Coursera-Machine Learning-Andrew Ng-Programming Exercise 1

【该系列心得】

良心入门课程,作业设计很用心,也挺有意思。

各作业中对Matlab脚本中公式的vectorization、索引技巧尤其有趣,值得反复思考。


【Exercise1 Linear Regression】

代码【第一部分】

ex1.m

-> 数据可视化

-> 加全一列 

-> 全零初始化 

-> 实现成本函数

-> 梯度下降法

--> 作拟合曲线

-> 单点预测

-> 成本函数可视化:曲面图、等高线图,画出最优点

%% 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);

plotData.m 

数据可视化,简单的plot调用

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.


figure; % open a new figure window


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


    plot(x,y,'rx','MarkerSize',10);


    ylabel('Profit in $10,000s'); % Set the y_axis label
    xlabel('Population of City in 10,000s'); % Set the x_axis label
% ============================================================


end

computeCost.m 

计算成本,套公式,注意vectorization

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.
    J=1/2/m*sum((X*theta-y).^2)
% =========================================================================


end

gradientDescent.m 

梯度下降,并记录每次迭代成本。套公式、向量化。

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


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


end


end

【第二部分】

ex1_multi.m

梯度下降法

-> 预处理:特征缩放、均值归一、加全一列

-> 全零初始化 梯度下降

-> 成本收敛图

-> 单点预测(同样要经过“特征缩放 均值归一 加全一列”的预处理)

正规公式法:

-> 加全一列 利用公式计算

-> 单点预测(加全一列的预处理,没有特征缩放和均值归一!因为整个方法从头就没有用)

%% 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];




%% ================ 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 = 1;
num_iters = 400;


% 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;
plot(1:numel(J_history), J_history, '-b', '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.
X_est=[1650 3];
X_est=(X_est-mu)./sigma;
X_est=[1 X_est];


price = 0; % You should change this
price = X_est*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;


%% ================ 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 ======================
X_est=[1650 3];
% 没有特征缩放和均值归一化!
X_est=[1 X_est];


price = 0; % You should change this
price = X_est*theta


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


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



featureNormalize.m

特征缩放、均值归一

同时输出均值标准差(供预测时用,预测时对输入数据进行相同的预处理)(以后在测试集cv集也能用到)。

套公式。

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);
    sigma=std(X);
    X_norm=(X-mu)./sigma;
% ============================================================

end

gradientDescentMulti.m

梯度下降法

由于向量化带来的通用性,与单变量完全相同

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);

end

end


normalEqn.m

正规公式法

套公式

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


未计分部分:

不同学习率下的成本收敛曲线:

Coursera-Machine Learning-Andrew Ng-Programming Exercise 1_第1张图片

3-1






你可能感兴趣的:(ML)