斯坦福机器学习Coursera课程:第二周作业--一元和多元线性回归

一. 一元线性回归

在Gradient Descent Algorithm中，我们利用不断推导得到两个对此算法非常重要的公式，一个是J(θ)的求解公式，另一个是θ的求解公式：





一元回归中，直接使用这两个公式，来绘制J(θ)的分布曲面，以及θ的求解路径。

命题为：我们为一家连锁餐饮企业新店开张的选址进行利润估算，手中掌握了该连锁集团所辖店铺当地人口数据，及利润金额，需要使用线性回归算法来建立人口与利润的关系，进而为新店进行利润估算，以评估店铺运营前景。

首先我们将该企业的数据绘制在坐标图上，如下图所示，我们需要建立的模型是一条直线，能够在最佳程度上，拟合population与profit之间的关系。其模型为：





[align=left]在逼近θ的过程中，我们如下实现梯度下降：进行了1500次的迭代（相当于朝着最佳拟合点行走1500步），我们在1500步后，得到θ=[-3.630291,1.166362]。[/align]



下图是J(θ)的分布曲面：



接来下是我们求得的最佳θ值在等高线图上所在的位置，和上一张图其实可以重合在一起：



在作业框架基础上，主要要实现的代码部分如下 ：

1. 计算成本函数 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 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;

e=h-y;

J=e'*e/(2*m)

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

end

2、梯度下降算法 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.

    %
      h=X*theta;
      e=h-y;
      theta=theta-alpha*(X'*e)/m;

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

    % Save the cost J in every iteration    

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

最后在octave交互窗口中输入 source ex1.m 即可输出和显示如上的图形。

本例中，根据回归得到的theta, 计算X为35, 75时的Y。从图中我们可以看出，X在[5，10]之间的训练数据比较密集，超过10的已很稀疏，这里计算35，75时的数据只能作以参考，与实际可能偏差较大。

二. 多元线性回归

题目为：使用有多个变量的线性回归来预测房屋的价格。ex1data2.txt 房屋价格的训练集. 第一列为房屋面积，第二列为房屋中的房间数量，第三列为实际的价格。预测X_p=[1650 3];的价格

1. 梯度下降法

a. 首先由于x1和x2相差太多，所以需要使用feature scalling. 这里采用mean derivation. 或者是max-min

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;

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.

%       
m=size(X,1)

mu=mean(X);

sigma=std(X);

disp('mu'),disp(mu);

disp('sigma'),disp(sigma);

for i=1:m;
X_norm(i,:)=(X(i,:)-mu)./sigma;

end;

%disp('X_norm'), disp(X_norm);

b. 在X最前面加上一列，构成最后需要使用的X, 然后初始化alpha = 0.01; num_iters = 400;theta = zeros(3, 1);
[theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters);

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 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.

    %

h=zeros(m,1);

disp('computeCostMulti');

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

h=X*theta;

tmp1=zeros(size(X,2),1);

for i=1:m

tmp1=tmp1+(h(i)-y(i)).*X(i,:)';

end;

theta=theta-(alpha/m)*tmp1;

disp(J_history(iter));

disp(theta);

end;

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

    % Save the cost J in every iteration    

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

期间会调用到computeCostMulti, 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 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=zeros(m,1);

h=X*theta;

J=(1/(2*m))*sum((h-y).^2);

disp('J'),disp(J);

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

函数实现完后，还要在ex1_multi.m中找到调用梯度下降法后面的变量price，这里也要作以实现，原来只定义了变量price=0;

price = 0; % You should change this

X_P=[1650 3];

X_P=(X_P - mu)./sigma;

X_P=[1 X_P];

price = theta' * X_P';   // 这里theta和X_P都是3*1的，都转置为1*3进行乘法计算。

2. 正规方程法

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.
%
theta=pinv(X'*X)*X'*y;
% ---------------------- Sample Solution ----------------------

end

在ex1_multi.m中增加对price的处理，输入为1650和3.

price = 0; % You should change this
X_P=[1650 3];

X_P=[1 X_P];

disp(X_P');

price = theta' * X_P';

octave命令行中执行  source ex1_multi.m ，梯度下降法中成本函数和迭代次数关系如下，300次后，成本函数变化已较小，趋于平行。

这个例子中，梯度下降法和正规方程法的结果如下：

Theta computed from gradient descent:
 334302.063993

 100087.116006

 3673.548451

Predicted price of a 1650 sq-ft, 3 br house (using gradient descent):

 $289314.620338

Program paused. Press enter to continue.

Solving with normal equations...

Theta computed from the normal equations:
 89597.909542

 139.210674

 -8738.019112

Predicted price of a 1650 sq-ft, 3 br house (using normal equations):

 $293081.464335

至此运行结束，命令行中执行submit，按要求填写即可。没法上传附件，代码就不上传了。