机器学习编程作业（exe1）总结

川长思鸟来 2022-05-18 00:39 214阅读 0赞

# 本博客内容是对Coursera上Andrew Ng的《机器学习》的编程作业的总结，本博文是在Matlab基础上对第一次编程作业的总结，随后的博文会对所有的编程作业进行一一总结。    本博文主要对作业中需要实现的代码进行讲解。    脚本文件ex1.m用来执行单变量线性回归，ex1\_multi.m用来执行多变量线性回归。在这里先看ex1.m中的代码。 #

**单变量线性回归**

%% Initialization
    clear ; close all; clc

初始化部分：clear 清除工作区的所有变量，还可以后面跟变量名来清除某个变量；close all 关闭所有窗口（显示图像的figure窗口）；clc 清除命令窗口的内容（就是命令界面以前的命令）；两个百分号%%是matlab中用来表示代码块的注释，从%%开始到下一个%%之间会作为一个代码块，在matlab中查看时会用黄白相间显示。

%% ==================== 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;

fprintf和c语言中的printf用法类似，也支持%d等占位符，也可以直接输出字符串，\\n表示换行符，Matlab中字符串用单引号括起来；pause表示暂停。   
中间调用了warmUpExercise函数，对应warmUpExercise.m，这个函数要求输出一个5\*5的单位矩阵，直接使用eye函数就可以了。   
warmUpExercise.m实现：

function A = warmUpExercise()
    %WARMUPEXERCISE Example function in octave
    %   A = WARMUPEXERCISE() is an example function that returns the 5x5 identity matrix
    
    A = [];
    % ============= YOUR CODE HERE ==============
    % Instructions: Return the 5x5 identity matrix 
    %               In octave, we return values by defining which variables
    %               represent the return values (at the top of the file)
    %               and then set them accordingly. 
    A =eye(5);
    
    % ===========================================
    end

运行后结果：

![70][]

warmUpExercise函数无输入参数，返回值为A。

%% ======================= Part 2: Plotting =======================
    fprintf('Plotting Data ...\n')
    data = load('ex1data1.txt');
    X = data(:, 1); y = data(:, 2);%X为第一列数据 y为第二列数据
    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;

这一部分先通过load从文件读取数据，然后调用plotData来画图。

运行结果：

![70 1][]  
plotData.m实现:

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);
    
    figure; % open a new figure window
    plot(x,y,'rx','MarkerSize',10);% 画点的自带函数 x存放了第一列数据 y存放第二列数据
    ylabel('Profit in $10,000s');%x为x轴的标题
    xlabel('Population of City in 10,000s');%y为y轴的标题
    
    % ============================================================
    
    end

plot用来画点的自带函数，X存放了数据第一列，y存放了数据的第二列，’rx’表示红色x型；’MarkerSize’, 10 表示大小是10 ，plot的用法非常灵活，可以通过help plot命令查看使用说明。xlabel和ylabel用于设置坐标说明。

%% =================== Part 3: Cost and Gradient descent ===================
    %这个公式中有n+1个参数和n个变量，为了使得公式能够简化一些，引入x0=1，
    % 则公式转化为：hθ(x)=θ0*x0+θ1*x1+θ2*x2+...+θn*xn
    %任何一个训练实例也都是n+1n+1维的向量
    X = [ones(m, 1), data(:,1)]; % Add a column of ones to x
    
    %初始化选择合适的参数（parameters）θ0θ0 和 θ1θ1
    theta = zeros(2, 1); % initialize fitting parameters
    
    % Some gradient descent settings
    iterations = 1500;%迭代数
    alpha = 0.01;%学习率
    
    fprintf('\nTesting the cost function ...\n')
    % compute and display initial cost
    J = computeCost(X, y, theta);%代价函数
    
    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]);%代价函数
    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
    %梯度下降算法
    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
    %用X的输入特征来预测房价 这里最终结果需要扩大1000倍
    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;

这一部分主要用来计算梯度和代价，其中computeCost(X, y, theta)用来计算代价，gradientDescent(X, y, theta, alpha, iterations)用来计算梯度。   
computeCost.m实现：

function J = computeCost(X, y, theta)
    %线性回归算法
    %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 - y; %X*theta 为Hypothesis 
    J = 1/(2*m) * sum(h.^2);%Cost Function
    
    % =========================================================================
    
    end

根据代价函数公式实现代价函数。   
![这里写图片描述][20160902220519813]

再说一下gradientDescent.m的实现：

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);%初始化代价函数
    %% num_iters为迭代数
    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.
        %
        % ============================================================
        %向量化求值
        %假设函数hθ(x) = θ0·x0 + θ1·x1 ，用向量表示成：hθ(x) = θT·x
        %其中，下面公式的向量表示就是：[XT · (X·θ - y)]/m，用Matlab表示就是：X'*(X*theta-y) / m
        theta = theta - (alpha/m)*X'*(X*theta-y); % theta 就是用上面的向量表示法的 matlab 语言实现
    %    遍历求值
    %    for iter = 1:num_iters
    %        for j = 1 : n
    %            theta_new(j) = theta(j) - (alpha / m) * sum((X * theta - y) .* X(:, j));
    %        end
    %        theta = theta_new;
    
        % Save the cost J in every iteration    
        J_history(iter) = computeCost(X, y, theta);
    end
    
    end

根据梯度公式实现gradientDescent函数。   
![这里写图片描述][20160902220747595]

运行结果：

![70 2][]

![70 3][]

![70 4][]

**可视化*J*(θ)**:

使用**表面图**进行可视化：

%% ============= 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);%生成范围在[-10,10]之间100个点的线性行矢量，即维数为1*100的矩阵
    theta1_vals = linspace(-1, 4, 100);%生成范围在[-1,4]之间100个点的线性行矢量，即维数为1*100的矩阵
    
    % initialize J_vals to a matrix of 0's
    J_vals = zeros(length(theta0_vals), length(theta1_vals));%对应的代价函数值，维数为100*100
    
    % Fill out J_vals
    for i = 1:length(theta0_vals)      %计算代价函数值
        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函数的特性，必须进行转置。其实就是因为θ0和θ1要和行列坐标x，y对齐
    % Surface plot 3
    figure;
    surf(theta0_vals, theta1_vals, J_vals)    %绘制表面图
    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);

![70 5][]

![70 6][]

实现featureNormalize.m函数：

> 解决的方法是尝试将所有特征的尺度都尽量缩放到-1到1之间。如图：

![b8167ff0926046e112acf789dba98057.png][]

> 最简单的方法是令：

![03aaa898d71a031856abf52ece9281e7.png][]

其中 μnμn是平均值，snsn是标准差。

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);%均值
    sigma = std(X);%标准差
    
    %梯度下降法实践1-特征缩放公式
    for i = 1:size(X,1)
        X_norm(i, :) = (X(i, :) - mu) ./sigma;
    end
    % ============================================================
    
    end

求出每列的均值（mean）和标准差（std），之后可以对每行进行缩放，通过help命令查看帮助文档。   
对于多变量线性回归中的代价函数和损失函数的实现与单变量线性回归一样，在这里就不进行解释。   
下面来说说用Normai Equations求解theta的过程，   
根据Normal Equations的公式   
![这里写图片描述][20160902223704171]

![70 7][]

实现normalEqn.m函数：

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.
    % 正规方程是通过求解下面的方程来找出使得代价函数最小的参数的：??θjJ(θj)=0??θjJ(θj)=0 
    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 ----------------------
    % 利用正规方程解出向量
    theta=pinv(X'*X)*X'*y;
    
    % -------------------------------------------------------------
    % ============================================================
    
    end

**多变量线性回归**   
特征放缩主要通过放缩公式：

![这里写图片描述][20160902222725158]

参考博客[Coursera-AndrewNg(吴恩达)机器学习笔记——第二周编程作业（线性回归）][Coursera-AndrewNg]

修改ex1\_multi.m部分代码：

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

% Estimate the price of a 1650 sq-ft, 3 br house
    % ====================== YOUR CODE HERE ======================
    price = 0; % You should change this
    
    price = [1 1650 3] * theta;
    
    % ============================================================

**多变量线性回归** ：(代码)

%% 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
    % 特征缩放
    fprintf('Normalizing Features ...\n');
    
    [X mu sigma] = featureNormalize(X);
    
    % Add intercept term to X
    %这个公式中有n+1个参数和n个变量，为了使得公式能够简化一些，引入x0=1，
    % 则公式转化为：hθ(x)=θ0*x0+θ1*x1+θ2*x2+...+θn*xn
    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 = 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);
    
    % 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.
    price = 0; % You should change this
    % 选取其中一个输入特征来预测
    
    t = [1650 3];
    t = (t - mu) ./sigma;
    price = [1 t] * 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 ======================
    price = 0; % You should change this
    
    price = [1 1650 3]*theta;
    % ============================================================
    
    fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...
             '(using normal equations):\n $%f\n'], price);

![70 8][]

![70 9][]

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);
    n=length(theta);
    theta_t=theta;
    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.
        %
        %    遍历求值
        for j=1 : n
             theta_t(j,1) = theta(j,1)-alpha*sum((X*theta-y).*X(:,j))/m
    
          end
    
        theta=theta_t;
    
        % ============================================================
    
        % Save the cost J in every iteration    
        J_history(iter) = computeCostMulti(X, y, theta);
    
    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 = 0;
    
    % ====================== YOUR CODE HERE ======================
    % Instructions: Compute the cost of a particular choice of theta
    %               You should set J to the cost.
        J=sum((X*theta-y).^2)/(2*m)
    % =========================================================================
    
    end

[70]: /images/20220518/8ced7a0f2fa049ae80ec0f44ec7a8912.png
[70 1]: /images/20220518/3787cd70762c42feb6516e8c0802ba29.png
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[20160902222725158]: /images/20220518/7c9d24e5d03f427a870df5c8e8c4671e.png
[Coursera-AndrewNg]: https://www.cnblogs.com/LoganGo/p/8532018.html
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[70 9]: /images/20220518/3508ef893587495187d8267dc45a6655.png