Coursera-吴恩达-机器学习-第二周-编程作业: Linear Regression

Programming Exercise 1: Linear Regression.

 

0 Introduction

本片文章内容：

Coursera吴恩达机器学习课程，第二周编程作业。语言是Matlab。

 

2 Linear regression with one variable

 

Initialization：

%% Initialization
clear ; close all; clc

初始化部分：

clear 清除工作区的所有变量，还可以后面跟变量名来清除某个变量；

close all 关闭所有窗口（显示图像的figure窗口）；

clc 清除命令窗口的内容（就是命令界面以前的命令）

Part 1: Basic Function

%% ==================== 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;

fprintf 将数据写入文本文件将多个数值和字面文本输出到屏幕。

\n 表示换行符，Matlab中字符串用单引号括起来。

pause 表示暂停。 
warmUpExercise() 调用warmUpExercise函数，对应warmUpExercise.m，这个函数要求输出一个5*5的单位矩阵，直接使用eye函数。 函数在下面讲解。

Part 2: Plotting

%% ======================= 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.m
plotData(X, y);

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

load从文件读取数据；

调用plotData()函数画图，函数在下面讲解。

Part 3: Cost and Gradient descent

%% =================== 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('\nTesting the cost function ...\n')
% compute and display initial cost
J = computeCost(X, y, theta);
fprintf('With theta = [0 ; 0]\nCost computed = %f\n', J);
fprintf('Expected cost value (approx) 32.07\n');

% further testing of the cost function
J = computeCost(X, y, [-1 ; 2]);
fprintf('\nWith theta = [-1 ; 2]\nCost computed = %f\n', J);
fprintf('Expected cost value (approx) 54.24\n');

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

fprintf('\nRunning Gradient Descent ...\n')
% run gradient descent
theta = gradientDescent(X, y, theta, alpha, iterations);

% print theta to screen
fprintf('Theta found by gradient descent:\n');
fprintf('%f\n', theta);
fprintf('Expected theta values (approx)\n');
fprintf(' -3.6303\n  1.1664\n\n');

% 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\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;

这一部分主要用来计算梯度和代价,并在figure中画出求出的the hypothesis hθ(x)。

调用computeCost(X, y, theta)用来计算代价，函数在下面讲解。

 

the cost function：

 

where the hypothesis hθ(x) is given by the linear model：

 

调用 gradientDescent(X, y, theta, alpha, iterations)函数，用来计算梯度。 函数在下面讲解。

In batch gradient descent, each iteration performs the update：

Implementation Note: We store each example as a row in the the X matrix in Octave/MATLAB. To take into account the intercept term (theta0), we add an additional rst column to X and set it to all ones. This allows us to treat theta0 as simply another `feature'.

 

Part 4: Visualizing J(theta_0, theta_1)

%% ============= Part 4: Visualizing J(theta_0, theta_1) =============
fprintf('Visualizing J(theta_0, theta_1) ...\n')

% 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('\theta_0'); ylabel('\theta_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('\theta_0'); ylabel('\theta_1');
hold on;
plot(theta(1), theta(2), 'rx', 'MarkerSize', 10, 'LineWidth', 2);

可视化theta_0, theta_1，画出3D图和等高线图。

function函数部分汇总

warmUpExercise()

function A = warmUpExercise()
%WARMUPEXERCISE Example function in octave
%   A = WARMUPEXERCISE() is an example function that returns the 5x5 identity matrix

A = [];
% ============= 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

plotData()

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

% ====================== 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);

plot(x, y, 'rx ','Markersize', 10);
ylabel('Profit in $10,000s');
xlabel('Population do city in 10,000s');



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

end

computeCost(X, y, theta)

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 = 1/(2*m) * sum(  (X*theta - y).^2  );


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

end

 gradientDescent(X, y, theta, alpha, iterations)

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.
    %

    % =X * theta;
    %key = 1/m * (hx - y)' * X;
    %theta = theta -alpha * key;
    
    beta =  X' * (X * theta-y);
    theta = theta - (alpha/m) * beta;

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

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

end

end

Result：

 

3 Linear regression with multiple variables

Part 1: Initialization and Feature Normalization

%% 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(:, 1:2);
y = data(:, 3);
m = length(y);

% Print out some data points
fprintf('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 mean
fprintf('Normalizing Features ...\n');

[X mu sigma] = featureNormalize(X);

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

这里调用函数featureNormalize()， 函数在下面讲解。

特征放缩公式：

求出均值（mean）和标准差（std），之后可以对每行进行缩放，我使用的是矩阵操作

另一个需要注意的地方，做了特征缩放之后，在预测时，输入同样也需要特征缩放。

% ====================== 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
estimate_house1 = [1650 3];
estimate_house1 = (estimate_house1 - mu) ./ sigma;
price = [1 estimate_house1] * theta;

 

Part 2: Gradient Descent

%% ================ 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 value
alpha = 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 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: \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
estimate_house1 = [1650 3];
estimate_house1 = (estimate_house1 - mu) ./ sigma;
price = [1 estimate_house1] * theta;

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

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;

这里使用了梯度下降法，

对于多变量线性回归中的代价函数和损失函数的实现与单变量线性回归一样。

 

Part 3: Normal Equations 

Normal Equations的公式，the closed-form solution to linear regression is

这里调用normalEqn(X, y)函数，函数在下面讲解。

%% ================ 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 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: \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

price = [1 1650 3] * theta;

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

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

function函数部分汇总：

featureNormalize()

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));
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.
%       


v1 = ones(size(X_norm, 1), 1);
mu = mean(X_norm);
sigma = std(X_norm);
X_norm = ( X_norm- (v1 * mu) ) ./(v1* sigma);



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

end

normalEqn(X, y)

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

Result：