Linear regression with multiple variables(多特征的线型回归)算法实例_梯度下降解法(Gradient DesentMulti)以及正规方程解法(Normal Equation)

%第一列为 size of House(feet^2),第二列为 number of bedroom，第三列为 price of House
 1 2104,3,399900
 2 1600,3,329900
 3 2400,3,369000
 4 1416,2,232000
 5 3000,4,539900
 6 1985,4,299900
 7 1534,3,314900
 8 1427,3,198999
 9 1380,3,212000
10 1494,3,242500
11 1940,4,239999
12 2000,3,347000
13 1890,3,329999
14 4478,5,699900
15 1268,3,259900
16 2300,4,449900
17 1320,2,299900
18 1236,3,199900
19 2609,4,499998
20 3031,4,599000
21 1767,3,252900
22 1888,2,255000
23 1604,3,242900
24 1962,4,259900
25 3890,3,573900
26 1100,3,249900
27 1458,3,464500
28 2526,3,469000
29 2200,3,475000
30 2637,3,299900
31 1839,2,349900
32 1000,1,169900
33 2040,4,314900
34 3137,3,579900
35 1811,4,285900
36 1437,3,249900
37 1239,3,229900
38 2132,4,345000
39 4215,4,549000
40 2162,4,287000
41 1664,2,368500
42 2238,3,329900
43 2567,4,314000
44 1200,3,299000
45 852,2,179900
46 1852,4,299900
47 1203,3,239500

 1 %  Exercise 1: Linear regression with multiple variables
 2 
 3 %% Initialization
 4 
 5 %% ================ Part 1: Feature Normalization ================
 6 
 7 %% Clear and Close Figures
 8 clear ; close all; clc
 9 
10 fprintf(‘Loading data ...\n‘);
11 
12 %% Load Data
13 data = load(‘ex1data2.txt‘);
14 X = data(:, 1:2);
15 y = data(:, 3);
16 m = length(y);
17 
18 % Print out some data points
19 fprintf(‘First 10 examples from the dataset: \n‘);
20 fprintf(‘ x = [%.0f %.0f], y = %.0f \n‘, [X(1:10,:) y(1:10,:)]‘);
21 
22 fprintf(‘Program paused. Press enter to continue.\n‘);
23 pause;
24 
25 % Scale features and set them to zero mean
26 fprintf(‘Normalizing Features ...\n‘);
27 
28 [X, mu, sigma] = featureNormalize(X);

 1 %featureNormalize(X)函数实现
 2 function [X_norm, mu, sigma] = featureNormalize(X)
 3 X_norm = X;                      % X是需要正规化的矩阵
 4 mu = zeros(1, size(X, 2));       % 生成 1x3 的全0矩阵
 5 sigma = zeros(1, size(X, 2));    % 同上
 6 
 7 % Instructions: First, for each feature dimension, compute the mean
 8 %               of the feature and subtract it from the dataset,
 9 %               storing the mean value in mu. Next, compute the 
10 %               standard deviation of each feature and divide
11 %               each feature by it‘s standard deviation, storing
12 %               the standard deviation in sigma. 
13 %
14 %               Note that X is a matrix where each column is a 
15 %               feature and each row is an example. You need 
16 %               to perform the normalization separately for 
17 %               each feature. 
18 %
19 % Hint: You might find the ‘mean‘ and ‘std‘ functions useful.
20 
21 % std，均方差，std(X,0,1)求列向量方差，std(X,0,2)求行向量方差。
22 
23 mu = mean(X, 1);                 %求每列的均值--即一种特征的所有样本的均值
24 sigma = std(X);                  %默认同std(X,0,1)求列向量方差
25 %fprintf(‘Debug....\n‘); disp(sigma);  
26 i = 1;
27 len = size(X,2);                 %行数
28 while i <= len,
29     %对每列的所有行上的样本进行normalization(归一化):(每列的所有行-该列均值)/(该列的标准差)
30     X_norm(:,i) = (X(:,i) - mu(1,i)) / (sigma(1,i));
31     i = i + 1;
32 end

 1 % Add intercept term to X 
 2 X = [ones(m, 1) X];        
 3 
 4 
 5 %% ================ Part 2: Gradient Descent ================
 6 
 7 % ====================== YOUR CODE HERE ======================
 8 % Instructions: We have provided you with the following starter
 9 %               code that runs gradient descent with a particular
10 %               learning rate (alpha). 
11 %
12 %               Your task is to first make sure that your functions - 
13 %               computeCost and gradientDescent already work with 
14 %               this starter code and support multiple variables.
15 %
16 %               After that, try running gradient descent with 
17 %               different values of alpha and see which one gives
18 %               you the best result.
19 %
20 %               Finally, you should complete the code at the end
21 %               to predict the price of a 1650 sq-ft, 3 br house.
22 %
23 % Hint: By using the ‘hold on‘ command, you can plot multiple
24 %       graphs on the same figure.
25 %
26 % Hint: At prediction, make sure you do the same feature normalization.
27 %
28 
29 fprintf(‘Running gradient descent ...\n‘);
30 
31 % Choose some alpha value
32 alpha = 0.03;                         % learning rate - 可尝试0.01,0.03,0.1,0.3...
33 num_iters = 400;                      % 迭代次数
34 
35 % Init Theta and Run Gradient Descent 
36 theta = zeros(3, 1);                  % 3x1的全零矩阵
37 [theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters);

% gradientDescentMulti()函数实现
 1 function [theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters)
 2  3 %   theta = GRADIENTDESCENTMULTI(x, y, theta, alpha, num_iters) updates theta by
 4 %   taking num_iters gradient steps with learning rate alpha
 5 
 6 % Initialize some useful values
 7 m = length(y);                    % number of training examples
 8 feature_number = size(X,2);       % number of feature
 9 
10 J_history = zeros(num_iters, 1);
11 temp = zeros(feature_number, 1);
12 
13 for iter = 1 : num_iters
14     predictions = X * theta;
15     sqrError = (predictions - y);
16     for i = 1 : feature_number    % Simultneously update theta(i) （同时更新）
17         temp(i) = theta(i) - (alpha / m) * sum(sqrError .* X(:,i));
18     end
19     
20     for j = 1 : feature_number
21         theta(j) = temp(j);
22     end
23 
24     % ====================== YOUR CODE HERE ======================
25     % Instructions: Perform a single gradient step on the parameter vector
26     %               theta. 
27     %
28     % Hint: While debugging, it can be useful to print out the values
29     %       of the cost function (computeCostMulti) and gradient here.
30     %
31 
32     % ============================================================
33 
34     % Save the cost J in every iteration    
35     J_history(iter) = computeCostMulti(X, y, theta);
36     % disp(J_history(iter));
37 
38 end
39 
40 end

 1 % Plot the convergence graph
 2 figure;
 3 plot(1:numel(J_history), J_history, ‘-b‘, ‘LineWidth‘, 2); % ‘-b‘--用蓝线绘制图像,线宽为2
 4 xlabel(‘Number of iterations‘);
 5 ylabel(‘Cost J‘);
 6 
 7 % Display gradient descent‘s result
 8 fprintf(‘Theta computed from gradient descent: \n‘);
 9 fprintf(‘ %f \n‘, theta);
10 fprintf(‘\n‘);

Tip:
To compare how dierent learning learning
rates aect convergence, it‘s helpful to plot J for several learning rates
on the same gure. In Octave/MATLAB, this can be done by perform-
ing gradient descent multiple times with a `hold on‘ command between
plots. Concretely, if you‘ve tried three dierent values of alpha (you should
probably try more values than this) and stored the costs in J1, J2 and
J3, you can use the following commands to plot them on the same gure:
plot(1:50, J1(1:50), `b‘);
hold on;
plot(1:50, J2(1:50), `r‘);
plot(1:50, J3(1:50), `k‘);
The nal arguments `b‘, `r‘, and `k‘ specify dierent colors for the
plots.

 1 % 如，可以添加本段代码进行比较 不同的learning rate
 2 figure;
 3 plot(1:100, J_history(1:100), ‘-b‘, ‘LineWidth‘, 2);
 4 xlabel(‘Number of iterations‘);
 5 ylabel(‘Cost J‘);
 6 
 7 % Compare learning rate
 8 hold on;
 9 alpha = 0.03;
10 theta = zeros(3, 1);
11 [theta, J_history1] = gradientDescentMulti(X, y, theta, alpha, num_iters);
12 plot(1:100, J_history1(1:100), ‘r‘, ‘LineWidth‘, 2);
13 
14 hold on;
15 alpha = 0.1;
16 theta = zeros(3, 1);
17 [theta, J_history2] = gradientDescentMulti(X, y, theta, alpha, num_iters);
18 plot(1:100, J_history2(1:100), ‘g‘, ‘LineWidth‘, 2);

 1 % 利用梯度下降算法预测新值
 2 price = [1, X(1:2)] * theta;   %利用矩阵乘法--预测多特征下的price
 3 
 4 % ============================================================
 5 
 6 fprintf([‘Predicted price of a 1650 sq-ft, 3 br house ‘ ...
 7          ‘(using gradient descent):\n $%f\n‘], price);
 8 
 9 fprintf(‘Program paused. Press enter to continue.\n‘);
10 pause;

 1 %% ================ Part 3: Normal Equations ================
 2 %利用正规方程预测新值(Normal Equation)
 3 fprintf(‘Solving with normal equations...\n‘);
 4 
 5 %% Load Data
 6 data = csvread(‘ex1data2.txt‘);
 7 X = data(:, 1:2);
 8 y = data(:, 3);
 9 m = length(y);
10 
11 % Add intercept term to X
12 X = [ones(m, 1) X];
13 
14 % Calculate the parameters from the normal equation
15 theta = normalEqn(X, y);

 % normalEquation的实现
 1 function [theta] = normalEqn(X, y)
 2 
 3 theta = zeros(size(X, 2), 1);
 4
 6 % Instructions: Complete the code to compute the closed form solution
 7 %               to linear regression and put the result in theta.
 8 
 9 theta = pinv(X‘ * X) * X‘ * y;
10 
11 end

 1 % Display normal equation‘s result
 2 fprintf(‘Theta computed from the normal equations: \n‘);
 3 fprintf(‘ %f \n‘, theta);
 4 fprintf(‘\n‘);
 5 
 6 
 7 % Estimate the price of a 1650 sq-ft, 3 br house
 8 
 9 price = 0; 
10 price = [1, X(1:2)] * theta;    %利用正规方程预测新值
11 
12 
13 fprintf([‘Predicted price of a 1650 sq-ft, 3 br house ‘ ...
14          ‘(using normal equations):\n $%f\n‘], price);