34.62365962451697,78.0246928153624,0 30.28671076822607,43.89499752400101,0 35.84740876993872,72.90219802708364,0 60.18259938620976,86.30855209546826,1 79.0327360507101,75.3443764369103,1 45.08327747668339,56.3163717815305,0 61.10666453684766,96.51142588489624,1 75.02474556738889,46.55401354116538,1 76.09878670226257,87.42056971926803,1 84.43281996120035,43.53339331072109,1 95.86155507093572,38.22527805795094,0 75.01365838958247,30.60326323428011,0 82.30705337399482,76.48196330235604,1 69.36458875970939,97.71869196188608,1 39.53833914367223,76.03681085115882,0 53.9710521485623,89.20735013750205,1 69.07014406283025,52.74046973016765,1 67.94685547711617,46.67857410673128,0 70.66150955499435,92.92713789364831,1 76.97878372747498,47.57596364975532,1 67.37202754570876,42.83843832029179,0 89.67677575072079,65.79936592745237,1 50.534788289883,48.85581152764205,0 34.21206097786789,44.20952859866288,0 77.9240914545704,68.9723599933059,1 62.27101367004632,69.95445795447587,1 80.1901807509566,44.82162893218353,1 93.114388797442,38.80067033713209,0 61.83020602312595,50.25610789244621,0 38.78580379679423,64.99568095539578,0 61.379289447425,72.80788731317097,1 85.40451939411645,57.05198397627122,1 52.10797973193984,63.12762376881715,0 52.04540476831827,69.43286012045222,1 40.23689373545111,71.16774802184875,0 54.63510555424817,52.21388588061123,0 33.91550010906887,98.86943574220611,0 64.17698887494485,80.90806058670817,1 74.78925295941542,41.57341522824434,0 34.1836400264419,75.2377203360134,0 83.90239366249155,56.30804621605327,1 51.54772026906181,46.85629026349976,0 94.44336776917852,65.56892160559052,1 82.36875375713919,40.61825515970618,0 51.04775177128865,45.82270145776001,0 62.22267576120188,52.06099194836679,0 77.19303492601364,70.45820000180959,1 97.77159928000232,86.7278223300282,1 62.07306379667647,96.76882412413983,1 91.56497449807442,88.69629254546599,1 79.94481794066932,74.16311935043758,1 99.2725269292572,60.99903099844988,1 90.54671411399852,43.39060180650027,1 34.52451385320009,60.39634245837173,0 50.2864961189907,49.80453881323059,0 49.58667721632031,59.80895099453265,0 97.64563396007767,68.86157272420604,1 32.57720016809309,95.59854761387875,0 74.24869136721598,69.82457122657193,1 71.79646205863379,78.45356224515052,1 75.3956114656803,85.75993667331619,1 35.28611281526193,47.02051394723416,0 56.25381749711624,39.26147251058019,0 30.05882244669796,49.59297386723685,0 44.66826172480893,66.45008614558913,0 66.56089447242954,41.09209807936973,0 40.45755098375164,97.53518548909936,1 49.07256321908844,51.88321182073966,0 80.27957401466998,92.11606081344084,1 66.74671856944039,60.99139402740988,1 32.72283304060323,43.30717306430063,0 64.0393204150601,78.03168802018232,1 72.34649422579923,96.22759296761404,1 60.45788573918959,73.09499809758037,1 58.84095621726802,75.85844831279042,1 99.82785779692128,72.36925193383885,1 47.26426910848174,88.47586499559782,1 50.45815980285988,75.80985952982456,1 60.45555629271532,42.50840943572217,0 82.22666157785568,42.71987853716458,0 88.9138964166533,69.80378889835472,1 94.83450672430196,45.69430680250754,1 67.31925746917527,66.58935317747915,1 57.23870631569862,59.51428198012956,1 80.36675600171273,90.96014789746954,1 68.46852178591112,85.59430710452014,1 42.0754545384731,78.84478600148043,0 75.47770200533905,90.42453899753964,1 78.63542434898018,96.64742716885644,1 52.34800398794107,60.76950525602592,0 94.09433112516793,77.15910509073893,1 90.44855097096364,87.50879176484702,1 55.48216114069585,35.57070347228866,0 74.49269241843041,84.84513684930135,1 89.84580670720979,45.35828361091658,1 83.48916274498238,48.38028579728175,1 42.2617008099817,87.10385094025457,1 99.31500880510394,68.77540947206617,1 55.34001756003703,64.9319380069486,1 74.77589300092767,89.52981289513276,1
ex2data1.txt
0.051267,0.69956,1 -0.092742,0.68494,1 -0.21371,0.69225,1 -0.375,0.50219,1 -0.51325,0.46564,1 -0.52477,0.2098,1 -0.39804,0.034357,1 -0.30588,-0.19225,1 0.016705,-0.40424,1 0.13191,-0.51389,1 0.38537,-0.56506,1 0.52938,-0.5212,1 0.63882,-0.24342,1 0.73675,-0.18494,1 0.54666,0.48757,1 0.322,0.5826,1 0.16647,0.53874,1 -0.046659,0.81652,1 -0.17339,0.69956,1 -0.47869,0.63377,1 -0.60541,0.59722,1 -0.62846,0.33406,1 -0.59389,0.005117,1 -0.42108,-0.27266,1 -0.11578,-0.39693,1 0.20104,-0.60161,1 0.46601,-0.53582,1 0.67339,-0.53582,1 -0.13882,0.54605,1 -0.29435,0.77997,1 -0.26555,0.96272,1 -0.16187,0.8019,1 -0.17339,0.64839,1 -0.28283,0.47295,1 -0.36348,0.31213,1 -0.30012,0.027047,1 -0.23675,-0.21418,1 -0.06394,-0.18494,1 0.062788,-0.16301,1 0.22984,-0.41155,1 0.2932,-0.2288,1 0.48329,-0.18494,1 0.64459,-0.14108,1 0.46025,0.012427,1 0.6273,0.15863,1 0.57546,0.26827,1 0.72523,0.44371,1 0.22408,0.52412,1 0.44297,0.67032,1 0.322,0.69225,1 0.13767,0.57529,1 -0.0063364,0.39985,1 -0.092742,0.55336,1 -0.20795,0.35599,1 -0.20795,0.17325,1 -0.43836,0.21711,1 -0.21947,-0.016813,1 -0.13882,-0.27266,1 0.18376,0.93348,0 0.22408,0.77997,0 0.29896,0.61915,0 0.50634,0.75804,0 0.61578,0.7288,0 0.60426,0.59722,0 0.76555,0.50219,0 0.92684,0.3633,0 0.82316,0.27558,0 0.96141,0.085526,0 0.93836,0.012427,0 0.86348,-0.082602,0 0.89804,-0.20687,0 0.85196,-0.36769,0 0.82892,-0.5212,0 0.79435,-0.55775,0 0.59274,-0.7405,0 0.51786,-0.5943,0 0.46601,-0.41886,0 0.35081,-0.57968,0 0.28744,-0.76974,0 0.085829,-0.75512,0 0.14919,-0.57968,0 -0.13306,-0.4481,0 -0.40956,-0.41155,0 -0.39228,-0.25804,0 -0.74366,-0.25804,0 -0.69758,0.041667,0 -0.75518,0.2902,0 -0.69758,0.68494,0 -0.4038,0.70687,0 -0.38076,0.91886,0 -0.50749,0.90424,0 -0.54781,0.70687,0 0.10311,0.77997,0 0.057028,0.91886,0 -0.10426,0.99196,0 -0.081221,1.1089,0 0.28744,1.087,0 0.39689,0.82383,0 0.63882,0.88962,0 0.82316,0.66301,0 0.67339,0.64108,0 1.0709,0.10015,0 -0.046659,-0.57968,0 -0.23675,-0.63816,0 -0.15035,-0.36769,0 -0.49021,-0.3019,0 -0.46717,-0.13377,0 -0.28859,-0.060673,0 -0.61118,-0.067982,0 -0.66302,-0.21418,0 -0.59965,-0.41886,0 -0.72638,-0.082602,0 -0.83007,0.31213,0 -0.72062,0.53874,0 -0.59389,0.49488,0 -0.48445,0.99927,0 -0.0063364,0.99927,0 0.63265,-0.030612,0
ex2data2.txt
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.\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;
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\n‘, cost); fprintf(‘Gradient at initial theta (zeros): \n‘); fprintf(‘ %f \n‘, grad); 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(‘theta: \n‘);
fprintf(‘ %f \n‘, 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(‘\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\n‘], prob);
% Compute accuracy on our training set
p = predict(theta, X);
fprintf(‘Train Accuracy: %f\n‘, mean(double(p == y)) * 100);
fprintf(‘\nProgram paused. Press enter to continue.\n‘);
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\n‘, cost); 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);
样本：
逻辑回归：
预测结果：为83.050847
时间： 2024-10-23 03:55:23