吴恩达_Machine Learning_Programming Exercise 3: Multi-class Classification and Neural Networks

1 Multi-class Classification

1.1 Dataset

读取数据：

% Load saved matrices from file
load('ex3data1.mat');
% The matrices X and y will now be in your Octave environment

ex3data1.mat 中有 5000 个训练样例，其中每个训练样例是一个 20 像素乘 20 像素的数字灰度图像。 每个像素由一个浮点数表示，表示该位置的灰度强度。 20 x 20 像素网格被“展开”成一个 400 维向量。 这些训练示例中的每一个都成为我们数据矩阵 X 中的单行。这为我们提供了一个 5000 x 400 矩阵 X，其中每一行都是手写数字图像的训练示例。

训练集的第二部分是一个 5000 维的向量 y，其中包含训练集的标签。没有零索引，我们将数字零映射到值十。 因此，数字“0”被标记为“10”，而数字“1”到“9”按照自然顺序标记为“1”到“9”。 

1.2 Visualizing the data

取一百行，打印出来图像：

1.3 Vectorizing Logistic Regression

1.3.1 Vectorizing the cost function

---

(unregularized) logistic regression, cost function：

令：

Xθ：

1.3.2 Vectorizing the gradient

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the gradient of the (unregularized) logistic regression cost 

 

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1.3.3 Vectorizing regularized logistic regression

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regularized logistic regression, the cost function：

regularized logistic regression cost for θj：

---

lrCostFunction.m：

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)
%

theta_1=[0;theta(2:end,1)];
J=1/m*(-y'*log(sigmoid(X*theta))-(1-y)'*log(1-sigmoid(X*theta)))+lambda/(2*m)*(theta_1'*theta_1);
grad=1/m*X'*(sigmoid(X*theta)-y)+lambda/m*theta_1;


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

grad = grad(:);

end

1.4 One-vs-all Classification

多类分类器的函数：

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);

% You need to return the following variables correctly 
all_theta = zeros(num_labels, n + 1);

% 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 c=1:num_labels
    initial_theta = zeros(n + 1, 1);
    options = optimset('GradObj', 'on', 'MaxIter', 50);
    [theta] = fmincg (@(t)(lrCostFunction(t, X, (y == c), lambda)), ...
        initial_theta, options);
    all_theta(c,:)=theta';


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


end

1.4.1 One-vs-all Prediction

预测，使用上面训练好的参数theta做一个多分类的工作：

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.
%       


[q,p]=max((X*all_theta'),[],2);




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


end

2 Neural Networks

2.1 Model representation

导入数据：

% Load saved matrices from file
load('ex3weights.mat');
% The matrices Theta1 and Theta2 will now be in your Octave
% environment
% Theta1 has size 25 x 401
% Theta2 has size 10 x 26

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.
%

p = zeros(size(X, 1), 1);
X = [ones(m,1) X];
z_2 = X* Theta1';
a_2 = sigmoid(z_2);
m_2 = size(a_2, 1);
a_2 = [ones(m_2,1) a_2];
z_3 = a_2* Theta2';
a_3 = sigmoid(z_3);

[~, p] = max(a_3,[],2);

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


end

最后一行[~, p] = max(a_3,[],2);中，a_3维度为5000×10，max(a_3,[],2)返回a_3中每行最大值以及这个最大值所在的列数，也就是它的label。最大值这项赋空（因为~），所在位置赋给p。