Andrew Ng機器學習 一： Linear Regression

一：單變量線性迴歸（Linear regression with one variable）

　　背景：在某城市開辦飯館，咱們有這樣的數據集ex1data1.txt，第一列表明某個城市的人口，第二列表明在該城市開辦飯館的利潤。

 

　　　　咱們將數據集顯示在可視圖，能夠看出跟某個線性方程有關，而此數據只有單個變量（某城市人口），故接下來咱們就使用單變量線性迴歸擬合出一條近似知足於上數據的直線。

　　1，單變量的腳本ex1.m:

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

ex1.m

 

　　2，單變量代價函數（cost function）:

　　　　$J(\theta_0,\theta_1)=J(\theta)=\frac{1}{2m}\sum_{i=1}^{m}(h_\theta(x^{(i)})-y^{(i)})^2$ 

　　其中假設(預測)函數：$  h_{\theta}(x)=\theta^T * x=\theta_0*x_0+\theta_1*x_1$，其中$x_0=1$

function J = computeCost(X, y, theta) m = length(y); J = 0; ans=X*theta; %X*theta計算hθ(x) ans=(ans-y).^2; %計算平方差成本函數 J=sum(ans)/(2*m); %計算全部樣本m的代價 end

computeCost.m

   

　　3，單變量梯度降低（Gradient descent）：

　　$  \theta_j:=\theta_j- \frac{\alpha}{m}\sum_{i=1}^{m}[(h_\theta(x^{(i)})-y^{(i)})x^{(i)}_j]$ (同時更新$large  \theta_j$(all j))，其中$x^{(0)}=1$　

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. %

     % ans1=theta(1)-sum((sum(X*theta,2)-y).*X(:,1))*alpha/m; % ans2=theta(2)-sum((sum(X*theta,2)-y).*X(:,2))*alpha/m; %  theta(1)=ans1; %  theta(2)=ans2; %梯度降低，X爲(m,2)，hθ(x)-y爲(m,1)，先將X轉置 theta=theta-((X')*(X*theta-y)).*(alpha/m); 


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

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

gradientDescent.m

　　梯度降低算出來theta參數值後，咱們就能夠預測了。

　　假設咱們想預測城市人口爲35000，在該城市開辦飯館預計能得到多少利潤。

　　利潤=（[1,3.5]*theta)*10000。(咱們將數據進行了特徵縮放10000)。

　　

 

二：多變量線性迴歸（Linear regression with multiple variables）

　　背景：預測房價，如今有一些數據集ex1data2.txt，第一列爲房子大小(平方英尺)，第二列爲該房子臥室數量，第三列爲該房子的價值。

　　　　咱們觀察該數據集的可視圖後，用多變量線性迴歸去擬合該數據集，注意此處的數據集ex1data2.txt並無進行特徵縮放，故咱們首先對該數據集進行特徵縮放。

　　

　　1，多變量的腳本ex1_multi.m:

%% 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 = 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]; %而外增長一列截距項爲1的數據列 %% ================ 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); ##alpha2 = 0.1; ##theta2 = zeros(3, 1); ##[theta2, J2] = gradientDescentMulti(X, y, theta2, alpha2, num_iters); ##alpha3 = 0.9; ##theta3 = zeros(3, 1); ##[theta3, J3] = gradientDescentMulti(X, y, theta3, alpha3, num_iters); % Plot the convergence graph figure; plot(1:numel(J_history), J_history, '-b', 'LineWidth', 2); ##plot(1:50, J_history(1:50), '-b', 'LineWidth', 2); xlabel('Number of iterations'); ylabel('Cost J'); ##hold on; ##plot(1:50,J2(1:50),'r'); ## ##hold on; ##plot(1:50,J3(1:50),'k'); % 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. %預測房子1650平方，3個房間的價格 predict1=[1650 3]; predict1=(predict1.-mu)./sigma; %用測試數據的特徵縮放的平均值與標準差的值 predict1=[ones(1,1) predict1]; fprintf(' %f \n', predict1); price = predict1 *theta; % 預測價格 % ============================================================ 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 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 ====================== predict1=[1 1650 3]; price = predict1*theta; % 預測價格 % ============================================================ fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ... '(using normal equations):\n $%f\n'], price);

ex1_multi.m

 

　　2，特徵縮放：X進行縮放前，不需添加截距項，縮放後添加

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. % mu=mean(X); %計算X每列的平均值 sigma=std(X); %計算X每列的標準差 X_norm=(X.-mu)./sigma; %特徵縮放 % ============================================================ end

featureNormalize.m

　　

　　3，多變量代價函數（cost function）:

　　　　　　$J(\theta_0,\theta_1,...,\theta_n)=J(\theta)=\frac{1}{2m}\sum_{i=1}^{m}(h_\theta(x^{(i)})-y^{(i)})^2$

　　其中假設(預測)函數： $h_{\theta}(x)=\theta^T * x=\theta_0*x_0+\theta_1*x_1+...+\theta_n*x_n$ ，其中$x_0=1$。

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.

   ans=X*theta-y; %計算hθ(x)-y
   ans=(ans')*ans;  %計算(hθ(x)-y)^2
   % ans=ans.^2; %或者是這樣計算
   J=sum(ans)./(2*m); %計算代價函數

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

end

computeCostMulti.m

 

 

 　　4，多變量梯度降低（Gradient descent）：

　　$  \theta_j:=\theta_j- \frac{\alpha}{m}\sum_{i=1}^{m}[(h_\theta(x^{(i)})-y^{(i)})x^{(i)}_j]$ (同時更新$large  \theta_j$(all j))，其中$x^{(0)}=1$

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-((X')*(X*theta-y)).*(alpha/m);

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

    % Save the cost J in every iteration %每一次迭代計算代價函數保存，之後可做爲可視化圖 J_history(iter) = computeCostMulti(X, y, theta); end end

gradientDescentMulti.m

　

　　梯度降低算出來theta參數值後，咱們就能夠預測了。

　　假設咱們想預測某所房子房屋面積爲1650平方英尺，3個房間，該所房子的價值大約是多少？

　　predict1=([1650 3]-mu)./sigma，先進行縮放，使用訓練集的平均值與標準差。

　　predict1=[1 predict1] ，增長一個截距項1。

　　價值=predict1*theta。

 　　

　　咱們能夠選擇不一樣的學習速率alpha，試試那個學習速率收斂得更快，通常咱們每次選擇爲前一個選擇的3倍，例如：0.01,0.03,0.1,0.3,0.9......

 

三：正規方程(Normal Equations)求解線性迴歸：

　　用正規方程求解線性迴歸方程更便捷，它不須要一直迭代，只需求一個式子就好了：

　　$\theta=(X^TX)^{-1}X^Ty$  $X$爲該數據集的變量(已添加了截距項)，$y$爲該數據集的結果。

　　最後預測值就爲：predict1*theta。

　　

總結：

　　梯度降低與正規方程的比較：

1，須要選擇$\alpha$                                    1，不須要選擇$\alpha$ 

2，須要迭代次數　　　　　　　　　　　   2，不須要迭代　　　　

3，當n很大時，也能很好的運行                3，只需計算一次$(X^TX)^{-1}$

　　　　　　　　　　　　　　　　　　      4，當n很大時，會很慢(由於要逆運算),通常n不大於10000

 

 

個人便籤：作個有情懷的程序員。