作业说明
Exercise 2，Week 3，使用Octave实现逻辑回归模型。数据集 ex2data1.txt ，ex2data2.txt
实现 Sigmoid 、代价函数计算Computing Cost 和 梯度下降Gradient Descent。
文件清单
- ex2.m - Octave/MATLAB script that steps you through the exercise
- ex2 reg.m - Octave/MATLAB script for the later parts of the exercise
- ex2data1.txt - Training set for the first half of the exercise
- ex2data2.txt - Training set for the second half of the exercise
- submit.m - Submission script that sends your solutions to our servers
- mapFeature.m - Function to generate polynomial features
- plotDecisionBoundary.m - Function to plot classifier’s decision boundary
- [*] plotData.m - Function to plot 2D classification data
- [*] sigmoid.m - Sigmoid Function
- [*] costFunction.m - Logistic Regression Cost Function
- [*] predict.m - Logistic Regression Prediction Function
- [*] costFunctionReg.m - Regularized Logistic Regression Cost
* 为必须要完成的
结论
正则化不涉及第一个 θ0
逻辑回归
背景：大学管理员，想要根据两门课的历史成绩记录来每个是否被允许入学。
ex2data1.txt ，前两列是两门课的成绩，第三列是y值 0 和 1。
一、绘制数据图
plotData.m：
positive = find(y == );
negative = find(y == ); plot(X(positive,),X(positive,),'k+','MarkerFaceColor','g',
'MarkerSize',);
hold on;
plot(X(negative,),X(negative,),'ko','MarkerFaceColor','y',
'MarkerSize',);
运行效果如下：
二、sigmoid 函数
function g = sigmoid(z)
% Instructions: Compute the sigmoid of each value of z (z can be a matrix,
% vector or scalar).
g = ./ ( + exp(-z));
end
三、代价函数
costFunction.m：
function [J, grad] = costFunction(theta, X, y) m = length(y); % number of training examples part1 = - * y' * log(sigmoid(X * theta));
part2 = ( - y)' * log(1 - sigmoid(X * theta));
J = / m * (part1 - part2); grad = / m * X' *((sigmoid(X * theta) - y)); end
四、预测函数
输入X和theta，返回预测结果向量。每个值是 0 或 1
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 % 最开始没有四舍五入，导致错误
p = round(sigmoid(X * theta)); end
五、进行逻辑回归
ex1.m 中的调用：
加载数据：
data = load('ex2data1.txt');
X = data(:, [, ]); y = data(:, );
[m, n] = size(X);
% Add intercept term to x and X_test
X = [ones(m, ) X];
initial_theta = zeros(n + , );
调用 fminunc 函数
options = optimset('GradObj', 'on', 'MaxIter', );
[theta, cost] = ...
fminunc(@(t)(costFunction(t, X, y)), initial_theta, options);
四、绘制边界线
plotDecisionBoundary.m
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
% ) Mx3 matrix, where the first column is an all-ones column for the
% intercept.
% ) MxN, N> matrix, where the first column is all-ones % Plot Data
plotData(X(:,:), y);
hold on if size(X, ) <=
% Only need points to define a line, so choose two endpoints
plot_x = [min(X(:,))-, max(X(:,))+]; % Calculate the decision boundary line
plot_y = (-./theta()).*(theta().*plot_x + theta()); % Plot, and adjust axes for better viewing
plot(plot_x, plot_y) % Legend, specific for the exercise
legend('Admitted', 'Not admitted', 'Decision Boundary')
axis([, , , ])
else
% Here is the grid range
u = linspace(-, 1.5, );
v = linspace(-, 1.5, ); z = zeros(length(u), length(v));
% Evaluate z = theta*x over the grid
for i = :length(u)
for j = :length(v)
z(i,j) = mapFeature(u(i), v(j))*theta;
end
end
z = z'; % important to transpose z before calling contour % Plot z =
% Notice you need to specify the range [, ]
contour(u, v, z, [, ], 'LineWidth', )
end
hold off end
正则化逻辑回归
背景：预测来自制造工厂的微芯片是否通过质量保证（QA）。 在QA期间，每个微芯片都经过两个测试以确保其正常运行。
ex2data2.txt ，前两列是测试结果的成绩，第三列是y值 0 和 1。
只有两个feature，使用直线不能划分。
为了让数据拟合的更好，使用mapFeature函数，将x1，x2两个feature扩展到六次方。
六次方曲线复杂，容易造成过拟合，所以需要正则化。
mapFeature.m
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
二、代价函数
注意：θ0不参与正则化。
正则化逻辑回归的代价函数如下，分为三项：
梯度下降算法如下：
coatFunctionReg.m 如下：
function [J, grad] = costFunctionReg(theta, X, y, lambda)
m = length(y); % number of training examples
% theta0 不参与正则化。直接让变量等于theta，将第一个元素置为0，再参与和 λ 的运算
t = theta; t() = ; % 第一项
part1 = -y' * log(sigmoid(X * theta));
% 第二项
part2 = ( - y)' * log(1 - sigmoid(X * theta)); % 正则项
regTerm = lambda / / m * t' * t;
J = / m * (part1 - part2) + regTerm; % 梯度
grad = / m * X' *((sigmoid(X * theta) - y)) + lambda / m * t; end
em2_reg.m 里的调用
% 加载数据
data = load('ex2data2.txt');
X = data(:, [, ]); y = data(:, );
% mapfeature
X = mapFeature(X(:,), X(:,)); % Initialize fitting parameters
initial_theta = zeros(size(X, ), );
lambda = ;
% 调用 fminunc方法
options = optimset('GradObj', 'on', 'MaxIter', );
[theta, J, exit_flag] = ...
fminunc(@(t)(costFunctionReg(t, X, y, lambda)), initial_theta, options);
三、参数调整
（1）使用正则化之前，决策边界曲线如下，可以看到存在过拟合现象：
（2）当 λ = 1，决策边界曲线如下。此时训练集预测准确率为 83.05%
（3）当 λ = 100，曲线如下。此时训练集预测准确率为 61.01%
https://github.com/madoubao/coursera_machine_learning/tree/master/homework/machine-learning-ex2/ex2