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  and the standard deviation

 %   is . 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(, size(X, ));

 sigma = zeros(, size(X, ));

 num_fea=size(X,);

 % ====================== 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.

 %

 for i=:num_fea

     mu(i)=mean(X(:,i));

     sigma(i)=std(X(:,i));

     X_norm(:,i)=(X(:,i)-mu(i))./sigma(i);

 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 = ;

 % ====================== YOUR CODE HERE ======================

 % Instructions: Compute the cost of a particular choice of theta

 %               You should set J to the cost.

 temp=X*theta-y;

 J=/(*m)*temp'*temp;

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

 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, );

 for iter = :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.

     %

     h_error=X*theta-y;

     error=(alpha/m).*(h_error'*X);

     theta=theta-error';

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

     % Save the cost J in every iteration

     J_history(iter) = computeCostMulti(X, y, theta);

 end

 end

%% Machine Learning Online Class

%  Exercise : 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 : Feature Normalization ================

%% Clear and Close Figures

clear ; close all; clc

fprintf('Loading data ...\n');

%% Load Data

data = load('ex1data2.txt');

X = data(:, :);

y = data(:, );

m = length(y);

% Print out some data points

fprintf('First 10 examples from the dataset: \n');

fprintf(' x = [%.0f %.0f], y = %.0f \n', [X(:,:) y(:,:)]');

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, ) X];

%% ================ Part : 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  sq-ft,  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 = ;

% Init Theta and Run Gradient Descent

theta = zeros(, );

[theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters);

% Plot the convergence graph

figure;

plot(:numel(J_history), J_history, '-b', 'LineWidth', );

xlabel('Number of iterations');

ylabel('Cost J');

hold on;

% Display gradient descent's result

fprintf('Theta computed from gradient descent: \n');

fprintf(' %f \n', theta);

fprintf('\n');

% Estimate the price of a  sq-ft,  br house

% ====================== YOUR CODE HERE ======================

% Recall that the first column of X is all-ones. Thus, it does

% not need to be normalized.

x=[  ];

x()=(x()-mu())/sigma();

x()=(x()-mu())/sigma();

price = x*theta; %这里要注意，因为梯度下降使用了特征缩放，这里测试时也一定记得要做同样的特征缩放。

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

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 : 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  sq-ft,  br house.

%

%% Load Data

data = csvread('ex1data2.txt');

X = data(:, :);

y = data(:, );

m = length(y);

% Add intercept term to X

X = [ones(m, ) 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  sq-ft,  br house

% ====================== YOUR CODE HERE ======================

price = [  ]*theta; % You should change this

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

fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...

         '(using normal equations):\n $%f\n'], price);