Hint: You can use y == c to obtain a vector of 1's and 0's that tell you
%       whether the ground truth is true/false for this class.
%
% Note: For this assignment, we recommend using fmincg to optimize the cost
%       function. It is okay to use a for-loop (for c = 1:num_labels) to
%       loop over the different classes.
%
%       fmincg works similarly to fminunc, but is more efficient when we
%       are dealing with large number of parameters.
%
% Example Code for fmincg:
%
%     % Set Initial theta
%     initial_theta = zeros(n + 1, 1);
%     
%     % Set options for fminunc
%     options = optimset('GradObj', 'on', 'MaxIter', 50);
% 
%     % Run fmincg to obtain the optimal theta
%     % This function will return theta and the cost 
%     [theta] = ...
%         fmincg (@(t)(lrCostFunction(t, X, (y == c), lambda)), ...
%                 initial_theta, options);
%
for i=1:num_labels,
   initial_theta = zeros(n + 1, 1);
   options = optimset('GradObj', 'on', 'MaxIter', 50);
   [theta] =fmincg (@(t)(lrCostFunction(t, X, (y == i), lambda)),initial_theta, options);
   all_theta(i,:)=theta';
end

end
 

總結：這裡遇到的問題是如何求解theta。在我們求得J，grad之後，可以使用優化演算法一步步迭代求得theta。因此這裡其實就是每個分類呼叫優化演算法即可。但是要注意，優化演算法的引數在每一次分類呼叫時的初始化問題。因為我們在解析問題的時候，每一個分類對於其他分類來說是完全獨立的。

3.預測

function p = predictOneVsAll(all_theta, X)
%PREDICT Predict the label for a trained one-vs-all classifier. The labels 
%are in the range 1..K, where K = size(all_theta, 1). 
%  p = PREDICTONEVSALL(all_theta, X) will return a vector of predictions
%  for each example in the matrix X. Note that X contains the examples in
%  rows. all_theta is a matrix where the i-th row is a trained logistic
%  regression theta vector for the i-th class. You should set p to a vector
%  of values from 1..K (e.g., p = [1; 3; 1; 2] predicts classes 1, 3, 1, 2
%  for 4 examples) 

m = size(X, 1);
num_labels = size(all_theta, 1);

% You need to return the following variables correctly 
p = zeros(size(X, 1), 1);

% Add ones to the X data matrix
X = [ones(m, 1) X];

% ====================== YOUR CODE HERE ======================
% Instructions: Complete the following code to make predictions using
%               your learned logistic regression parameters (one-vs-all).
%               You should set p to a vector of predictions (from 1 to
%               num_labels).
%
% Hint: This code can be done all vectorized using the max function.
%       In particular, the max function can also return the index of the 
%       max element, for more information see 'help max'. If your examples 
%       are in rows, then, you can use max(A, [], 2) to obtain the max 
%       for each row.
% 
h=X*all_theta';
h_max=max(h,[],2);      
for i=1:size(X,1),
   for j=1:num_labels,
      if h_max(i)==h(i,j),
	     p(i)=j;
		 break;
    end
end
   
end
 

function p = predict(Theta1, Theta2, X)
%PREDICT Predict the label of an input given a trained neural network
%   p = PREDICT(Theta1, Theta2, X) outputs the predicted label of X given the
%   trained weights of a neural network (Theta1, Theta2)

% Useful values
m = size(X, 1);
num_labels = size(Theta2, 1);

% You need to return the following variables correctly 
p = zeros(size(X, 1), 1);

% ====================== YOUR CODE HERE ======================
% Instructions: Complete the following code to make predictions using
%               your learned neural network. You should set p to a 
%               vector containing labels between 1 to num_labels.
%
% Hint: The max function might come in useful. In particular, the max
%       function can also return the index of the max element, for more
%       information see 'help max'. If your examples are in rows, then, you
%       can use max(A, [], 2) to obtain the max for each row.
%
X=[ones(size(X,1),1),X];
a1=sigmoid(X*Theta1');
a1=[ones(size(a1,1),1),a1];
a2=sigmoid(a1*Theta2');
a3=max(a2,[],2);
for i=1:size(X,1),
   for j=1:num_labels,
       if a3(i)==a2(i,j),
	       p(i)=j;
		   break;
    end
end

end

總結：多分類問題，就是選取n個分類總概率最大的那一個作為最終結果。max[A,[],2]是求解A矩陣中每一行最大的值。A(1,2)表示第一行第二列對應的值。