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 2: 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(:, [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.
']);
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('
Program paused. Press enter to continue.
');
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
', cost);
fprintf('Expected cost (approx): 0.693
');
fprintf('Gradient at initial theta (zeros):
');
fprintf(' %f
', grad);
fprintf('Expected gradients (approx):
-0.1000
-12.0092
-11.2628
');
% 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('
Cost at test theta: %f
', cost);
fprintf('Expected cost (approx): 0.218
');
fprintf('Gradient at test theta:
');
fprintf(' %f
', grad);
fprintf('Expected gradients (approx):
0.043
2.566
2.647
');
fprintf('
Program paused. Press enter to continue.
');
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
', cost);
fprintf('Expected cost (approx): 0.203
');
fprintf('theta:
');
fprintf(' %f
', theta);
fprintf('Expected theta (approx):
');
fprintf(' -25.161
0.206
0.201
');
% 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('
Program paused. Press enter to continue.
');
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
'], prob);
fprintf('Expected value: 0.775 +/- 0.002
');
% Compute accuracy on our training set
p = predict(theta, X);
fprintf('Train Accuracy: %f
', mean(double(p == y)) * 100);
fprintf('Expected accuracy (approx): 89.0
');
fprintf('
');
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==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','MarkerFaceColor','y','MarkerSize',7);
% =========================================================================
hold off;
end
3，逻辑回归的逻辑函数（Sigmoid Function/Logistic Function）：
$h_{ heta}(x)=g( heta^{T}x)$ ：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=1./(1+exp(-z));
% =============================================================
end
4，逻辑回归的代价函数：
$J( heta)=-frac{1}{m}sum_{i=1}^{m}[y^{(i)}log(h_ heta(x^{(i)}))+(1-y^{(i)})log(1-h_{ heta}(x^{(i)}))]$
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 = 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
%
h=sigmoid(X*theta); %求hθ(x)
J=-sum(y.*log(h)+(1-y).*log(1-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(1-h))/m;
##grad=((h-y)'*X)/m;
% =============================================================
end
5，带学习速率$alpha$的梯度下降：
$ heta_j:= heta_j-frac{alpha}{m }sum_{i=1}^{m}[(h_ heta(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)：
$frac{partial J( heta)}{partial heta_j}=frac{1}{m}sum_{i=1}^{m}[(h_ heta(x^{(i)})-y^{(i)})x^{(i)}_j]$
使用内置fminunc函数来拟合参数$ heta$，之前我们是使用梯度下降来拟合参数$ heta$的，在这同样也能使用，不过我们这里使用内置fminunc函数来去拟合，它会自动选择学习速率$alpha$，不需要我们手工选择，我们只需要给定一个迭代次数，一个写好的代价函数，初始化$ heta$，最后它会为我们找到最优的$ heta$，它像可以加强版的梯度下降法。
options = optimset('GradObj', 'on', 'MaxIter', 400);
[theta, cost] = ...
fminunc(@(t)(costFunction(t, X, y)), initial_theta, options);//自己写好的costFunction函数
6，根据拟合好的参数$ heta$，预测数据，例如我们想预测某学生第一次分数为45，第二次分数为85，该学生能入学的概率为：
prob = sigmoid([1 45 85] * theta); %入学的概率
预测样本X，我们可以看到预测的准确率为89%。
function p = predict(theta, X)
%PREDICT Predict whether the label is 0 or 1 using learned logistic
%regression parameters theta
% p = PREDICT(theta, X) computes the predictions for X using a
% threshold at 0.5 (i.e., if sigmoid(theta'*x) >= 0.5, predict 1)
m = size(X, 1); % Number of training examples
% You need to return the following variables correctly
p = zeros(m, 1);
% ====================== YOUR CODE HERE ======================
% Instructions: Complete the following code to make predictions using
% your learned logistic regression parameters.
% You should set p to a vector of 0's and 1's
%
%第一种
for i=1:m
p(i,1)=sigmoid(X(i,:)*theta)>=0.5; %预测每一个样本的结果，大于0.5为正向类
end;
%第二种
%
## ans=sigmoid(X*theta);
## for i=1:m
## if(ans(i,1)>=0.5)
## p(i,1)=1;
## else
## p(i,1)=0;
## end
% =========================================================================
end
二：正则化逻辑回归（Regularized logistic regression）：
背景：假如你是某所工厂的管理员，该工厂生产芯片，每片芯片要经过两次测试后，达到标准方可通过，现在有一组以前的数据集ex2data2.txt，第一列为第一次测试的结果，第二列为第二次测试的结果，第三列1表示该芯片合格，0表示不合格。现在要你通过这些数据，拟合出一个模型，这个模型将作为以后判断芯片是否合格的标准。
我们通过可视化这些数据集，发现其与某条复杂的曲线方程有关，而数据集只有两个特征$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.^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
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 2: 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(:, [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)); %c从原来的二维变成了28(27+1截距项)维，m*28
% 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
', cost);
fprintf('Expected cost (approx): 0.693
');
fprintf('Gradient at initial theta (zeros) - first five values only:
');
fprintf(' %f
', grad(1:5));
fprintf('Expected gradients (approx) - first five values only:
');
fprintf(' 0.0085
0.0188
0.0001
0.0503
0.0115
');
fprintf('
Program paused. Press enter to continue.
');
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('
Cost at test theta (with lambda = 10): %f
', cost);
fprintf('Expected cost (approx): 3.16
');
fprintf('Gradient at test theta - first five values only:
');
fprintf(' %f
', grad(1:5));
fprintf('Expected gradients (approx) - first five values only:
');
fprintf(' 0.3460
0.1614
0.1948
0.2269
0.0922
');
fprintf('
Program paused. Press enter to continue.
');
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
', mean(double(p == y)) * 100);
fprintf('Expected accuracy (with lambda = 1): 83.1 (approx)
');
2，正则化逻辑回归代价函数(忽略偏差项$ heta_0$的正则化)：
$J( heta)=-frac{1}{m}sum_{i=1}^{m}[y^{(i)}log(h_ heta(x^{(i)}))+(1-y^{(i)})log(1-h_{ heta}(x^{(i)}))]+frac{lambda }{2m}sum_{j=1}^{n} heta_j^{2}$
3，梯度下降：
带学习速率：
$ heta_0:= heta_0-alpha frac{1}{m }sum_{i=1}^{m}[(h_ heta(x^{(i)})-y^{(i)})x^{(i)}_0]$ for $j=0$
$ heta_j:= heta_j-alpha (frac{1}{m }sum_{i=1}^{m}[(h_ heta(x^{(i)})-y^{(i)})x^{(i)}_j]+frac{lambda }{m} heta_j)$ for $jgeq 1$
不带学习速率(给之后fminunc作为梯度下降使用)：
$frac{partial J( heta)}{partial heta_0}=frac{1}{m}sum_{i=1}^{m}[(h_ heta(x^{(i)})-y^{(i)})x^{(i)}_0]$ for $j=0$
$frac{partial J( heta)}{partial heta_j}=(frac{1}{m}sum_{i=1}^{m}[(h_ heta(x^{(i)})-y^{(i)})x^{(i)}_j])+frac{lambda }{m} heta_j $ for $jgeq 1$
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 = 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
h=sigmoid(X*theta);
n=size(X,2);
J=(-(y')*log(h)-(1-y)'*log(1-h))/m+(lambda/(2*m))*sum(theta([2:n],:).^2); %忽略偏差项θ(0)的影响
grad(1,1)=((X(:,1)')*(h-y))/m; %梯度下降
grad([2:n],:)=(X(:,[2:n])')*(h-y)./m+(theta([2:n],:)).*(lambda/m);
##h=sigmoid(X*theta);
##theta(1,1)=0;
##J=sum(-y'*log(h)-(1-y)'*log(1-h))/m+lambda/2/m*sum(power(theta,2));
##grad=((h-y)'*X)/m+lambda/m*theta';
% =============================================================
end
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。