单变量线性回归：
- ex1.m
- plotData.m
- computerCost.m
- gradientDescent.m
多变量线性回归：
- ex1_multi.m
- featureNormalize.m
- computerCostMulti.m
- gredientDescentMulti.m
- normalEqn.m
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
%
%% Initialization
clear ; close all; clc
%% ==================== 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: Cost and 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;
fprintf('
Testing the cost function ...
')
% compute and display initial cost
J = computeCost(X, y, theta);
fprintf('With theta = [0 ; 0]
Cost computed = %f
', J);
fprintf('Expected cost value (approx) 32.07
');
% further testing of the cost function
J = computeCost(X, y, [-1 ; 2]);
fprintf('
With theta = [-1 ; 2]
Cost computed = %f
', J);
fprintf('Expected cost value (approx) 54.24
');
fprintf('Program paused. Press enter to continue.
');
pause;
fprintf('
Running Gradient Descent ...
')
% run gradient descent
theta = gradientDescent(X, y, theta, alpha, iterations);
% print theta to screen
fprintf('Theta found by gradient descent:
');
fprintf('%f
', theta);
fprintf('Expected theta values (approx)
');
fprintf(' -3.6303
1.1664
');
% 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);
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.
figure; % open a new figure window
plot(x, y, 'rx', 'MarkerSize', 10);
ylabel('Profit in $10,000s');
xlabel('Population of City in 10,000s');
% ====================== 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);
% ============================================================
end
computerCost.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 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. h = X * theta; J = 1/(2*m) * sum((h-y).^2) % ========================================================================= end
gradientDescent.m
function [theta, J_history] = gradientDescent(X, y, theta, alpha, num_iters)
%GRADIENTDESCENT Performs gradient descent to learn theta
% theta = GRADIENTDESCENT(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 (computeCost) and gradient here.
%
tempTheta = theta;
theta(1) = tempTheta(1) - alpha / m * sum(X * tempTheta - y);
theta(2) = tempTheta(2) - alpha / m * sum((X * tempTheta - y) .*X(:,2));
% ============================================================
% Save the cost J in every iteration
J_history(iter) = computeCost(X, y, theta);
end
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 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);
% 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, '-g', '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.
a = (1650 - mu(1,1)) / sigma(1,1)
b = (3 - mu(1,2)) / sigma(1,2)
price = [1,a,b] * theta; % You should change this
% ============================================================
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 ; % You should change this
% ============================================================
fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...
'(using normal equations):
$%f
'], price);
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 correctly X_norm = X; size(X, 2) mu = zeros(1, size(X, 2)); sigma = zeros(1, size(X, 2)); % ====================== 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); sigma = std(X) % for i = 1:size(X_norm, 2); % X_norm(:,i) = (X_norm(:,i) - mu(:,i))/sigma(:,i) X_norm = (X - repmat(mu,size(X,1),1)) ./ repmat(sigma,size(X,1),1); %X_norm = (X - mu) ./ sigma % ============================================================ end
computerCostMulti.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 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); % J = 1 / (2 * m) * sum((X * theta - y).^2); % ========================================================================= end
gredientDescentMulti.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 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.
%
tempTheta = theta;
for i = 1:size(theta,1)
theta(i) = tempTheta(i) - alpha / m * sum((X * tempTheta - y) .* X(:,i));
end
% ============================================================
% Save the cost J in every iteration
J_history(iter) = computeCostMulti(X, y, theta);
end
end
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 = (X' * X) X' * y; % ------------------------------------------------------------- % ============================================================ end