目录
序言
无正则化
1、主函数：
2、sigmoid函数
3、plotData绘图
4、costFunction代价函数
5、predict预测函数
6、submit函数
正则化
1、主函数
2、plotData绘图结果展示
2、costFunctionReg代价函数
3、plotDecisionBoundary函数
4、mapFeature函数
5、说明
序言
逻辑回归是解决分类问题的一种分类器，是在线性回归的基础上进行了非线性的调整，即引入了sigmoid函数。虽说是对线性回归的调整，但是并不是说二者均是解决同一类问题，线性回归解决的是预测问题，其代价函数是采用最小二乘法进行求解，而逻辑回归的代价函数则是引入了概率的思想。有关逻辑回归的详细内容请参考如下介绍，博客内容仅供参考，如有不当之处欢迎大家在评论区批评指正。
无正则化
1、主函数：
首先、关于逻辑回归的模型如下图所示。输入的每一项与对应权重相乘，得到的结果进入sigmoid函数进行非线性处理，将输出限制在（0，1）范围内。再利用输出值和实际值求出代价函数，再对代价函数求偏导求解出梯度grad进而更新权重theta。
%% 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);
X = data(:,[1,2])';
y = data(:,3)';
%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(1, n) ;X];%加偏置项
%Y = [y0;y1;y2;.......]
% Initialize fitting parameters
initial_theta = zeros(1,m + 1);%初始化 偏置项置为全0 1 X n+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);
fprintf('Train Accuracy: %f\n', mean(double(p == y)) * 100);
fprintf('Expected accuracy (approx): 89.0\n');
fprintf('\n');
初始化theta为0的时候计算代价和梯度结果如下所示，红色为实验所得结果，蓝色为参考值。
分类结果示意图如下所示：
正确率为：
2、sigmoid函数
sigmoid函数也称激活函数，当一个神经元的激活函数是一个 Sigmoid函数时，这个单元的输出保证总是介于0和1之间。此外，由于 Sigmoid是一个非线性函数，这个单元的输出将是一个非线性函数的加权和的输入。这种以Sigmoid函数为激活函数的神经元被称为sigmoid unit ，其图像如下图所示。
function g = sigmoid(z) %SIGMOID Compute sigmoid function % g = SIGMOID(z) computes the sigmoid of z. % You need to return the following variables correctly % ====================== 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
3、plotData绘图
关于plotData函数，需要先对数据进行处理。根据标签将数据集处理，利用find函数找出对应类别数据集的索引下标将数据集分为pos和neg两类，最后绘制两类数据的图像，如下图所示。
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. % %data数据说明：第一列横坐标，第二列纵坐标，第三列标签标志（1：pos；2：neg） %找出y = 0和1的下标 （y为标签标志） pos= find(y==1); neg = find(y==0); %画出对应点（x，y） plot(X(1,pos), X(2,pos), 'k+','LineWidth', 2, 'MarkerSize', 7); plot(X(1,neg), X(2,neg), 'ko', 'MarkerFaceColor', 'y', 'MarkerSize', 7); % plot(X(1,pos), X(2,pos), 'k+','LineWidth', 2, 'MarkerSize', 7); % plot(X(1,neg), X(2,neg), 'ko', 'MarkerFaceColor', 'y', 'MarkerSize', 7); % ========================================================================= hold off; end
4、costFunction代价函数
代价函数，其特点是：实际值为 1 且预测值也为 1 时误差为 0 ，当实际值为 1 预测值不为1时，误差随预测值减小而增大。当实际值为 0 且预测值也为 0 时，误差为 0 ，当实际值为 0 但预测值不为 0 时，误差随预测值增大而增大，此过程可以参考sigmoid函数图像理解。
计算过程也较为简单，只需求出预测值h_theta，带入如下公式即可得出代价J。得到J后，我们便可以通过梯度下降法求出梯度grad。
有关如何求导便不在此过多介绍，有兴趣的可以自行推导。
无正则化的代价函数部分，并没有易错点，不过在编写代码的时候仍需注意，避免粗心导致报错。
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_theta = sigmoid(theta*X); J = -sum(y.*log(h_theta)+(1-y).*log((1-h_theta)))/m; %%%%%%%%%%%%%%%%%方法二%%%%%%%%%%%%%%%%%% %不用sum 矩阵乘法包含有求和，可以不需要sum 和点乘的方式；因此将y = y'； % y = y';%100 * 1 log(h_theta) 1*100 % J = -(log(h_theta)*y+log((1-h_theta))*(1-y))/m; %%%%%%%%%%%%%%%%二者时间复杂度比较？？？？？？%%%%%%% grad = sum((h_theta-y).*X,2)/m; %更新theta % theta = theta - alpha.*grad; % ============================================================= end
5、predict预测函数
预测函数，即先根据预测值对预测输出的数据集进行分类，分类标准即sigmoid函数的区分标准，以0.5为界，大于0.5则判为1，小于0.5判为0。此过程之后，我们便可得到数据集的逻辑输出，非真实的线性组合输出，经过sigmoid函数后，输出值便变成了逻辑值中的0和1，这可以看出逻辑回归实际上是一个分类器，而非做预测。
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
%
Matrix = sigmoid(theta*X);
num_cols= size(Matrix,2);
for i = 1 : num_cols
if Matrix(1,i)>=0.5
p(i) = 1;
else
p(i) = 0;
end
end
% =========================================================================
end
6、submit函数
function submit()
addpath('./lib');
conf.assignmentSlug = 'logistic-regression';
conf.itemName = 'Logistic Regression';
conf.partArrays = { ...
{ ...
'1', ...
{ 'sigmoid.m' }, ...
'Sigmoid Function', ...
}, ...
{ ...
'2', ...
{ 'costFunction.m' }, ...
'Logistic Regression Cost', ...
}, ...
{ ...
'3', ...
{ 'costFunction.m' }, ...
'Logistic Regression Gradient', ...
}, ...
{ ...
'4', ...
{ 'predict.m' }, ...
'Predict', ...
}, ...
{ ...
'5', ...
{ 'costFunctionReg.m' }, ...
'Regularized Logistic Regression Cost', ...
}, ...
{ ...
'6', ...
{ 'costFunctionReg.m' }, ...
'Regularized Logistic Regression Gradient', ...
}, ...
};
conf.output = @output;
submitWithConfiguration(conf);
end
function out = output(partId, auxstring)
% Random Test Cases
X = [ones(20,1) (exp(1) * sin(1:1:20))' (exp(0.5) * cos(1:1:20))'];
y = sin(X(:,1) + X(:,2)) > 0;
if partId == '1'
out = sprintf('%0.5f ', sigmoid(X));
elseif partId == '2'
out = sprintf('%0.5f ', costFunction([0.25 0.5 -0.5]', X, y));
elseif partId == '3'
[cost, grad] = costFunction([0.25 0.5 -0.5]', X, y);
out = sprintf('%0.5f ', grad);
elseif partId == '4'
out = sprintf('%0.5f ', predict([0.25 0.5 -0.5]', X));
elseif partId == '5'
out = sprintf('%0.5f ', costFunctionReg([0.25 0.5 -0.5]', X, y, 0.1));
elseif partId == '6'
[cost, grad] = costFunctionReg([0.25 0.5 -0.5]', X, y, 0.1);
out = sprintf('%0.5f ', grad);
end
end
正则化
有关正则化过拟合欠拟合问题将在后续博客中讨论，本文仅讨论如何对代价函数进行正则化。
1、主函数
%% 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,:));%已加偏置项
% Initialize fitting parameters
initial_theta = zeros(1,size(X, 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(1,size(X,1));%1*28
[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(1,size(X, 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');
2、plotData绘图结果展示
2、costFunctionReg代价函数
正则化处理的代价函数，需要注意的地方有两个。一个是theta0是不参与正则化的，所以在计算代价函数J的时候不应将其进算在内。第二个地方是，代价函数对theta求偏导的时候，需要分成两个部分进行：偏置项部分和剩余部分。在对偏置项的权重求偏导时，由于偏置项为常数，求偏导后无X(i)这一项。另外，值得注意的是theta0_grad的值为一个数！因此，在进行计算的时候，一定要注意各参数的维度。
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. %theta 28*1====1*28 X 28*118 y 1*118 % theta = theta'; % 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_theta = sigmoid(theta*X); %%%theta0 即偏置量权重不必正则化 %theta0_grad = sum((h_theta(1,1)-y(1,1))*X(1,:)')/m; theta0_grad = ((h_theta-y)*X(1,:)')/m; %theta_grad = sum((h_theta-y).*X(2:end,:)')/m +lambda.*theta(1,2:end)/m; theta_grad = ((h_theta-y)*X(2:end,:)')/m +lambda.*theta(1,2:end)/m; grad = [theta0_grad theta_grad]; alpha = 0.003; theta = theta - alpha.*grad; % grad = sum((h_theta-y).*X,2)/m ; J = -sum(y.*log(h_theta)+(1-y).*log((1-h_theta)))/m + lambda.*sum(theta(1,2:end).^2)/(2*m); % ============================================================= end
结果展示，红色框线表示训练结果，蓝色框线表示参考结果。本实验所得结果与参考值差异不大，表明代码无误。
当正则化因子为10的时候，所得的代价和梯度结果。
3、plotDecisionBoundary函数
此函数作用就是画出边界线，用于区分两类数据。
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
if size(X,1) <= 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(1,3)).*(theta(1,2).*plot_x + theta(1,1));
% 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(1,i), v(1,j))*theta;%%%%%%%%%?
w = mapFeature(u(1,i), v(1,j));%%test 28*1
z(i,j) = theta*mapFeature(u(1,i), v(1,j));%theta 118*1
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
4、mapFeature函数
此函数作用是将输入参数进行排列组合，组合出更多的特征。
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);%28*?
end
end
end 5、说明
两个实例会用到一些共同的函数，因此在第二个实例中并未重复列出。如果需要单独的实例二的朋友，可以根据主函数中的函数按需自取。
正则化后的正确率如下：