%测试数据 'ex1data1.txt', 第一列为 population of City in 10,000s, 第二列为 Profit in $10,000s 1 6.1101,17.592
 2 5.5277,9.1302
 3 8.5186,13.662
 4 7.0032,11.854
 5 5.8598,6.8233
 6 8.3829,11.886
 7 7.4764,4.3483
 8 8.5781,12
 9 6.4862,6.5987
10 5.0546,3.8166
11 5.7107,3.2522
12 14.164,15.505
13 5.734,3.1551
14 8.4084,7.2258
15 5.6407,0.71618
16 5.3794,3.5129
17 6.3654,5.3048
18 5.1301,0.56077
19 6.4296,3.6518
20 7.0708,5.3893
21 6.1891,3.1386
22 20.27,21.767
23 5.4901,4.263
24 6.3261,5.1875
25 5.5649,3.0825
26 18.945,22.638
27 12.828,13.501
28 10.957,7.0467
29 13.176,14.692
30 22.203,24.147
31 5.2524,-1.22
32 6.5894,5.9966
33 9.2482,12.134
34 5.8918,1.8495
35 8.2111,6.5426
36 7.9334,4.5623
37 8.0959,4.1164
38 5.6063,3.3928
39 12.836,10.117
40 6.3534,5.4974
41 5.4069,0.55657
42 6.8825,3.9115
43 11.708,5.3854
44 5.7737,2.4406
45 7.8247,6.7318
46 7.0931,1.0463
47 5.0702,5.1337
48 5.8014,1.844
49 11.7,8.0043
50 5.5416,1.0179
51 7.5402,6.7504
52 5.3077,1.8396
53 7.4239,4.2885
54 7.6031,4.9981
55 6.3328,1.4233
56 6.3589,-1.4211
57 6.2742,2.4756
58 5.6397,4.6042
59 9.3102,3.9624
60 9.4536,5.4141
61 8.8254,5.1694
62 5.1793,-0.74279
63 21.279,17.929
64 14.908,12.054
65 18.959,17.054
66 7.2182,4.8852
67 8.2951,5.7442
68 10.236,7.7754
69 5.4994,1.0173
70 20.341,20.992
71 10.136,6.6799
72 7.3345,4.0259
73 6.0062,1.2784
74 7.2259,3.3411
75 5.0269,-2.6807
76 6.5479,0.29678
77 7.5386,3.8845
78 5.0365,5.7014
79 10.274,6.7526
80 5.1077,2.0576
81 5.7292,0.47953
82 5.1884,0.20421
83 6.3557,0.67861
84 9.7687,7.5435
85 6.5159,5.3436
86 8.5172,4.2415
87 9.1802,6.7981
88 6.002,0.92695
89 5.5204,0.152
90 5.0594,2.8214
91 5.7077,1.8451
92 7.6366,4.2959
93 5.8707,7.2029
94 5.3054,1.9869
95 8.2934,0.14454
96 13.394,9.0551
97 5.4369,0.61705

%绘制实际数据图像——人口和利润的关系图
fprintf('Plotting Data ...\n')
data = load('ex1data1.txt');
X = data(:, 1); y = data(:, 2);
m = length(y); % number of training examples% Plot Data
% Note: You have to complete the code in plotData.m
plotData(X, y);fprintf('Program paused. Press enter to continue.\n');
pause;

 1 %plotData()函数实现
 2
 3 function plotData(x, y)
 4
 5 figure;                                           % open a new figure window
 6 plot(x, y, 'rx', 'MarkerSize', 10);     %Set the size of Points('MarkerSize', 10)
 7 ylabel('profit in $10,1000s');
 8 xlabel('population of City in 10,000s');
 9
10 end

 1 %% =================== Part 3: Gradient descent
 2
 3 fprintf('Running Gradient Descent ...\n')
 4
 5 X = [ones(m, 1), data(:,1)]; % Add a column of ones to x
 6
 7 theta = zeros(2, 1); % initialize fitting parameters
 8
 9 % Some gradient descent settings
10 iterations = 1500;       %迭代次数
11 alpha = 0.01;              %learning rate
12
13 % compute and display initial cost
14 computeCost(X, y, theta)      %y是真实的值

 1 % Compute Cost for linear regression
 2 % cost Function函数实现___利用矩阵操作进行！！
 3 function J = computeCost(X, y, theta)
 4
 5 % Initialize some useful values
 6 m = length(y); % number of training examples
 7 J = 0;
 8
 9 % Instructions: Compute the cost of a particular choice of theta
10 %               You should set J to the cost.
11
12 % X = [ ones(m, 1), data(:, 1) ], theta = [ th1; th2]
13 predictions = X * theta;             %矩阵操作--预测函数
14 sqrError = (predictions - y).^2;
15 J = sum(sqrError) / (2*m);
16
17 end

1 %运行梯度下降算法
2 % run gradient descent
3
4 theta = gradientDescent(X, y, theta, alpha, iterations);
5
6 % print theta to screen
7 fprintf('Theta found by gradient descent: ');
8 fprintf('%f %f \n', theta(1), theta(2));

 1 %梯度下降算法实现 gradientDescent(X, y, theta, alpha, iterations)    %X-training example,y-实际数值,alpha-learning rate
 2 function [theta, J_history] = gradientDescent(X, y, theta, alpha, num_iters)
 3
 4 %   theta = GRADIENTDESENT(X, y, theta, alpha, num_iters) updates theta by
 5 %   taking num_iters gradient steps with learning rate alpha
 6
 7 % Initialize some useful values
 8 m = length(y); % number of training examples
 9 J_history = zeros(num_iters, 1);
10
11 for iter = 1:num_iters
12     predictions = X * theta;         %预测值h(xi)--利用了矩阵运算
13     sqrError = (predictions - y);    %预测值 - 实际值
14
15     % Simultaneously update(同时更新thetaj) thetaj for all j.
16     % alpha - learning rate, '.*'---是内积(矩阵对应元素相乘)
17     theta1 = theta(1) - alpha * (1/m) * sum(sqrError .* X(:,1));
18     theta2 = theta(2) - alpha * (1/m) * sum(sqrError .* X(:,2));
19     theta(1) = theta1;
20     theta(2) = theta2;
21
22
23     % Save the cost J in every iteration
24     J_history(iter) = computeCost(X, y, theta);
25
26   %disp(J_history);  %增加输出语句，方便调试
27
28 end
29
30 end

1 %绘制拟合曲线
2 % Plot the linear fit
3 hold on;              % keep previous plot visible
4 plot(X(:,2), X*theta, '-')
5 legend('Training data', 'Linear regression')    %添加图例
6 hold off              % don't overlay any more plots on this figure

 1 % Predict values for population sizes of 35,000 and 70,0002 %利用求出的拟合参数--预测新值，利用矩阵运算
 3 predict1 = [1, 3.5] *theta;
 4 fprintf('For population = 35,000, we predict a profit of %f\n',...
 5     predict1*10000);
 6
 7 predict2 = [1, 7] * theta;
 8 fprintf('For population = 70,000, we predict a profit of %f\n',...
 9     predict2*10000);
10
11 fprintf('Program paused. Press enter to continue.\n');
12 pause;

 1 %计算不同 theta参数下， J(θ)值的变化, 绘制图像2 %% ============= Part 4: Visualizing J(theta_0, theta_1) =============
 3
 4 fprintf('Visualizing J(theta_0, theta_1) ...\n')
 5
 6 % Grid over which we will calculate J
 7 %linspace(x, y, n)--在(x,y)区间内均匀生成n个数
 8 theta0_vals = linspace(-10, 10, 100);
 9 theta1_vals = linspace(-1, 4, 100);
10
11 % initialize J_vals to a matrix of 0's
12 J_vals = zeros(length(theta0_vals), length(theta1_vals));
13
14 % Fill out J_vals
15 for i = 1:length(theta0_vals)
16     for j = 1:length(theta1_vals)
17       t = [theta0_vals(i); theta1_vals(j)];
18       J_vals(i,j) = computeCost(X, y, t);
19     end
20 end
21
22
23 % Because of the way meshgrids work in the surf command, we need to
24 % transpose J_vals before calling surf, or else the axes will be flipped
25
26 J_vals = J_vals';
27 % Surface plot
28 figure;
29
30 %surf(X,Y,Z)--creates the surface plot from corresponding(对应值) value in X, Y，Z (default: color is proportional(成正比) to surface height.)
31
32 surf(theta0_vals, theta1_vals, J_vals)
33 xlabel('\theta_0'); ylabel('\theta_1');

1 % Contour plot----轮廓图的绘制
2 figure;
3
4 % Plot J_vals as 15 contours spaced logarithmically between 0.01 and 100
5
6 contour(theta0_vals, theta1_vals, J_vals, logspace(-2, 3, 20))
7 xlabel('\theta_0'); ylabel('\theta_1');
8 hold on;
9 plot(theta(1), theta(2), 'rx', 'MarkerSize', 10, 'LineWidth', 2);