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
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 1 %  Exercise 1: Linear regression with multiple variables

 %% Initialization

 %% ================ Part 1: Feature Normalization ================

 %% Clear and Close Figures
 clear ; close all; clc

 fprintf('Loading data ...\n');

 %% Load Data
 data = load('ex1data2.txt');
 X = data(:, :);
 y = data(:, );
 m = length(y);

 % Print out some data points
 fprintf('First 10 examples from the dataset: \n');
 fprintf(' x = [%.0f %.0f], y = %.0f \n', [X(:,:) y(:,:)]');

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

 % Scale features and set them to zero mean
 fprintf('Normalizing Features ...\n');

 [X, mu, sigma] = featureNormalize(X);

 1 %featureNormalize(X)函数实现
 function [X_norm, mu, sigma] = featureNormalize(X)
 X_norm = X;                      % X是需要正规化的矩阵
 mu = zeros(, size(X, ));       % 生成 1x3 的全0矩阵
 sigma = zeros(, size(X, ));    % 同上

 % Instructions: First, for each feature dimension, compute the mean
 %               of the feature and subtract it from the dataset,
 %               storing the mean value in mu. Next, compute the
 %               standard deviation of each feature and divide
 %               each feature by it's standard deviation, storing
 %               the standard deviation in sigma.
 %
 %               Note that X is a matrix where each column is a
 %               feature and each row is an example. You need
 %               to perform the normalization separately for
 %               each feature.
 %
 % Hint: You might find the 'mean' and 'std' functions useful.

 % std，均方差，std(X,,)求列向量方差，std(X,,)求行向量方差。

 mu = mean(X, );                 %求每列的均值--即一种特征的所有样本的均值
 sigma = std(X);                  %默认同std(X,,)求列向量方差
 %fprintf('Debug....\n'); disp(sigma);
 i = ;
 len = size(X,);                 %行数
 while i <= len,
     %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)
     X_norm(:,i) = (X(:,i) - mu(,i)) / (sigma(,i));
     i = i + ;
 end

 1 % Add intercept term to X
 2 X = [ones(m, 1) X];        

 %% ================ Part : Gradient Descent ================

 % Instructions: We have provided you with the following starter
 %               code that runs gradient descent with a particular
 %               learning rate (alpha).
 %
 %               Your task is to first make sure that your functions -
 %               computeCost and gradientDescent already work with
 %               this starter code and support multiple variables.
 %
 %               After that, try running gradient descent with
 %               different values of alpha and see which one gives
 %               you the best result.
 %
 %               Finally, you should complete the code at the end
 %               to predict the price of a  sq-ft,  br house.
 %
 % Hint: By using the 'hold on' command, you can plot multiple
 %       graphs on the same figure.
 %
 % Hint: At prediction, make sure you do the same feature normalization.
 %

 fprintf('Running gradient descent ...\n');

 % Choose some alpha value
 alpha = 0.03;                         % learning rate - 可尝试0.,0.03,0.1,0.3...
 num_iters = ;                      % 迭代次数

 % Init Theta and Run Gradient Descent
 theta = zeros(, );                  % 3x1的全零矩阵
 [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)

 % Initialize some useful values
 m = length(y);                    % number of training examples
 feature_number = size(X,);       % number of feature

 J_history = zeros(num_iters, );
 temp = zeros(feature_number, );

 for iter =  : num_iters
     predictions = X * theta;
     sqrError = (predictions - y);
     for i =  : feature_number    % Simultneously update theta(i) （同时更新）
         temp(i) = theta(i) - (alpha / m) * sum(sqrError .* X(:,i));
     end

     for j =  : feature_number
         theta(j) = temp(j);
     end

     % Instructions: Perform a single gradient step on the parameter vector
     %               theta.
     %
     % Hint: While debugging, it can be useful to print out the values
     %       of the cost function (computeCostMulti) and gradient here.
     %

     % ============================================================

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

 end

 end

 1 % Plot the convergence graph
 figure;
 plot(:numel(J_history), J_history, '-b', 'LineWidth', ); % '-b'--用蓝线绘制图像,线宽为2
 xlabel('Number of iterations');
 ylabel('Cost J');

 % Display gradient descent's result
 fprintf('Theta computed from gradient descent: \n');
 fprintf(' %f \n', theta);
 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 % 上面的Tip实现如： 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
 price = [, X(:)] * theta;   %利用矩阵乘法--预测多特征下的price

 % ============================================================

 fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...
          '(using gradient descent):\n $%f\n'], price);

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

 1 %% ================ Part 3: Normal Equations ================
 2 %利用正规方程预测新值(Normal Equation)
 fprintf('Solving with normal equations...\n');

 %% Load Data
 data = csvread('ex1data2.txt');
 X = data(:, :);
 y = data(:, );
 m = length(y);

 % Add intercept term to X
 X = [ones(m, ) X];

 % Calculate the parameters from the normal equation
 theta = normalEqn(X, y);

 % normalEquation的实现
 1 function [theta] = normalEqn(X, y)

 theta = zeros(size(X, ), );

 % Instructions: Complete the code to compute the closed form solution
 %               to linear regression and put the result in theta.

 theta = pinv(X' * X) * X' * y;

 end

 1 % Display normal equation's result
 fprintf('Theta computed from the normal equations: \n');
 fprintf(' %f \n', theta);
 fprintf('\n');

 % Estimate the price of a  sq-ft,  br house

 price = ;
 price = [, X(:)] * theta;    %利用正规方程预测新值

 fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...
          '(using normal equations):\n $%f\n'], price);