%% Initializationclear ; close all; clc%% Load Data%  The first two columns contains the exam scores and the third column%  contains the label.data = load('ex2data1.txt');X = data(:, [1, 2]); y = data(:, 3);%% ==================== Part 1: Plotting ====================%  We start the exercise by first plotting the data to understand the %  the problem we are working with.fprintf(['Plotting data with + indicating (y = 1) examples and o ' ...         'indicating (y = 0) examples./n']);plotData(X, y);% Put some labels hold on;% Labels and Legendxlabel('Exam 1 score')ylabel('Exam 2 score')% Specified in plot orderlegend('Admitted', 'Not admitted')hold off;fprintf('/nProgram paused. Press enter to continue./n');pause;%% ============ Part 2: Compute Cost and Gradient ============%  In this part of the exercise, you will implement the cost and gradient%  for logistic regression. You neeed to complete the code in %  costFunction.m%  Setup the data matrix appropriately, and add ones for the intercept term[m, n] = size(X);% Add intercept term to x and X_testX = [ones(m, 1) X];% Initialize fitting parametersinitial_theta = zeros(n + 1, 1);% Compute and display initial cost and gradient[cost, grad] = costFunction(initial_theta, X, y);fprintf('Cost at initial theta (zeros): %f/n', cost);fprintf('Gradient at initial theta (zeros): /n');fprintf(' %f /n', grad);fprintf('/nProgram paused. Press enter to continue./n');pause;%% ============= Part 3: Optimizing using fminunc  =============%  In this exercise, you will use a built-in function (fminunc) to find the%  optimal parameters theta.%  Set options for fminuncoptions = optimset('GradObj', 'on', 'MaxIter', 400);%  Run fminunc to obtain the optimal theta%  This function will return theta and the cost [theta, cost] = ...    fminunc(@(t)(costFunction(t, X, y)), initial_theta, options);% Print theta to screenfprintf('Cost at theta found by fminunc: %f/n', cost);fprintf('theta: /n');fprintf(' %f /n', theta);% Plot BoundaryplotDecisionBoundary(theta, X, y);% Put some labels hold on;% Labels and Legendxlabel('Exam 1 score')ylabel('Exam 2 score')% Specified in plot orderlegend('Admitted', 'Not admitted')hold off;fprintf('/nProgram paused. Press enter to continue./n');pause;%% ============== Part 4: Predict and Accuracies ==============%  After learning the parameters, you'll like to use it to predict the outcomes%  on unseen data. In this part, you will use the logistic regression model%  to predict the probability that a student with score 45 on exam 1 and %  score 85 on exam 2 will be admitted.%%  Furthermore, you will compute the training and test set accuracies of %  our model.%%  Your task is to complete the code in predict.m%  Predict probability for a student with score 45 on exam 1 %  and score 85 on exam 2 prob = sigmoid([1 45 85] * theta);fprintf(['For a student with scores 45 and 85, we predict an admission ' ...         'probability of %f/n/n'], prob);% Compute accuracy on our training setp = predict(theta, X);fprintf('Train Accuracy: %f/n', mean(double(p == y)) * 100);fprintf('/nProgram paused. Press enter to continue./n');pause;首先可以在feature（x_1,x_2）平面上visualize出訓練數據集， 對正實例和負實例用不同記號表示如下
圖中橫縱坐標對應兩門課程成績，兩類點分別對應錄取的學生和不錄取的學生。 畫圖的code如下
[plain] view plain copy print?function plotData(X, y)  %PLOTDATA Plots the data points X and y into a new figure   %   PLOTDATA(x,y) plots the data points with + for the positive examples  %   and o for the negative examples. X is assumed to be a Mx2 matrix.    % Create New Figure  figure; hold on;  % ====================== YOUR CODE HERE ======================  % Instructions: Plot the positive and negative examples on a  %               2D plot, using the option ‘k+’ for the positive  %               examples and ‘ko’ for the negative examples.    % find all the indices of postive and negtive training example  pos = find(y == 1); neg = find(y == 0);   plot(X(pos,1), X(pos,2), ‘k+’, ‘LineWidth’, 2, ‘MarkerSize’, 7);  plot(X(neg,1), X(neg,2), ‘ko’, ‘LineWidth’, 2, ‘MarkerSize’, 7, ‘MarkerFaceColor’, ‘y’);  % =========================================================================  hold off;    end  
function plotData(X, y)%PLOTDATA Plots the data points X and y into a new figure %   PLOTDATA(x,y) plots the data points with + for the positive examples%   and o for the negative examples. X is assumed to be a Mx2 matrix.% Create New Figurefigure; hold on;% ====================== YOUR CODE HERE ======================% Instructions: Plot the positive and negative examples on a%               2D plot, using the option 'k+' for the positive%               examples and 'ko' for the negative examples.% find all the indices of postive and negtive training examplepos = find(y == 1); neg = find(y == 0); plot(X(pos,1), X(pos,2), 'k+', 'LineWidth', 2, 'MarkerSize', 7);plot(X(neg,1), X(neg,2), 'ko', 'LineWidth', 2, 'MarkerSize', 7, 'MarkerFaceColor', 'y');% =========================================================================hold off;end然后可以實現sigmoid函數如下，用于將feature的線性組合或者非線性組合映射到[0.1]之間
[plain] view plain copy print?function g = sigmoid(z)  %SIGMOID Compute sigmoid functoon  %   J = SIGMOID(z) computes the sigmoid of z.    % You need to return the following variables correctly   g = zeros(size(z));    % ====================== YOUR CODE HERE ======================  % Instructions: Compute the sigmoid of each value of z (z can be a matrix,  %               vector or scalar).  g = 1.0 ./ (1.0 + exp(-z));  % =============================================================    end  
function g = sigmoid(z)%SIGMOID Compute sigmoid functoon%   J = SIGMOID(z) computes the sigmoid of z.% You need to return the following variables correctly g = zeros(size(z));% ====================== YOUR CODE HERE ======================% Instructions: Compute the sigmoid of each value of z (z can be a matrix,%               vector or scalar).g = 1.0 ./ (1.0 + exp(-z));% =============================================================endsigmoid函數中z與g(z)是正相關的，z越大,g(z)越接近1，反之g(z)越接近0.
下面我們可以實現Logistic Regression的代價函數（cost function） 和梯度函數（gradient），分別如下
cost function可以認為是生成訓練數據的negative log likelihood, 最小化cost function等價于最大似然估計MLE。下面梯度函數的推導過程詳見本文第二部分。這里用的是batch gradient descent，即每更新一個theta_j都需要掃描所有m個訓練樣本。代碼實現如下
[plain] view plain copy print?function [J, grad] = costFunction(theta, X, y)  %COSTFUNCTION Compute cost and gradient for logistic regression  %   J = COSTFUNCTION(theta, X, y) computes the cost of using theta as the  %   parameter for 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   % ====================== 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  %  % Note: grad should have the same dimensions as theta  % define cost function and gradient   % use fminunc function to solve the minimul value   hx = sigmoid(X * theta);   J = (1.0/m) * sum(-y .* log(hx) - (1.0 - y) .* log(1.0 - hx));   grad = (1.0/m) .* X’ * (hx - y);     % =============================================================    end  
function [J, grad] = costFunction(theta, X, y)%COSTFUNCTION Compute cost and gradient for logistic regression%   J = COSTFUNCTION(theta, X, y) computes the cost of using theta as the%   parameter for logistic regression and the gradient of the cost%   w.r.t. to the parameters.% Initialize some useful valuesm = length(y); % number of training examples% You need to return the following variables correctly % ====================== 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%% Note: grad should have the same dimensions as theta% define cost function and gradient % use fminunc function to solve the minimul value hx = sigmoid(X * theta); J = (1.0/m) * sum(-y .* log(hx) - (1.0 - y) .* log(1.0 - hx)); grad = (1.0/m) .* X' * (hx - y);% =============================================================end可以求出初始情況下（initial_theta = zeros(n + 1, 1)）的cost function 和gradient。
[plain] view plain copy print?Cost at initial theta (zeros): 0.693147  Gradient at initial theta (zeros):    -0.100000    -12.009217    -11.262842     Program paused. Press enter to continue.  
Cost at initial theta (zeros): 0.693147Gradient at initial theta (zeros):  -0.100000  -12.009217  -11.262842 Program paused. Press enter to continue.然后用Matlab內置的fminunc函數來求解cost function的最小值和對應的參數值/theta.這是一個無約束的優化問題，可以用fminunc函數求解，不必自己實現gradient descent（自己實現也很容易但是用這個函數更方便）。查閱下doc，其支持如下的輸入輸出
[x,fval] = fminunc(fun,x0,options)
input 中 fun是定義待優化的cost func和gradient(可選)的函數，x0是自變量初始值，options是選項設置，在Logistic Regression的實現中相關語句是
[plain] view plain copy print?%  Set options for fminunc  options = optimset(‘GradObj’, ‘on’, ‘MaxIter’, 400);    %  Run fminunc to obtain the optimal theta  %  This function will return theta and the cost   [theta, cost] = fminunc(@(t)(costFunction(t, X, y)), initial_theta, options);  
%  Set options for fminuncoptions = optimset('GradObj', 'on', 'MaxIter', 400);%  Run fminunc to obtain the optimal theta%  This function will return theta and the cost [theta, cost] = fminunc(@(t)(costFunction(t, X, y)), initial_theta, options);
optimset提供了兩個參數選項分別是是否提供gradient和迭代次數。下面調用fminunc是第一個參數@(t)是一個匿名函數的函數句柄，后面costFunction(t, X, y)是函數定義，具體實現在對應的.m函數文件中。
output中是最優參數值x以及最優cost function的值。程序輸出如下
[plain] view plain copy print?Local minimum possible.    fminunc stopped because the final change in function value relative to   its initial value is less than the default value of the function tolerance.    <stopping criteria details>    Cost at theta found by fminunc: 0.203506  theta:    -24.932905    0.204407    0.199617   
Local minimum possible.fminunc stopped because the final change in function value relative to its initial value is less than the default value of the function tolerance.<stopping criteria details>Cost at theta found by fminunc: 0.203506theta:  -24.932905  0.204407  0.199617 這樣就找到了最優的參數theta值和對應的cost function值。將最優參數值對應的decision boundary畫出來就是
下面可以對新的學生測試樣本做預測。并且，可以利用decision對所有訓練樣本中的學生的錄取結果做預測，然后對比實際錄取情況計算train accuracy 。給定一個學生的兩門課程成績的feature，我們可以得到其被錄取的概率即g(/theta’ * x),然后大于等于0.5的認為應該錄取，否則不錄取。實現如下
[plain] view plain copy print?function p = predict(theta, X)  %PREDICT Predict whether the label is 0 or 1 using learned logistic   %regression parameters theta  %   p = PREDICT(theta, X) computes the predictions for X using a   %   threshold at 0.5 (i.e., if sigmoid(theta’*x) >= 0.5, predict 1)    m = size(X, 1); % Number of training examples    % You need to return the following variables correctly  p = zeros(m, 1);    % ====================== YOUR CODE HERE ======================  % Instructions: Complete the following code to make predictions using  %               your learned logistic regression parameters.   %               You should set p to a vector of 0’s and 1’s  %    p = sigmoid(X* theta);  index_1 = find(p >= 0.5);  index_0 = find(p < 0.5);    p(index_1) = ones(size(index_1));  p(index_0) = zeros(size(index_0));    % =========================================================================  end  
function p = predict(theta, X)%PREDICT Predict whether the label is 0 or 1 using learned logistic %regression parameters theta%   p = PREDICT(theta, X) computes the predictions for X using a %   threshold at 0.5 (i.e., if sigmoid(theta'*x) >= 0.5, predict 1)m = size(X, 1); % Number of training examples% You need to return the following variables correctlyp = zeros(m, 1);% ====================== YOUR CODE HERE ======================% Instructions: Complete the following code to make predictions using%               your learned logistic regression parameters. %               You should set p to a vector of 0's and 1's%p = sigmoid(X* theta);index_1 = find(p >= 0.5);index_0 = find(p < 0.5);p(index_1) = ones(size(index_1));p(index_0) = zeros(size(index_0));% =========================================================================end程序輸出如下
[plain] view plain copy print?For a student with scores 45 and 85, we predict an admission probability of 0.774322    Train Accuracy: 89.000000  
For a student with scores 45 and 85, we predict an admission probability of 0.774322Train Accuracy: 89.000000可以知道對于取得45和85的學生錄取的概率是0.774322，train出的model對89%的訓練樣本的預測結果是正確的。
3.2 Regularized Logistic Regression的Matlab實現
現在我們換一個數據集，這個數據集的特征是訓練樣本不再線性可分。比如某公司生產的芯片需要經過兩次質量檢查，然后質檢員根據兩次質量檢查的結果決定是否通過質檢。給定一些歷史的質檢結果和是否通過質檢的決策結果，需要我們對新的測試樣本是否可以通過質檢做預測。這個問題中，芯片也是用兩個feature來描述，即兩次檢查的結果。我們同樣可以基于訓練數據train 一個Logistic Regression model，然后根據model對測試樣本做預測。主程序如下
[plain] view plain copy print?%% Initialization  clear ; close all; clc    %% Load Data  %  The first two columns contains the X values and the third column  %  contains the label (y).    data = load(‘ex2data2.txt’);  X = data(:, [1, 2]); y = data(:, 3);    plotData(X, y);    % Put some labels   hold on;    % Labels and Legend  xlabel(‘Microchip Test 1’)  ylabel(‘Microchip Test 2’)    % Specified in plot order  legend(‘y = 1’, ‘y = 0’)  hold off;      %% =========== Part 1: Regularized Logistic Regression ============  %  In this part, you are given a dataset with data points that are not  %  linearly separable. However, you would still like to use logistic   %  regression to classify the data points.   %  %  To do so, you introduce more features to use – in particular, you add  %  polynomial features to our data matrix (similar to polynomial  %  regression).  %    % Add Polynomial Features    % Note that mapFeature also adds a column of ones for us, so the intercept  % term is handled  X = mapFeature(X(:,1), X(:,2));    % Initialize fitting parameters  initial_theta = zeros(size(X, 2), 1);    % Set regularization parameter lambda to 1  lambda = 1;    % Compute and display initial cost and gradient for regularized logistic  % regression  [cost, grad] = costFunctionReg(initial_theta, X, y, lambda);    fprintf(‘Cost at initial theta (zeros): %f/n’, cost);    fprintf(‘/nProgram paused. Press enter to continue./n’);  pause;    %% ============= Part 2: Regularization and Accuracies =============  %  Optional Exercise:  %  In this part, you will get to try different values of lambda and   %  see how regularization affects the decision coundart  %  %  Try the following values of lambda (0, 1, 10, 100).  %  %  How does the decision boundary change when you vary lambda? How does  %  the training set accuracy vary?  %    % Initialize fitting parameters  initial_theta = zeros(size(X, 2), 1);    % Set regularization parameter lambda to 1 (you should vary this)  lambda = 100;    % Set Options  options = optimset(‘GradObj’, ‘on’, ‘MaxIter’, 400);    % Optimize  [theta, J, exit_flag] = …      fminunc(@(t)(costFunctionReg(t, X, y, lambda)), initial_theta, options);    % Plot Boundary  plotDecisionBoundary(theta, X, y);  hold on;  title(sprintf(‘lambda = %g’, lambda))    % Labels and Legend  xlabel(‘Microchip Test 1’)  ylabel(‘Microchip Test 2’)    legend(‘y = 1’, ‘y = 0’, ‘Decision boundary’)  hold off;    % Compute accuracy on our training set  p = predict(theta, X);    fprintf(‘Train Accuracy: %f/n’, mean(double(p == y)) * 100);  
%% Initializationclear ; close all; clc%% Load Data%  The first two columns contains the X values and the third column%  contains the label (y).data = load('ex2data2.txt');X = data(:, [1, 2]); y = data(:, 3);plotData(X, y);% Put some labels hold on;% Labels and Legendxlabel('Microchip Test 1')ylabel('Microchip Test 2')% Specified in plot orderlegend('y = 1', 'y = 0')hold off;%% =========== Part 1: Regularized Logistic Regression ============%  In this part, you are given a dataset with data points that are not%  linearly separable. However, you would still like to use logistic %  regression to classify the data points. %%  To do so, you introduce more features to use -- in particular, you add%  polynomial features to our data matrix (similar to polynomial%  regression).%% Add Polynomial Features% Note that mapFeature also adds a column of ones for us, so the intercept% term is handledX = mapFeature(X(:,1), X(:,2));% Initialize fitting parametersinitial_theta = zeros(size(X, 2), 1);% Set regularization parameter lambda to 1lambda = 1;% Compute and display initial cost and gradient for regularized logistic% regression[cost, grad] = costFunctionReg(initial_theta, X, y, lambda);fprintf('Cost at initial theta (zeros): %f/n', cost);fprintf('/nProgram paused. Press enter to continue./n');pause;%% ============= Part 2: Regularization and Accuracies =============%  Optional Exercise:%  In this part, you will get to try different values of lambda and %  see how regularization affects the decision coundart%%  Try the following values of lambda (0, 1, 10, 100).%%  How does the decision boundary change when you vary lambda? How does%  the training set accuracy vary?%% Initialize fitting parametersinitial_theta = zeros(size(X, 2), 1);% Set regularization parameter lambda to 1 (you should vary this)lambda = 100;% Set Optionsoptions = optimset('GradObj', 'on', 'MaxIter', 400);% Optimize[theta, J, exit_flag] = ...    fminunc(@(t)(costFunctionReg(t, X, y, lambda)), initial_theta, options);% Plot BoundaryplotDecisionBoundary(theta, X, y);hold on;title(sprintf('lambda = %g', lambda))% Labels and Legendxlabel('Microchip Test 1')ylabel('Microchip Test 2')legend('y = 1', 'y = 0', 'Decision boundary')hold off;% Compute accuracy on our training setp = predict(theta, X);fprintf('Train Accuracy: %f/n', mean(double(p == y)) * 100);首先同樣是visualize 數據如下
可以看出此時兩類樣本不再線性可分。即找不到一條直線很好的區分正樣本和負樣本，如果直接應用Logistic Regression，效果肯定會不好，得出的cost function最優值會比較大。此時，就需要增加feature dimension， 參數/theta的dimension也同樣增加，更多的參數可以描述更豐富的樣本信息。首先做feature mapping，比如基于x1和x2生成階為6以下的所有多項式組合，一共有28項（1+2+3+ … + 7 = 28）如下
實現如下
[plain] view plain copy print?function out = mapFeature(X1, X2)  % MAPFEATURE Feature mapping function to polynomial features  %  %   MAPFEATURE(X1, X2) maps the two input features  %   to quadratic features used in the regularization exercise.  %  %   Returns a new feature array with more features, comprising of   %   X1, X2, X1.^2, X2.^2, X1*X2, X1*X2.^2, etc..  %  %   Inputs X1, X2 must be the same size  %    degree = 6;  out = ones(size(X1(:,1)));  for i = 1:degree      for j = 0:i          out(:, end+1) = (X1.^(i-j)).*(X2.^j);      end  end    end  
function out = mapFeature(X1, X2)% MAPFEATURE Feature mapping function to polynomial features%%   MAPFEATURE(X1, X2) maps the two input features%   to quadratic features used in the regularization exercise.%%   Returns a new feature array with more features, comprising of %   X1, X2, X1.^2, X2.^2, X1*X2, X1*X2.^2, etc..%%   Inputs X1, X2 must be the same size%degree = 6;out = ones(size(X1(:,1)));for i = 1:degree    for j = 0:i        out(:, end+1) = (X1.^(i-j)).*(X2.^j);    endendend這樣通過feature mapping，原始2維feature被映射到的新的28維feature空間，相應的/theta也變成了29維(包括theta_0).在這樣更高維的feature vector上面訓練出來的Logistic Regression Classifier可以有更復雜的decision boundary，不再是一條直線。然而，更多的參數更復雜的decision boundary也容易造成model over-fitting，即過擬合，造成model的泛化能力差。因此我們需要在cost function中引入 regularization term正則項來解決over fitting的問題。
帶regularization term的Logistic Regression的cost function 定義如下
我們在原始cost function后面多加了一項，即參數/theta_j的平方和除以2m，參數lambda控制regularization term的權重。lambda越大，regularization term權重越大，越容易under fit;反之，regularization term權重越小，越容易over fit.但是我們不應該對參數/theta_0進行regularization。cost function的梯度是一個n+1維向量(含j_0),每一維的計算公式是
可以發現僅僅是對j>=1的情況后面多加了一項regularization term對/theta_j求導的結果。因此帶regularization的Logistic Regression的cost function和gradient 實現如下
[plain] view plain copy print?function [J, grad] = costFunctionReg(theta, X, y, lambda)  %COSTFUNCTIONREG Compute cost and gradient for logistic regression with regularization  %   J = COSTFUNCTIONREG(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   hx = sigmoid(X * theta);   J = (1.0/m) * sum(-y .* log(hx) - (1.0 - y) .* log(1.0 - hx)) + lambda / (2 * m) * norm(theta([2:end]))^2;      reg = (lambda/m) .* theta;   reg(1) = 0;   grad = (1.0/m) .* X’ * (hx - y) + reg;  % =============================================================    end  
function [J, grad] = costFunctionReg(theta, X, y, lambda)%COSTFUNCTIONREG Compute cost and gradient for logistic regression with regularization%   J = COSTFUNCTIONREG(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 valuesm = 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 hx = sigmoid(X * theta); J = (1.0/m) * sum(-y .* log(hx) - (1.0 - y) .* log(1.0 - hx)) + lambda / (2 * m) * norm(theta([2:end]))^2; reg = (lambda/m) .* theta; reg(1) = 0; grad = (1.0/m) .* X' * (hx - y) + reg;% =============================================================end其中norm是對向量theta的后面n維求模，用于計算cost function 中的regularization term。cost function的minimize求解仍然使用fminunc函數，與3.1部分一樣。當lambda = 1時，得到的decision boundary如下
Train Accuracy是 83.050847。現在我們把lambda調小，比如設為0.0001,也就是減小regularization term的權重，就會發現分類器幾乎可以把所有training data分類正確，但是得到一條很復雜的decision boundary，因此overfitting
Train Accuracy是86.440678但是模型的泛化能力變差。比如對于x =(0.25,1.5)的芯片會被預測為通過，這顯然和traning數據表現出的特征不符合。相反，如果lambda太大，比如100，那么regularization term權重過大，model容易under fit,如下圖所示，Train Accuracy只有61.016949。
因此選取合適的regularization term的weight對于得到既fit data又有良好泛化能力的model是很重要的。