【转载】用OCTAVE实现一元线性回归的梯度下降算法

原文地址：http://www.cnblogs.com/KID-XiaoYuan/p/7247481.html

STEP1 PLOTTING THE DATA

在处理数据之前，我们通常要了解数据，对于这次的数据集合，我们可以通过离散的点来描绘它，在一个2D的平面里把它画出来。

 

我们把ex1data1中的内容读取到X变量和y变量中，用m表示数据长度。

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data = load('ex1data1.txt'); X = data(:,1); y = data(:,2); m = length(y);

接下来通过图像描绘出来。

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plot(x,y,'rx','MakerSize',10); ylabel('Profit in $10,000s'); xlabel('Population of City in 10,000s');

　　现在我们得到图像如图所示，就是原始的数据的直观表示。

STEP2 GRADIENT DESCENT

现在，我们通过梯度下降法对参数θ进行线性回归。

依照我们之前所得出步骤方法

迭代更新

计算θ值函数：

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function J = computeCost(X, y, theta) %COMPUTECOST Compute cost for linear regression %   J = COMPUTECOST(X, y, theta) computes the cost of using theta as the %   parameter for linear regression to fit the data points in X and y   % Initialize some useful values m = length(y); % number of training examples   % You need to return the following variables correctly J = 0;   % ====================== YOUR CODE HERE ====================== % Instructions: Compute the cost of a particular choice of theta %               You should set J to the cost. J = sum((X * theta - y).^2) / (2*m);     % X(79,2)  theta(2,1)           % =========================================================================   end

　　接下来是梯度下降函数

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function [theta, J_history] = gradientDescent(X, y, theta, alpha, num_iters) %GRADIENTDESCENT Performs gradient descent to learn theta %   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, 1); theta_s=theta;   for iter = 1:num_iters       % ====================== YOUR CODE HERE ======================     % 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 (computeCost) and gradient here.     %     theta(1) = theta(1) - alpha / m * sum(X * theta_s - y);           theta(2) = theta(2) - alpha / m * sum((X * theta_s - y) .* X(:,2));     % 必须同时更新theta(1)和theta(2)，所以不能用X * theta,而要用theta_s存储上次结果。     theta_s=theta;             % ============================================================       % Save the cost J in every iteration        J_history(iter) = computeCost(X, y, theta);   end J_history end

绘图函数：

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function plotData(x, y) %PLOTDATA Plots the data points x and y into a new figure % PLOTDATA(x,y) plots the data points and gives the figure axes labels of % population and profit.   % ====================== YOUR CODE HERE ====================== % Instructions: Plot the training data into a figure using the % "figure" and "plot" commands. Set the axes labels using % the "xlabel" and "ylabel" commands. Assume the % population and revenue data have been passed in % as the x and y arguments of this function. % % Hint: You can use the 'rx' option with plot to have the markers % appear as red crosses. Furthermore, you can make the % markers larger by using plot(..., 'rx', 'MarkerSize', 10);   figure; % open a new figure window plot(x, y, 'rx', 'MarkerSize', 10); % Plot the data ylabel('Profit in $10,000s'); % Set the y axis label xlabel('Population of City in 10,000s'); % Set the x axis label             % ============================================================   end

　　　　根据以上函数，我们进行线性回归：

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<br>%% Machine Learning Online Class - Exercise 1: Linear Regression   %  Instructions %  ------------ % %  This file contains code that helps you get started on the %  linear exercise. You will need to complete the following functions %  in this exericse: % %     warmUpExercise.m %     plotData.m %     gradientDescent.m %     computeCost.m %     gradientDescentMulti.m %     computeCostMulti.m %     featureNormalize.m %     normalEqn.m % %  For this exercise, you will not need to change any code in this file, %  or any other files other than those mentioned above. % % x refers to the population size in 10,000s % y refers to the profit in $10,000s %     %% ==================== Part 1: Basic Function ==================== % Complete warmUpExercise.m fprintf('Running warmUpExercise ... '); fprintf('5x5 Identity Matrix: '); warmUpExercise()   fprintf('Program paused. Press enter to continue. '); pause;     %% ======================= Part 2: Plotting ======================= fprintf('Plotting Data ... ') 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. '); pause;   %% =================== Part 3: Gradient descent =================== fprintf('Running Gradient Descent ... ')   X = [ones(m, 1), data(:,1)]; % Add a column of ones to x theta = zeros(2, 1); % initialize fitting parameters   % Some gradient descent settings iterations = 1500; alpha = 0.01;   % compute and display initial cost computeCost(X, y, theta)   % run gradient descent theta = gradientDescent(X, y, theta, alpha, iterations);   % print theta to screen fprintf('Theta found by gradient descent: '); fprintf('%f %f ', theta(1), theta(2));   % Plot the linear fit hold on; % keep previous plot visible plot(X(:,2), X*theta, '-') legend('Training data', 'Linear regression') hold off % don't overlay any more plots on this figure   % Predict values for population sizes of 35,000 and 70,000 predict1 = [1, 3.5] *theta; fprintf('For population = 35,000, we predict a profit of %f ',...     predict1*10000); predict2 = [1, 7] * theta; fprintf('For population = 70,000, we predict a profit of %f ',...     predict2*10000);   fprintf('Program paused. Press enter to continue. '); pause;   %% ============= Part 4: Visualizing J(theta_0, theta_1) ============= fprintf('Visualizing J(theta_0, theta_1) ... ')   % Grid over which we will calculate J theta0_vals = linspace(-10, 10, 100); theta1_vals = linspace(-1, 4, 100);   % initialize J_vals to a matrix of 0's J_vals = zeros(length(theta0_vals), length(theta1_vals));   % Fill out J_vals for i = 1:length(theta0_vals)     for j = 1: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(theta0_vals, theta1_vals, J_vals) xlabel(' heta_0'); ylabel(' heta_1');   % Contour plot figure; % Plot J_vals as 15 contours spaced logarithmically between 0.01 and 100 contour(theta0_vals, theta1_vals, J_vals, logspace(-2, 3, 20)) xlabel(' heta_0'); ylabel(' heta_1'); hold on; plot(theta(1), theta(2), 'rx', 'MarkerSize', 10, 'LineWidth', 2);

　　

如图所示，绘制出线性回归函数。

这时所绘制2D等高线图梯度下降表面图：

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function [X_norm, mu, sigma] = featureNormalize(X) %FEATURENORMALIZE Normalizes the features in X %   FEATURENORMALIZE(X) returns a normalized version of X where %   the mean value of each feature is 0 and the standard deviation %   is 1. This is often a good preprocessing step to do when %   working with learning algorithms.   % You need to set these values correctly X_norm = X; mu = zeros(1, size(X, 2));      % mean value 均值   size(X,2)  列数 sigma = zeros(1, size(X, 2));   % standard deviation  标准差   % ====================== YOUR CODE HERE ====================== % 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. %         mu = mean(X);       %  mean value   sigma = std(X);     %  standard deviation   X_norm  = (X - repmat(mu,size(X,1),1)) ./  repmat(sigma,size(X,1),1);                  % ============================================================   end function [theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters) %GRADIENTDESCENTMULTI Performs gradient descent to learn theta %   theta = GRADIENTDESCENTMULTI(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, 1);   for iter = 1:num_iters       % ====================== YOUR CODE HERE ======================     % 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.     %     theta = theta - alpha / m * X' * (X * theta - y);         % ============================================================       % Save the cost J in every iteration        J_history(iter) = computeCostMulti(X, y, theta);   end   end function J = computeCostMulti(X, y, theta) %COMPUTECOSTMULTI Compute cost for linear regression with multiple variables %   J = COMPUTECOSTMULTI(X, y, theta) computes the cost of using theta as the %   parameter for linear regression to fit the data points in X and y   % Initialize some useful values m = length(y); % number of training examples   % You need to return the following variables correctly J = 0;   % ====================== YOUR CODE HERE ====================== % Instructions: Compute the cost of a particular choice of theta %               You should set J to the cost. J = sum((X * theta - y).^2) / (2*m);            % =========================================================================   end function [theta] = normalEqn(X, y) %NORMALEQN Computes the closed-form solution to linear regression %   NORMALEQN(X,y) computes the closed-form solution to linear %   regression using the normal equations.   theta = zeros(size(X, 2), 1);   % ====================== YOUR CODE HERE ====================== % Instructions: Complete the code to compute the closed form solution %               to linear regression and put the result in theta. %   % ---------------------- Sample Solution ----------------------   theta = pinv( X' * X ) * X' * y;     % -------------------------------------------------------------     % ============================================================   end %% Machine Learning Online Class %  Exercise 1: Linear regression with multiple variables % %  Instructions %  ------------ % %  This file contains code that helps you get started on the %  linear regression exercise. % %  You will need to complete the following functions in this %  exericse: % %     warmUpExercise.m %     plotData.m %     gradientDescent.m %     computeCost.m %     gradientDescentMulti.m %     computeCostMulti.m %     featureNormalize.m %     normalEqn.m % %  For this part of the exercise, you will need to change some %  parts of the code below for various experiments (e.g., changing %  learning rates). %   %% Initialization   %% ================ Part 1: Feature Normalization ================   %% Clear and Close Figures clear ; close all; clc   fprintf('Loading data ... ');   %% Load Data data = load('ex1data2.txt'); X = data(:, 1:2); y = data(:, 3); m = length(y);   % Print out some data points fprintf('First 10 examples from the dataset: '); fprintf(' x = [%.0f %.0f], y = %.0f ', [X(1:10,:) y(1:10,:)]');   fprintf('Program paused. Press enter to continue. '); pause;   % Scale features and set them to zero mean fprintf('Normalizing Features ... ');   [X mu sigma] = featureNormalize(X);      % 均值0，标准差1   % Add intercept term to X X = [ones(m, 1) X];     %% ================ Part 2: Gradient Descent ================   % ====================== YOUR CODE HERE ====================== % 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 1650 sq-ft, 3 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 ... ');   % Choose some alpha value alpha = 0.01; num_iters = 8500;   % Init Theta and Run Gradient Descent theta = zeros(3, 1); [theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters);   % Plot the convergence graph figure; plot(1:numel(J_history), J_history, '-b', 'LineWidth', 2); xlabel('Number of iterations'); ylabel('Cost J');   % Display gradient descent's result fprintf('Theta computed from gradient descent: '); fprintf(' %f ', theta); fprintf(' ');   % Estimate the price of a 1650 sq-ft, 3 br house % ====================== YOUR CODE HERE ====================== % Recall that the first column of X is all-ones. Thus, it does % not need to be normalized. price = [1 (([1650 3]-mu) ./ sigma)] * theta ; % ============================================================   fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...          '(using gradient descent): $%f '], price);   fprintf('Program paused. Press enter to continue. '); pause;   %% ================ Part 3: Normal Equations ================   fprintf('Solving with normal equations... ');   % ====================== YOUR CODE HERE ====================== % Instructions: The following code computes the closed form %               solution for linear regression using the normal %               equations. You should complete the code in %               normalEqn.m % %               After doing so, you should complete this code %               to predict the price of a 1650 sq-ft, 3 br house. %   %% Load Data data = csvread('ex1data2.txt'); X = data(:, 1:2); y = data(:, 3); m = length(y);   % Add intercept term to X X = [ones(m, 1) X];   % Calculate the parameters from the normal equation theta = normalEqn(X, y);   % Display normal equation's result fprintf('Theta computed from the normal equations: '); fprintf(' %f ', theta); fprintf(' ');     % Estimate the price of a 1650 sq-ft, 3 br house % ====================== YOUR CODE HERE ====================== price = [1 1650 3] * theta ;     % ============================================================   fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...          '(using normal equations): $%f '], price);

　　处理前：

处理后：

 回归过程如图所示：

至此，我们通过梯度下降法解决了此问题，我们还可以通过之前所说的数学方法来解决，但是对于数据太大的情况（通常大于10000），我们就会通过梯度下降法来解决了

　　根据以上函数，我们进行线性回归：