Linear regression with one variable算法实例讲解（绘制图像，cost_Function ,Gradient Desent, 拟合曲线, 轮廓图绘制）_矩阵操作

%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
 5.5277,9.1302
 8.5186,13.662
 7.0032,11.854
 5.8598,6.8233
 8.3829,11.886
 7.4764,4.3483
 8.5781,
 6.4862,6.5987
 5.0546,3.8166
 5.7107,3.2522
 14.164,15.505
 5.734,3.1551
 8.4084,7.2258
 5.6407,0.71618
 5.3794,3.5129
 6.3654,5.3048
 5.1301,0.56077
 6.4296,3.6518
 7.0708,5.3893
 6.1891,3.1386
 20.27,21.767
 5.4901,4.263
 6.3261,5.1875
 5.5649,3.0825
 18.945,22.638
 12.828,13.501
 10.957,7.0467
 13.176,14.692
 22.203,24.147
 5.2524,-1.22
 6.5894,5.9966
 9.2482,12.134
 5.8918,1.8495
 8.2111,6.5426
 7.9334,4.5623
 8.0959,4.1164
 5.6063,3.3928
 12.836,10.117
 6.3534,5.4974
 5.4069,0.55657
 6.8825,3.9115
 11.708,5.3854
 5.7737,2.4406
 7.8247,6.7318
 7.0931,1.0463
 5.0702,5.1337
 5.8014,1.844
 11.7,8.0043
 5.5416,1.0179
 7.5402,6.7504
 5.3077,1.8396
 7.4239,4.2885
 7.6031,4.9981
 6.3328,1.4233
 6.3589,-1.4211
 6.2742,2.4756
 5.6397,4.6042
 9.3102,3.9624
 9.4536,5.4141
 8.8254,5.1694
 5.1793,-0.74279
 21.279,17.929
 14.908,12.054
 18.959,17.054
 7.2182,4.8852
 8.2951,5.7442
 10.236,7.7754
 5.4994,1.0173
 20.341,20.992
 10.136,6.6799
 7.3345,4.0259
 6.0062,1.2784
 7.2259,3.3411
 5.0269,-2.6807
 6.5479,0.29678
 7.5386,3.8845
 5.0365,5.7014
 10.274,6.7526
 5.1077,2.0576
 5.7292,0.47953
 5.1884,0.20421
 6.3557,0.67861
 9.7687,7.5435
 6.5159,5.3436
 8.5172,4.2415
 9.1802,6.7981
 6.002,0.92695
 5.5204,0.152
 5.0594,2.8214
 5.7077,1.8451
 7.6366,4.2959
 5.8707,7.2029
 5.3054,1.9869
 8.2934,0.14454
 13.394,9.0551
 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(:, ); y = data(:, );
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;

 %plotData()函数实现

 function plotData(x, y)

 figure;                                           % open a new figure window
 plot(x, y, 'rx', 'MarkerSize', );     %Set the size of Points('MarkerSize', )
 ylabel('profit in $10,1000s');
 xlabel('population of City in 10,000s');

 end

 %% =================== Part : Gradient descent

 fprintf('Running Gradient Descent ...\n')

 X = [ones(m, ), data(:,)]; % Add a column of ones to x

 theta = zeros(, ); % initialize fitting parameters

 % Some gradient descent settings
 iterations = ;       %迭代次数
 alpha = 0.01;              %learning rate

 % compute and display initial cost
 computeCost(X, y, theta)      %y是真实的值

 % Compute Cost for linear regression
 % cost Function函数实现___利用矩阵操作进行！！
 function J = computeCost(X, y, theta)

 % Initialize some useful values
 m = length(y); % number of training examples
 J = ;

 % Instructions: Compute the cost of a particular choice of theta
 %               You should set J to the cost.

 % X = [ ones(m, ), data(:, ) ], theta = [ th1; th2]
 predictions = X * theta;             %矩阵操作--预测函数
 sqrError = (predictions - y).^;
 J = sum(sqrError) / (*m);

 end

 %运行梯度下降算法
 % run gradient descent

 theta = gradientDescent(X, y, theta, alpha, iterations);

 % print theta to screen
 fprintf('Theta found by gradient descent: ');
 fprintf('%f %f \n', theta(), theta());

 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
 function [theta, J_history] = gradientDescent(X, y, theta, alpha, num_iters)

 %   theta = GRADIENTDESENT(X, y, theta, alpha, num_iters) updates theta by
 %   taking num_iters gradient steps with learning rate alpha

 % Initialize some useful values
 m = length(y); % number of training examples
 J_history = zeros(num_iters, );

 for iter = :num_iters
     predictions = X * theta;         %预测值h(xi)--利用了矩阵运算
     sqrError = (predictions - y);    %预测值 - 实际值

     % Simultaneously update(同时更新thetaj) thetaj for all j.
     % alpha - learning rate, '.*'---是内积(矩阵对应元素相乘)
     theta1 = theta() - alpha * (/m) * sum(sqrError .* X(:,));
     theta2 = theta() - alpha * (/m) * sum(sqrError .* X(:,));
     theta() = theta1;
     theta() = theta2;

     % Save the cost J in every iteration
     J_history(iter) = computeCost(X, y, theta);

   %disp(J_history);  %增加输出语句，方便调试

 end

 end

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

 1 % Predict values for population sizes of 35,000 and 70,000
 2 %利用求出的拟合参数--预测新值，利用矩阵运算
 predict1 = [, 3.5] *theta;
 fprintf('For population = 35,000, we predict a profit of %f\n',...
     predict1*);   

 predict2 = [, ] * theta;
 fprintf('For population = 70,000, we predict a profit of %f\n',...
     predict2*);

 fprintf('Program paused. Press enter to continue.\n');
 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) =============

 fprintf('Visualizing J(theta_0, theta_1) ...\n')

 % Grid over which we will calculate J
 %linspace(x, y, n)--在(x,y)区间内均匀生成n个数
 theta0_vals = linspace(-, , );
 theta1_vals = linspace(-, , );

 % initialize J_vals to a matrix of 's
 J_vals = zeros(length(theta0_vals), length(theta1_vals));

 % Fill out J_vals
 for i = :length(theta0_vals)
     for j = :length(theta1_vals)
       t = [theta0_vals(i); theta1_vals(j)];
       J_vals(i,j) = computeCost(X, y, t);
     end
 end

 % Because of the way meshgrids work in the surf command, we need to
 % transpose J_vals before calling surf, or else the axes will be flipped

 J_vals = J_vals';
 % Surface plot
 figure;

 %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.)

 surf(theta0_vals, theta1_vals, J_vals)
 xlabel('\theta_0'); ylabel('\theta_1');

 % Contour plot----轮廓图的绘制
 figure;

 % Plot J_vals as  contours spaced logarithmically between 0.01 and 

 contour(theta0_vals, theta1_vals, J_vals, logspace(-, , ))
 xlabel('\theta_0'); ylabel('\theta_1');
 hold on;
 plot(theta(), theta(), 'rx', 'MarkerSize', , 'LineWidth', );

绘图效果如上。