Andrew Ng机器学习 二： Logistic Regression

一：逻辑回归（Logistic Regression）

　　背景：假设你是一所大学招生办的领导，你依据学生的成绩，给与他入学的资格。现在有这样一组以前的数据集ex2data1.txt，第一列表示第一次测验的分数，第二列表示第二次测验的分数，第三列1表示允许入学，0表示不允许入学。现在依据这些数据集，设计出一个模型，作为以后的入学标准。

　　

　　我们通过可视化这些数据集，发现其与某条直线方程有关，而结果又只有两类，故我们接下来使用逻辑回归去拟合该数据集。

　　

　　1，回归方程的脚本ex2.m:

%% Machine Learning Online Class - Exercise : Logistic Regression
%
%  Instructions
%  ------------
%
%  This file contains code that helps you get started on the logistic
%  regression exercise. You will need to complete the following functions
%  in this exericse:
%
%     sigmoid.m
%     costFunction.m
%     predict.m
%     costFunctionReg.m
%
%  For this exercise, you will not need to change any code in this file,
%  or any other files other than those mentioned above.
%

%% Initialization
clear ; 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(:, [, ]); y = data(:, );

%% ==================== Part : 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 Legend
xlabel('Exam 1 score')
ylabel('Exam 2 score')

% Specified in plot order
legend('Admitted', 'Not admitted')
hold off;

fprintf('\nProgram paused. Press enter to continue.\n');
pause;

%% ============ Part : 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_test
X = [ones(m, ) X];

% Initialize fitting parameters
initial_theta = zeros(n + , );

% Compute and display initial cost and gradient
[cost, grad] = costFunction(initial_theta, X, y);

fprintf('Cost at initial theta (zeros): %f\n', cost);
fprintf('Expected cost (approx): 0.693\n');
fprintf('Gradient at initial theta (zeros): \n');
fprintf(' %f \n', grad);
fprintf('Expected gradients (approx):\n -0.1000\n -12.0092\n -11.2628\n');

% Compute and display cost and gradient with non-zero theta
test_theta = [-; 0.2; 0.2];
[cost, grad] = costFunction(test_theta, X, y);

fprintf('\nCost at test theta: %f\n', cost);
fprintf('Expected cost (approx): 0.218\n');
fprintf('Gradient at test theta: \n');
fprintf(' %f \n', grad);
fprintf('Expected gradients (approx):\n 0.043\n 2.566\n 2.647\n');

fprintf('\nProgram paused. Press enter to continue.\n');
pause;

%% ============= Part : Optimizing using fminunc  =============
%  In this exercise, you will use a built-in function (fminunc) to find the
%  optimal parameters theta.

%  Set options for fminunc
options = optimset('GradObj', 'on', 'MaxIter', );

%  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 screen
fprintf('Cost at theta found by fminunc: %f\n', cost);
fprintf('Expected cost (approx): 0.203\n');
fprintf('theta: \n');
fprintf(' %f \n', theta);
fprintf('Expected theta (approx):\n');
fprintf(' -25.161\n 0.206\n 0.201\n');

% Plot Boundary
plotDecisionBoundary(theta, X, y);

% Put some labels
hold on;
% Labels and Legend
xlabel('Exam 1 score')
ylabel('Exam 2 score')

% Specified in plot order
legend('Admitted', 'Not admitted')
hold off;

fprintf('\nProgram paused. Press enter to continue.\n');
pause;

%% ============== Part : 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  on exam  and
%  score  on exam  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  on exam
%  and score  on exam  

prob = sigmoid([  ] * theta);
fprintf(['For a student with scores 45 and 85, we predict an admission ' ...
         'probability of %f\n'], prob);
fprintf('Expected value: 0.775 +/- 0.002\n\n');

% Compute accuracy on our training set
p = predict(theta, X);

fprintf('Train Accuracy: %f\n', mean(double(p == y)) * );
fprintf('Expected accuracy (approx): 89.0\n');
fprintf('\n');

ex2.m

　　

　　2,可视化数据plotData.m：

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.
%

   pos=find(y==);
  neg=find(y==);
  plot(X(pos,),X(pos,),'k+','LineWidth',,'MarkerSize',);
  plot(X(neg,),X(neg,),'ko','MarkerFaceColor','y','MarkerSize',);

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

hold off;

end

plotData.m

　　

　　3，逻辑回归的逻辑函数（Sigmoid Function/Logistic Function）：

　　$h_{\theta}(x)=g(\theta^{T}x)$ ：表示在输入为$x$，预测为$y=1$的概率

　　$g(z)=\frac{1}{1+e^{-z}}$　　

function g = sigmoid(z)
%SIGMOID Compute sigmoid function
%   g = 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=./(+exp(-z));

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

end

sigmoid.m

　　4，逻辑回归的代价函数：

　　$J(\theta)=-\frac{1}{m}\sum_{i=1}^{m}[y^{(i)}log(h_\theta(x^{(i)}))+(1-y^{(i)})log(1-h_{\theta}(x^{(i)}))]$

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
J = ;
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
%
% Note: grad should have the same dimensions as theta
%

  h=sigmoid(X*theta); %求hθ(x)
  J=-sum(y.*log(h)+(-y).*log(-h))/m; %代价函数

  grad=(X')*(h-y)./m; %梯度下降,没有学习速率α，之后给我们调用内置函数fminunc使用

##  h=sigmoid(X*theta);
##J=sum(-y'*log(h)-(1-y)'*log(-h))/m;
##grad=((h-y)'*X)/m;

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

end

costFunction.m

　　5，带学习速率$\alpha$的梯度下降：

　　$\theta_j:=\theta_j-\frac{\alpha}{m }\sum_{i=1}^{m}[(h_\theta(x^{(i)})-y^{(i)})x^{(i)}_j]$

　　

　　不带学习速率$\alpha$的梯度下降(给之后fminunc作为梯度下降使用)：

　　$\frac{\partial J(\theta)}{\partial \theta_j}=\frac{1}{m}\sum_{i=1}^{m}[(h_\theta(x^{(i)})-y^{(i)})x^{(i)}_j]$

　　使用内置fminunc函数来拟合参数$\theta$，之前我们是使用梯度下降来拟合参数$\theta$的，在这同样也能使用，不过我们这里使用内置fminunc函数来去拟合，它会自动选择学习速率$\alpha$，不需要我们手工选择，我们只需要给定一个迭代次数，一个写好的代价函数，初始化$\theta$，最后它会为我们找到最优的$\theta$，它像可以加强版的梯度下降法。

options = optimset('GradObj', 'on', 'MaxIter', );
[theta, cost] = ...
    fminunc(@(t)(costFunction(t, X, y)), initial_theta, options);//自己写好的costFunction函数

　　

　　6，根据拟合好的参数$\theta$，预测数据，例如我们想预测某学生第一次分数为45，第二次分数为85，该学生能入学的概率为：

prob = sigmoid([  ] * theta); %入学的概率

　　预测样本X，我们可以看到预测的准确率为89%。

function p = predict(theta, X)
%PREDICT Predict whether the label is  or  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, ); % Number of training examples

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

% ====================== 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 's and 1's
%

  %第一种
  for i=:m
    p(i,)=sigmoid(X(i,:)*theta)>=0.5; %预测每一个样本的结果，大于0.5为正向类
  end;

  %第二种
  %
##  ans=sigmoid(X*theta);
##  for i=:m
##      if(ans(i,)>=0.5)
##        p(i,)=;
##      else
##        p(i,)=;
##  end

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

end

predict.m

二：正则化逻辑回归（Regularized logistic regression）：

　　背景：假如你是某所工厂的管理员，该工厂生产芯片，每片芯片要经过两次测试后，达到标准方可通过，现在有一组以前的数据集ex2data2.txt，第一列为第一次测试的结果，第二列为第二次测试的结果，第三列1表示该芯片合格，0表示不合格。现在要你通过这些数据，拟合出一个模型，这个模型将作为以后判断芯片是否合格的标准。

　　

　　我们通过可视化这些数据集，发现其与某条复杂的曲线方程有关，而数据集只有两个特征$x_1$和$x_2$，显然是拟合不出曲线，那么我们可以通过原本的两个特征创造出更多的特征，将原本的特征映射为6次幂，这样我们就得到了28维的特征向量。当特征多了的话，很可能会出现过拟合，显然这不是我们想要的（即是它能很好的拟合原训练集，但预测新样本的能力会很低）。

构造更多的特征：

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.^, X2.^, X1*X2, X1*X2.^, etc..
%
%   Inputs X1, X2 must be the same size
%

degree = ;
out = ones(size(X1(:,)));
for i = :degree
    for j = :i
        out(:, end+) = (X1.^(i-j)).*(X2.^j);
    end
end

end

mapFeature.m

所以这时我们使用正则化(Regularization)来解决过拟合的问题。

　　1，正则化回归的脚本ex2.m:　

%% Machine Learning Online Class - Exercise : Logistic Regression
%
%  Instructions
%  ------------
%
%  This file contains code that helps you get started on the second part
%  of the exercise which covers regularization with logistic regression.
%
%  You will need to complete the following functions in this exericse:
%
%     sigmoid.m
%     costFunction.m
%     predict.m
%     costFunctionReg.m
%
%  For this exercise, you will not need to change any code in this file,
%  or any other files other than those mentioned above.
%

%% 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(:, [, ]); y = data(:, );

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 : 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(:,), X(:,)); %c从原来的二维变成了28(+1截距项)维，m*

% Initialize fitting parameters
initial_theta = zeros(size(X, ), );

% Set regularization parameter lambda to
lambda = ;

% 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('Expected cost (approx): 0.693\n');
fprintf('Gradient at initial theta (zeros) - first five values only:\n');
fprintf(' %f \n', grad(:));
fprintf('Expected gradients (approx) - first five values only:\n');
fprintf(' 0.0085\n 0.0188\n 0.0001\n 0.0503\n 0.0115\n');

fprintf('\nProgram paused. Press enter to continue.\n');
pause;

% Compute and display cost and gradient
% with all-ones theta and lambda =
test_theta = ones(size(X,),);
[cost, grad] = costFunctionReg(test_theta, X, y, );

fprintf('\nCost at test theta (with lambda = 10): %f\n', cost);
fprintf('Expected cost (approx): 3.16\n');
fprintf('Gradient at test theta - first five values only:\n');
fprintf(' %f \n', grad(:));
fprintf('Expected gradients (approx) - first five values only:\n');
fprintf(' 0.3460\n 0.1614\n 0.1948\n 0.2269\n 0.0922\n');

fprintf('\nProgram paused. Press enter to continue.\n');
pause;

%% ============= Part : 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 (, , , ).
%
%  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, ), );

% Set regularization parameter lambda to  (you should vary this)
lambda = ;

% Set Options
options = optimset('GradObj', 'on', 'MaxIter', );

% 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)) * );
fprintf('Expected accuracy (with lambda = 1): 83.1 (approx)\n');

ex2_reg.m

　　2，正则化逻辑回归代价函数(忽略偏差项$\theta_0$的正则化)：

　　$J(\theta)=-\frac{1}{m}\sum_{i=1}^{m}[y^{(i)}log(h_\theta(x^{(i)}))+(1-y^{(i)})log(1-h_{\theta}(x^{(i)}))]+\frac{\lambda }{2m}\sum_{j=1}^{n}\theta_j^{2}$

　　

　　3，梯度下降：

　　带学习速率：

　　　　$\theta_0:=\theta_0-\alpha \frac{1}{m }\sum_{i=1}^{m}[(h_\theta(x^{(i)})-y^{(i)})x^{(i)}_0]$   for $j=0$

　　　　$\theta_j:=\theta_j-\alpha (\frac{1}{m }\sum_{i=1}^{m}[(h_\theta(x^{(i)})-y^{(i)})x^{(i)}_j]+\frac{\lambda }{m}\theta_j)$  for $j\geq 1$

　　不带学习速率(给之后fminunc作为梯度下降使用)：

　　　　$\frac{\partial J(\theta)}{\partial \theta_0}=\frac{1}{m}\sum_{i=1}^{m}[(h_\theta(x^{(i)})-y^{(i)})x^{(i)}_0]$  for $j=0$

　　　　$\frac{\partial J(\theta)}{\partial \theta_j}=(\frac{1}{m}\sum_{i=1}^{m}[(h_\theta(x^{(i)})-y^{(i)})x^{(i)}_j])+\frac{\lambda }{m}\theta_j $ for $j\geq 1$

　　

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 = ;
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

  h=sigmoid(X*theta);
  n=size(X,);
  J=(-(y')*log(h)-(1-y)'*log(-h))/m+(lambda/(*m))*sum(theta([:n],:).^); %忽略偏差项θ()的影响

  grad(,)=((X(:,)')*(h-y))/m; %梯度下降
  grad([:n],:)=(X(:,[:n])')*(h-y)./m+(theta([2:n],:)).*(lambda/m);

##h=sigmoid(X*theta);
##theta(,)=;
##J=sum(-y'*log(h)-(1-y)'*log(-h))/m+lambda//m*sum(power(theta,));
##grad=((h-y)'*X)/m+lambda/m*theta';
% =============================================================

end

costFunctionReg.m

　　我们可以选择不同的$\lambda$大小去拟合数据集并可视化，选择一个较优的$lambda$。

　　4，预测方法跟逻辑回归差不多，只是现在加入要预测第一次分数为45，第二次分数为80时，要先将这两个特征放到mapFeature函数构造。

我的标签：做个有情怀的程序员。