[杂七杂八] 机器学习: 逻辑回归(Logistic Regression) 小项目

1. Introduction

逻辑回归(Logistic Regression), 在Wiki上的定义如下：

In statistics, logistic regression, or logit regression, or logit model[1] is a regression model where the dependent variable (DV) is categorical. This article covers the case of a binary dependent variable—that is, where it can take only two values, “0” and “1”, which represent outcomes such as pass/fail, win/lose, alive/dead or healthy/sick. Cases where the dependent variable has more than two outcome categories may be analysed in multinomial logistic regression, or, if the multiple categories are ordered, in ordinal logistic regression.[2] In the terminology of economics, logistic regression is an example of a qualitative response/discrete choice model.

以我的理解，逻辑回归也是数据拟合的一种工具，它用于解决分类问题，即结果是离散型的，而这类问题往往用线性回归不能很好的解决。因此，对于分类问题，我们应当选择逻辑回归而不是线性回归。

逻辑回归主要有两类：

• Logistic Regression with two outcome, it often called logistic regression.
• Logistic Regression with more than two outcome, it often call multinomial logistic Regression

对于逻辑回归，那么不得不提到Decision Boundary(决策边界)。决策边界是用于将不同类别的数据划分开来，它也有两种类型：

• Linear decision boundaries 线性决策边界
• Non-linear decision boundaries 非线性决策边界

通常，我们会通过梯度下降或者其它优化算法得到最终的theta。然后利用theta，划出决策边界。注意：决策边界是假设函数的属性，与训练集并没有直接关系，它是由假设函数的theta决定的。

逻辑回归的假设函数，Sigmoid函数，这个函数的特征如下：

• 当X为0时，函数值为0.5；
• 当X<0时，函数值< 0.5, 当X > 0时，函数值>0.5;
• 当X趋近于负无穷时，函数值趋近于0；
• 当X趋近于正无穷时，函数值趋近于1；
• 函数形状类似于一个S，所以称为S型函数。

假设函数的函数值代表的是是输出为1的概率，可以设置一个值如0.5：当函数值>=0.5时，output为1, 当函数值 < 0.5时， output为0.

对于损失函数，如果直接以代入假设函数求得，那么该损失函数将会是“non-convex”(非凸性)，这将对梯度下降的速率有很大的影响。所以便有了对数形式的损失函数。因为是分类问题，所以损失函数有两种情况，但通过一些技巧可以将它们合二为一。

在编写代码时，将式子以向量化(Vectorization)的形式也是很方便的。

而梯度，就是损失函数J(theta)对theta的偏导。

为了解决多结果的回归问题，我们要使用的是one-VS-all的算法。例如我的结果有三类（A、B、C），我可以用3个分类器实现（AB，AC，BC）。然后将X带入每个分类器，哪个值高选哪个。

对于拟合，有三类情况：

• Underfitting 欠拟合： 没有较好拟合数据，预测准确率不高
• Overfitting 过拟合 ：过分拟合数据，导致泛化程度不高，影响预测的准确率
• Oridinary 普通 ： 较好拟合数据

Underfitting, or high bias, is when the form of our hypothesis function h maps poorly to the trend of the data. It is usually caused by a function that is too simple or uses too few features. At the other extreme, overfitting, or high variance, is caused by a hypothesis function that fits the available data but does not generalize well to predict new data. It is usually caused by a complicated function that creates a lot of unnecessary curves and angles unrelated to the data.

解决Overfitting，主要有两种方法：

• Reduce the number of features;
  
  • Manually select which features to keep.
  • Use a model selection algorithm (studied later in the course).
• Regularization.
  
  • Keep all the features, but reduce the magnitude of parameters theta
  • Regularization works well when we have a lot of slightly useful features

下面谈谈Regularization (正规化)：

正规化就是通过添加多余的项以减少theta的大小，从而避免出现过拟合的情况。我们知道，当feature过多时，会导致过拟合的情况。为了解决这一问题，我们要减少feature的影响，即我们需要适当减少theta的值，极限情况下是对应theta等于0，那么feature也就不发挥作用了。

那如何衡量正规化的程度呢？有lambda参数。合适的lambda参数可以使过拟合变成正常拟合，但是太大的lambada值也有可能将过拟合变成欠拟合。所以选择一个合适的lambada值也是很重要的。待会project会有所涉及。

需要掌握：
1. 逻辑回归的假设函数、损失函数、梯度下降的迭代函数；
2. 逻辑回归的梯度计算；
3. 正规化后逻辑回归的假设函数、损失函数、梯度下降的迭代函数。

2. Logistic Regression

主函数

%% 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('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 thetatest_theta = [-24; 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 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('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 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'], prob);fprintf('Expected value: 0.775 +/- 0.002\n\n');% Compute accuracy on our training setp = predict(theta, X);% 如果矩阵中所有元素一一对应，则mean(double(p == y))的值为1% 如果不对应，则值相应减少。如四个中有三个一一对应，那么则其值为0.75fprintf('Train Accuracy: %f\n', mean(double(p == y)) * 100);fprintf('Expected accuracy (approx): 89.0\n');fprintf('\n');

Part1： 将训练集呈现在图中，“+”表示Admitted， “o”表示Not Admitted。而这一结果是由两个exam的分数决定的，即包含两个feature决定最后的结果。
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 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 Indices of Positive and Negative Examplepos = find(y == 1);neg = find(y == 0);% Plot Examples% Attention: cannot add '' to the number, such as '2', '7', it's wrongplot(X(pos, 1), X(pos, 2), 'k+', 'LineWidth', 2, 'MarkerSize', 7);plot(X(neg, 1), X(neg, 2), 'ko', 'MarkerFaceColor', 'y', 'MarkerSize', 7);% =========================================================================hold off;end

Part2：计算损失函数和梯度

costFunction.m

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 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%% Note: grad should have the same dimensions as theta%% Calulate the function h, we know it's sigmoid functionh = sigmoid(X*theta);% Calculate the cost functionJ = 1/m*(-y'*log(h)-(1-y)'*log(1-h));% Calculate the Gradientgrad = 1/m*X'*(h-y);% =============================================================end

Part3: 使用优化函数fminunc来得到使Cost Function函数值最小的theta, 然后使用该theta值画出决策边界。

plotDecisionBoundary.m
此处分了两类情况：如果feature小于等于2，则决策边界是线性的；如果feature大于2，则决策边界是非线性的。至于它为什么这么定义决策边界我就不大清楚了，如果有大神指点一下表示感谢耶~

function plotDecisionBoundary(theta, X, y)%PLOTDECISIONBOUNDARY Plots the data points X and y into a new figure with%the decision boundary defined by theta%   PLOTDECISIONBOUNDARY(theta, X,y) plots the data points with + for the %   positive examples and o for the negative examples. X is assumed to be %   a either %   1) Mx3 matrix, where the first column is an all-ones column for the %      intercept.%   2) MxN, N>3 matrix, where the first column is all-ones% Plot DataplotData(X(:,2:3), y);hold on% size < 3, it means that the feature is less than or equal to 2if size(X, 2) <= 3    % Only need 2 points to define a line, so choose two endpoints    plot_x = [min(X(:,2))-2,  max(X(:,2))+2];    % Calculate the decision boundary line    plot_y = (-1./theta(3)).*(theta(2).*plot_x + theta(1));    % Plot, and adjust axes for better viewing    plot(plot_x, plot_y)    % Legend, specific for the exercise    legend('Admitted', 'Not admitted', 'Decision Boundary')    axis([30, 100, 30, 100])else    % Here is the grid range    u = linspace(-1, 1.5, 50);    v = linspace(-1, 1.5, 50);    z = zeros(length(u), length(v));    % Evaluate z = theta*x over the grid    for i = 1:length(u)        for j = 1:length(v)            z(i,j) = mapFeature(u(i), v(j))*theta;        end    end    z = z'; % important to transpose z before calling contour    % Plot z = 0    % Notice you need to specify the range [0, 0]    contour(u, v, z, [0, 0], 'LineWidth', 2)endhold offend

mapFeature.m
这个函数适用于产生多项式项，即可以增加feature的数量，使得决策边界更贴近训练集。

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

Part4： 使用上述方法进行训练，进而对exam1(45分)、exam2(85分)进行预测并计算出预测准确率。

predict.m

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%% 这个函数的目的是：计算你的预测的准确率% 方法是用你得到的theta作为h(x)的参数，对训练集中的数据全都预测一遍% 然后与训练集中对应的y值进行对比，计算出准确率% 对训练集进行逐行便利，计算出每个训练样例的y值准确率% 如果大于或等于0.5，则将p对应位置设为1，否则为0for i = 1: m  if (sigmoid(X(i,:)*theta) >= 0.5)    p(i) = 1;  else    p(i) = 0;  end% =========================================================================end

输出结果：

3. Regularization

主函数：

%% 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 labelshold 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('Expected cost (approx): 0.693\n');fprintf('Gradient at initial theta (zeros) - first five values only:\n');fprintf(' %f \n', grad(1:5));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 = 10test_theta = ones(size(X,2),1);[cost, grad] = costFunctionReg(test_theta, X, y, 10);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(1:5));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 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 = 1;% 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);fprintf('Expected accuracy (with lambda = 1): 83.1 (approx)\n');

Part1: 正规化逻辑回归

训练集分布如下：

costFunctionReg.m

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% Calulate the function h, we know it's sigmoid functionh = sigmoid(X*theta);% Calculate the cost function% 因为theta1不参与正规化，所以应当把theta1去掉（而且式子中也是不包含这一项的）% 注意：式子中向量下标是从0开始的，而在Octave中，向量下标是从1开始的。J = 1/m*(-y'*log(h)-(1-y)'*log(1-h))+lambda/(2*m)*(sum(theta.^2)-theta(1)^2);% Calculate the Gradientgrad = 1/m*X'*(h-y)+lambda/m*theta;% It must start at 1grad(1) = (1/m*X'*(h-y))(1);% =============================================================end

Part2： 正规化与准确率

lambda为0：

lambda为1：

lambda为10：

lambda为100：

从lambda = 0与 lambda = 1对比可知，正规化削弱了过拟合；
从lambda = 10 与 lambda = 100可知，正规化参数过大会导致欠拟合。

输出结果：

3. Conclusion

本篇博文简要介绍了与逻辑回归的相关知识以及小项目，同时还介绍了Regularization来提高预测的准确性。虽然在项目中只对逻辑回归进行正规化，但是Regularization同样可以运用到线性回归、梯度下降、正规方程等中。

通过这个小项目，不仅对逻辑回归有了进一步的了解，还了解到了正规化的方法。不过这篇博文涉及的逻辑回归主要是只有两个outcomes的，对于多outcomes的没有涉及到，这也是这个project的局限性所在。以后如果接触了多outcomes的逻辑回归再进一步介绍。

以上内容皆为本人观点，欢迎大家提出批评和知道，我们一起探讨！