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本次算法的背景是，假如你是一个大学的管理者，你需要根据学生之前的成绩（两门科目）来预测该学生是否能进入该大学。
根据题意，我们不难分辨出这是一种二分类的逻辑回归，输入x有两种（科目1与科目2），输出有两种（能进入本大学与不能进入本大学）。输入测试样例以已经本文最前面贴出分别有两组数据。
我们在进行逻辑回归之前，通常想把数据数据更为直观的显示出来，那么我们根据输入样例绘制图像。
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 Examples pos = find(y == 1); neg = find(y == 0); % Plot Examples 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
如上代码所展示的是绘图函数，我们可以通过它把数据绘制出来
执行如下代码，绘制图像
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;
绘制结果入下图所示：
图中用+与O分别表示y = 1 与y = 0的两种结果。
在接触到真正的代价函数之前，我们通常假设函数是hΘ(x）= g（ΘTx）
是一S形函数，他可以很好的将0与1区分开。
S形函数的实现：
function g = sigmoid(z) %SIGMOID Compute sigmoid functoon % J = 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
现在我们可以对逻辑函数进行梯度下降，回归函数中的代价函数J(Θ)
代价函数代码实现为
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 % J= -1 * sum( y .* log( sigmoid(X*theta) ) + (1 - y ) .* log( (1 - sigmoid(X*theta)) ) ) / m ; grad = ( X' * (sigmoid(X*theta) - y ) )/ m ; % ============================================================= end
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
theta_1=[0;theta(2:end)];
J= -1 * sum( y .* log( sigmoid(X*theta) ) + (1 - y ) .* log( (1 - sigmoid(X*theta)) ) ) / m + lambda/(2*m) * theta_1' * theta_1 ;
grad = ( X' * (sigmoid(X*theta) - y ) )/ m + lambda/m * theta_1 ;
% =============================================================
end
预测函数：
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 % k = find(sigmoid( X * theta) >= 0.5 ); p(k)= 1; % p(sigmoid( X * theta) >= 0.5) = 1; % it's a more compat way. % ========================================================================= end
现在我们实现代价函数和他的梯度下降，并拟合出直线
%% ============ 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('Gradient at initial theta (zeros):
');
fprintf(' %f
', grad);
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('theta:
');
fprintf(' %f
', theta);
% 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);
% Compute accuracy on our training set
p = predict(theta, X);
fprintf('Train Accuracy: %f
', mean(double(p == y)) * 100);
fprintf('
Program paused. Press enter to continue.
');
pause;
实例2，对非线性函数进行逻辑回归，
实现步骤如下：
%% 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));
% 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('
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);
样本：
逻辑回归：
预测结果：为83.050847