Stanford 机器学习练习 Part 1 Linear Regression

warmUpExercise.m

function A = warmUpExercise()%WARMUPEXERCISE Example function in octave%   A = WARMUPEXERCISE() is an example function that returns the 5x5 identity matrixA = [];% ============= YOUR CODE HERE ==============% Instructions: Return the 5x5 identity matrix %               In octave, we return values by defining which variables%               represent the return values (at the top of the file)%               and then set them accordingly.     A = eye(5);% ===========================================end

computeCost.m

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 valuesm = 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

plotData.m

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 windowplot(x, y, 'rx', 'MarkerSize', 5);xlabel("x");ylabel("y");% ============================================================end

gradientDescent.m

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 valuesm = length(y); % number of training examplesJ_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 (computeCost) and gradient here.    %    temp = 0;    temp = temp + alpha/m * X' * (y - X * theta);    theta = theta + temp;    % ============================================================    % Save the cost J in every iteration        J_history(iter) = computeCost(X, y, theta);endend

ex1.m

%% 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%%% Initializationclear ; close all; clc%% ==================== Part 1: Basic Function ====================% Complete warmUpExercise.m fprintf('Running warmUpExercise ... \n');fprintf('5x5 Identity Matrix: \n');warmUpExercise()fprintf('Program paused. Press enter to continue.\n');pause;%% ======================= Part 2: Plotting =======================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.mplotData(X, y);fprintf('Program paused. Press enter to continue.\n');pause;%% =================== Part 3: Gradient descent ===================fprintf('Running Gradient Descent ...\n')X = [ones(m, 1), data(:,1)]; % Add a column of ones to xtheta = zeros(2, 1); % initialize fitting parameters% Some gradient descent settingsiterations = 1500;alpha = 0.01;% compute and display initial costcomputeCost(X, y, theta)% run gradient descenttheta = gradientDescent(X, y, theta, alpha, iterations);% print theta to screenfprintf('Theta found by gradient descent: ');fprintf('%f %f \n', theta(1), theta(2));% Plot the linear fithold on; % keep previous plot visibleplot(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 70,000predict1 = [1, 3.5] *theta;fprintf('For population = 35,000, we predict a profit of %f\n',...    predict1*10000);predict2 = [1, 7] * theta;fprintf('For population = 70,000, we predict a profit of %f\n',...    predict2*10000);fprintf('Program paused. Press enter to continue.\n');pause;%% ============= Part 4: Visualizing J(theta_0, theta_1) =============fprintf('Visualizing J(theta_0, theta_1) ...\n')% Grid over which we will calculate Jtheta0_vals = linspace(-10, 10, 100);theta1_vals = linspace(-1, 4, 100);% initialize J_vals to a matrix of 0'sJ_vals = zeros(length(theta0_vals), length(theta1_vals));% Fill out J_valsfor 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);    endend% 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 flippedJ_vals = J_vals';% Surface plotfigure;surf(theta0_vals, theta1_vals, J_vals)xlabel('\theta_0'); ylabel('\theta_1');% Contour plotfigure;% Plot J_vals as 15 contours spaced logarithmically between 0.01 and 100contour(theta0_vals, theta1_vals, J_vals, logspace(-2, 3, 20))xlabel('\theta_0'); ylabel('\theta_1');hold on;plot(theta(1), theta(2), 'rx', 'MarkerSize', 10, 'LineWidth', 2);

featureNormalize.m

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 correctlyX_norm = X;m = size(X, 2);mu = zeros(1, size(X, 2));mu = mean(X);sigma = std(X);for i = 1:m    X_norm(:,i) = (X(:,i).-mu(i))./sigma(i);end% ====================== 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.%       % ============================================================end

computeCostMulti.m

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 valuesm = 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 = 1/(2*m) * ( X * theta - y)' * (X*theta - y);% =========================================================================end

gradientDescentMulti.m

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 valuesm = length(y); % number of training examplesJ_history = zeros(num_iters, 1);temp = zeros(feature_number,1); for iter = 1:num_iters    temp = alpha/m * X' * (y - X*theta);    theta = theta + temp;    J_history(iter) = computeCostMulti(X, y, theta);     % ====================== 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.    %    % ============================================================    % Save the cost J in every iteration        J_history(iter) = computeCostMulti(X, y, theta);endend

normalEqn.m

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

ex1_multi.m

%% 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 Figuresclear ; close all; clcfprintf('Loading data ...\n');%% Load Datadata = load('ex1data2.txt');X = data(:, 1:2);y = data(:, 3);m = length(y);% Print out some data pointsfprintf('First 10 examples from the dataset: \n');fprintf(' x = [%.0f %.0f], y = %.0f \n', [X(1:10,:) y(1:10,:)]');fprintf('Program paused. Press enter to continue.\n');pause;% Scale features and set them to zero meanfprintf('Normalizing Features ...\n');[X mu sigma] = featureNormalize(X);% Add intercept term to XX = [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 ...\n');% Choose some alpha valuealpha = 0.01;num_iters = 400;% Init Theta and Run Gradient Descent theta = zeros(3, 1);[theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters);% Plot the convergence graphfigure;plot(1:numel(J_history), J_history, '-b', 'LineWidth', 2);xlabel('Number of iterations');ylabel('Cost J');% Display gradient descent's resultfprintf('Theta computed from gradient descent: \n');fprintf(' %f \n', theta);fprintf('\n');% 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 = 0; % You should change this% ============================================================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;%% ================ Part 3: Normal Equations ================fprintf('Solving with normal equations...\n');% ====================== 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 Datadata = csvread('ex1data2.txt');X = data(:, 1:2);y = data(:, 3);m = length(y);% Add intercept term to XX = [ones(m, 1) X];% Calculate the parameters from the normal equationtheta = normalEqn(X, y);% Display normal equation's resultfprintf('Theta computed from the normal equations: \n');fprintf(' %f \n', theta);fprintf('\n');% Estimate the price of a 1650 sq-ft, 3 br house% ====================== YOUR CODE HERE ======================price = 0; % You should change this% ============================================================fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...         '(using normal equations):\n $%f\n'], price);