吴恩达机器学习Ex1多元回归部分

多元线性回归
提交作业情况：

背景：预测房价
数据集：房屋大小，卧室的数量，房价。

Loading data ...
First 10 examples from the dataset: x = [2104 3], y = 399900 x = [1600 3], y = 329900 x = [2400 3], y = 369000 x = [1416 2], y = 232000 x = [3000 4], y = 539900 x = [1985 4], y = 299900 x = [1534 3], y = 314900 x = [1427 3], y = 198999 x = [1380 3], y = 212000 x = [1494 3], y = 242500
Program paused. Press enter to continue.

数值归一化

减去均值，然后除以标准差。
两者在matlab中有对应的函数，均值的函数名mean，标准差的函数名std
featureNormalize.m文件

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;%房子大小，卧室的数量
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);
X_norm=(X-mu)./sigma;
% ============================================================end

多元线性回归的代价函数

矩阵形式
J(θ)=12m(Xθ−y)T(Xθ−y)J(\theta)=\frac{1}{2m}(X\theta-y)^T(X\theta-y)J(θ)=2m1(Xθ−y)T(Xθ−y)

其中
X是一个矩阵，维度是(m×(n+1))(m\times (n+1))(m×(n+1)),
θ\thetaθ是一个向量，维度是((n+1)×1)((n+1)\times 1)((n+1)×1)
y是一个列向量，维度是(m×1)(m\times 1)(m×1)

computeCostMulti.m文件

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=1/(2*m)*(X*theta-y)'*(X*theta-y);% =========================================================================end

多元线性回归梯度

梯度下降法求θ\thetaθ:

θ=θ−1mXT(Xθ−y)\theta=\theta-\frac{1}{m}X^T(X\theta-y) θ=θ−m1XT(Xθ−y)
下面的代码主要用来实现此公式
gradientDescentMulti.m文件

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.%theta=theta-alpha/m*X'*(X*theta-y);% ============================================================% Save the cost J in every iteration    J_history(iter) = computeCostMulti(X, y, theta);endend

选择（这也是学习率和迭代次数比较合理的选择）

% Choose some alpha value
alpha = 0.09;
num_iters = 50;

得到的收敛图

% Choose some alpha value
alpha = 0.12;
num_iters = 50;

% Choose some alpha value
alpha = 0.15;
num_iters = 50;

得到的收敛图

part2完整代码

%% ================ 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 = 0.12;
num_iters = 50;% 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;
hold on;
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: \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
price=[1 ([1650 3]-mu)./sigma]*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;

梯度下降法预测房价的代码

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

得到的结果

Running gradient descent ...
Theta computed from gradient descent: 339842.312379 106499.134141 -2521.749858 Predicted price of a 1650 sq-ft, 3 br house (using gradient descent):$293411.151468

使用正规方程

正规方程求解θ\thetaθ 的公式为
θ=(XTX)−1XTy\theta=(X^TX)^{-1}X^T yθ=(XTX)−1XTy

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

part3完整代码

%% ================ 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]-mu)./sigma]*theta;
price=[1 1650 3]*theta;
% ============================================================fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...'(using normal equations):\n $%f\n'], price);

预测房价的代码

price=[1 1650 3]*theta;

这里不需要再进行归一化，

得到的结果```cpp
Solving with normal equations...
Theta computed from the normal equations: 89597.909544 139.210674 -8738.019113 Predicted price of a 1650 sq-ft, 3 br house (using normal equations):$293081.464335

最后两者的结果进行比较，得到不同的θ\thetaθ值，最后预测的房价结果差不多，使用梯度下降法预测的房价是 $293411.151468，使用正规方程预测的房价是$293081.464335。

Normalizing Features ...
Running gradient descent ...
Theta computed from gradient descent: 339842.312379 106499.134141 -2521.749858 Predicted price of a 1650 sq-ft, 3 br house (using gradient descent):$293411.151468
Program paused. Press enter to continue.
Solving with normal equations...
Theta computed from the normal equations: 89597.909544 139.210674 -8738.019113 Predicted price of a 1650 sq-ft, 3 br house (using normal equations):$293081.464335

请注意预测房价的代码差别

%梯度下降法
%需要归一化
price=[1 ([1650 3]-mu)./sigma]*theta;
%正规方程
price=[1 1650 3]*theta;

当特征数量不大（小于10000）时，通常采用正规方程方法来计算参数θ\thetaθ，而不是用梯度下降法。

图片来源：黄海广，机器学习笔记