机器学习之逻辑回归

根据吴恩达《机器学习》教程整理，不做原理讨论。题目及代码：链接，提取码：hkt9。可执行文件为：fromInt_regularizedLogistic.m 和 fromInt.m

 costCalculate.m

function [J,d_theta] = costCalculate(theta,x,y)

m =size(y,1);

a = sigmoid(x*theta);

cost = y.*log(a) + (1-y).*log(1-a);

J = -1/m*sum(cost);

d_theta = 1/m*(x'*(a-y));

end

 

 costFunctionReg_regularized.m

function [J, grad] = costFunctionReg_regularized(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

a = sigmoid(X*theta);

J = 1/m * sum(-y' * log(a) - (1 - y') * log(1 - a)) + lambda/2/m*sum(theta(2:end).^2);
 
grad(1,:) = 1/m * (X(:, 1)' * (a - y));
grad(2:size(theta), :) = 1/m * (X(:, 2:size(theta))' * (a - y)) + lambda/m*theta(2:size(theta), :); 
% =============================================================

end

 

 fromInt.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

clc
clear ; 
close all; 

%% 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;
 
%% ============ 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] = costCalculate(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] = costCalculate(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);%GradObj 设置为on ，告诉fminunc 函数返回成本和渐变，迭代次数400
 
%  Run fminunc to obtain the optimal theta （获得最佳θ）
%  This function will return theta and the cost 
[theta, cost] = ...
	fminunc(@(t)(costCalculate(t, X, y)), initial_theta, options);%@(t)(costFunction(t, X, y))，所需求解最小值的代价函数，@为函数句柄，@后面括号里的 t 表示函数的参数，也就是我们所需要求解最小代价的参数θ
 
% 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');

 

fromInt_regularizedLogistic.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))

% Initialize fitting parameters
initial_theta = zeros(size(X, 2), 1)

% Set regularization parameter lambda to 1
lambda = 1;% λ=1;当λ=0时表示不正则化(No regularization ),这时会出现overfitting;当λ=100时会出现Too much regularization(Underfitting)
 
% Compute and display initial cost and gradient for regularized logistic
% regression
[cost, grad] = costFunctionReg_regularized(initial_theta, X, y, lambda);%调用costFunctionReg.m文件中的costFunctionReg(theta, X, y, lambda)函数
 
fprintf('Cost at initial theta (zeros): %f\n', cost);%计算initial theta (zeros)时的cost 值
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(size(X,2),1);%生成一个与X列相同的矩阵，元素都为1
[cost, grad] = costFunctionReg_regularized(test_theta, X, y, 10);%lamda=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(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_regularized(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');

 

 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.
%a
%   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   %i从1到6
        for j = 0:i    %j从0到i
            out(:, end+1) = (X1.^(i-j)).*(X2.^j);
        end
    end
 
end

 

plotData.m 

function plotData(X,y)
    
    pos = find(y == 1);
    neg = find(y == 0);
    
    plot(X(pos,1),X(pos,2),'*b');
    hold on
    plot(X(neg,1),X(neg,2),'ob');
    
    legend('admission','no admission')

end

 

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 
%   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, 2) <= 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(3)).*(theta(2).*plot_x + theta(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(i), v(j))*theta;
        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

 

predict.m 

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);   %m行1列
 
% ====================== 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
 
p = round(sigmoid(X * theta));
 
% =========================================================================
end

 

sigmoid.m 

function [y_hat] = sigmoid(x)

y_hat = 1./(1+exp(-x));

end