斯坦福大学机器学习公开课---Programming Exercise 1: Linear Regression

1  Linear regression with one variable

In thispart of this exercise, you will implement linear regression with one variableto predict profits for a food truck. Suppose you are the CEO of a restaurant franchiseand are considering different cities for opening a new outlet. The chainalready has trucks in various cities and you have data for profits andpopulations from the cities.

You wouldlike to use this data to help you select which city to expand to next.

1.1  Plotting the Data

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);

% data = http://www.mamicode.com/load('ex1data1.txt'); % read comma separated data>



1.2  Gradient Descent
I will t the linear regression parameters  to our dataset using gradient descent.
1.2.1 Update Equations
cost


hypothesis


gradient descent

1.2.2 Implementation
2.2.3 Computing the cost J(theta)
<strong>function J = computeCost(X, y, theta)</strong>
%COMPUTECOST Compute cost for linear regression(Attention)

%   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, also m =size(X,1);

% 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.

m=size(X,1);                 % number of training examples m row X n colm
predictions=X*theta;         % predictions of hypothesis on all m examples
sqrErrors=(predictions-y).^2;% squared errors

J=1/(2*m)*sum(sqrErrors);    % cost function

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

end

1.2.4 Gradient descent
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);

%%%%%%%%%%%%%%%%%
n=length(X(1,:)); % number of features(include X0)
%initialize  vals to a matrix of 0's
%theta=zeros(size(X(1,:)'));
delta=zeros(n,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.
    %
 
predictions=X*theta;         % predictions of hypothesis on all m examples
errors=(predictions-y);      %  errors
sums=X'*errors;              %
delta=1/m *sums;

theta=theta-alpha*delta;     %iteration

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

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

end

end



1.3  Debugging
1.4  Visualizing J(theta)
%% 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 ... \n');
fprintf('5x5 Identity Matrix: \n');
warmUpExercise()

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


%% ======================= Part 2: Plotting =======================
fprintf('Plotting Data ...\n')
data = http://www.mamicode.com/load('ex1data1.txt');>

2   Linear regression with multiple variables
you will implement linear regression with multiple variables to predict the prices of houses. Suppose you are selling your house and you want to know what a good market price would be. One way to do this is to first collect information on recent houses sold and make a model of housing prices.

2.1  Feature Normalization
By looking at the values, note that house sizes are about 1000 times the number of bedrooms. When features diff er by orders of magnitude, first performing feature scaling can make gradient descent converge much more quickly.
Your task here is to complete the code in featureNormalize.m to
? Subtract the mean value of each feature from the dataset.
? After subtracting the mean, additionally scale (divide) the feature values by their respective \standard deviations."
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));     % 1 x n
sigma = zeros(1, size(X, 2));  %1 x n


n= size(X,2); % n
m=size(X,1);
% ====================== 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);   % 1 x n;
sigma=std(X) ;% 1 x n;

mu_temp=((mu')*ones(1,m))';  % m x n
sigma_temp=(sigma'*ones(1,m))' ; % m x n

X_norm=(X-mu_temp)./sigma_temp;

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

end

2.2  Gradient Descent
Previously, you implemented gradient descent on a univariate regression problem. The only di fference now is that there is one more feature in the matrix X.
I complete the code in computeCostMulti.m and gradientDescentMulti.m to implement the cost function and gradient descent for linear regression with multiple variables.

cost functions with multiple various
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.
m=size(X,1);                 % number of training examples m row X n colm
predictions=X*theta;         % predictions of hypothesis on all m examples
errors=predictions-y;% squared errors

J=1/(2*m)*(errors)'*errors;    % cost function

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

end

gradien  functions with multiple various
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
% 好像和单变量的一样
% Initialize some useful values
m = length(y); % number of training examples
J_history = zeros(num_iters, 1);

%%%%%%%%%%%%%%%%%%%%%%%%
n= size(X,2);

%theta=zeros(n,1);
delta= zeros(n);

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.
    %

predictions=X*theta;         % predictions of hypothesis on all m examples
errors=(predictions-y);      %  errors
sums=X'*errors;              %
delta=1/m *sums;

theta=theta-alpha*delta;     %iteration

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

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

end

end


2.2.1  Optional (ungraded) exercise: Selecting learning rates
 Trying values of the learning rate on a log-scale, at multiplicative steps of about 3 times the previous value (i.e., 0.3, 0.1, 0.03, 0.01 and so on).
2.3  Normal Equations
The closed-form solution to linear regression is

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));     % 1 x n
sigma = zeros(1, size(X, 2));  %1 x n


n= size(X,2); % n
m=size(X,1);
% ====================== 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);   % 1 x n;
sigma=std(X) ;% 1 x n;

mu_temp=((mu')*ones(1,m))';  % m x n
sigma_temp=(sigma'*ones(1,m))' ; % m x n

X_norm=(X-mu_temp)./sigma_temp;

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

end

Linear regression with multiple variables
%% 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 ...\n');

%% Load Data
data = http://www.mamicode.com/load('ex1data2.txt');>




斯坦福大学机器学习公开课---Programming Exercise 1: Linear Regression