机器学习: 逻辑回归(Logistic Regression) 小项目

该项目的所有代码在我的github上，欢迎有兴趣的同学与我探讨研究~
地址：Machine-Learning/machine-learning-ex2/

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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

%% 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(:, [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 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 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_test
X = [ones(m, 1) X];

% Initialize fitting parameters
initial_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 theta
test_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 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);

% 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 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 set
p = 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0.75

fprintf('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 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 Indices of Positive and Negative Example
pos = find(y == 1);
neg = find(y == 0);

% Plot Examples
% Attention: cannot add '' to the number, such as '2', '7', it's wrong
plot(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 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
%
% Note: grad should have the same dimensions as theta
%

% Calulate the function h, we know it's sigmoid function
h = sigmoid(X*theta);
% Calculate the cost function
J = 1/m*(-y'*log(h)-(1-y)'*log(1-h));
% Calculate the Gradient
grad = 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 Data
plotData(X(:,2:3), y);
hold on

% size < 3, it means that the feature is less than or equal to 2
if 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)
end
hold off

end

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);
    end
end

end

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 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
%

% 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0

for 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：

%% 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('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 = 10
test_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 parameters
initial_theta = zeros(size(X, 2), 1);

% Set regularization parameter lambda to 1 (you should vary this)
lambda = 1;

% 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);
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 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

% Calulate the function h, we know it's sigmoid function
h = 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 Gradient
grad = 1/m*X'*(h-y)+lambda/m*theta;
% It must start at 1
grad(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。

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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！

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