吴恩达机器学习第一次作业——线性回归
线性回归
- 一、单变量线性回归
- 1，线性回归
- 2，梯度下降算法（Gradient descent algorithm）
- 3，可视化实现
- 二、多变量线性回归
- 1，回归方程
- 2，缩小特征
- 3，梯度下降算法的实现
一、单变量线性回归
1，线性回归
使用plotData.m，完成了对已加载数据集ex1data1.txt的可视化，反应在二维坐标中。
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
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 xxaxis label
% ============================================================
end
平面中存在零散的数据集，寻找数据集之间的联系，可以将数据集拟合成一条直线。
假设单变量线性回归的函数为：
h θ ( x ) = θ 0 + θ 1 x h_{\theta }(x)=\theta _{0}+\theta_{1}x hθ(x)=θ0+θ1x
我们程序也需要一个机制去评估我们θ是否比较好，所以说需要对我们做出的h函数进行评估，描述h函数不好的程度，在下面，我们称这个函数为代价函数为：
J ( θ ) = 1 2 m ∑ i = 1 m ( h θ ( x ( i ) ) − y ( i ) ) 2 J(\theta )=\frac{1}{2m}\sum_{i=1}^{m} (h_\theta(x^{(i)})-y^{(i)})^{2} J(θ)=2m1∑i=1m(hθ(x(i))−y(i))2
自定义函数computeCost.m，编写这里分析的代价函数的公式
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 = sum((X*theta-y).^2)/(2*m);
% =========================================================================
end
希望得到最小的代价函数。此时得到的参数 θ \theta θ对应的线性拟合即为最佳拟合。
得到最小的代价函数的方法:梯度下降算法（Gradient descent algorithm）
2，梯度下降算法（Gradient descent algorithm）
循环下述赋值过程，直到 θ \theta θ收敛。
{
θ j : = θ j − α ∂ ∂ / θ j J ( θ ) \theta_j:=\theta_j-\alpha\frac{\partial }{\partial /\theta_j}J(\theta) θj:=θj−α∂/θj∂J(θ)
(for j=0 and j=1 ) }
其中 α \alpha α是机器学习效率。 α \alpha α的设置不能太大，只有当 α \alpha α足够小时，代价函数才会在经过每一次迭代后都会减小，知道收敛；但是 α \alpha α也不能太小， α \alpha α太小时，代价函数收敛的速度非常慢。
要求 θ 0 , θ 1 \theta_0,\theta_1 θ0,θ1同步更新。
更新过程为：
t e m p 0 : = θ 0 − α ∂ ∂ / θ 0 J ( θ ) temp_0:=\theta_0-\alpha\frac{\partial }{\partial /\theta_0}J(\theta) temp0:=θ0−α∂/θ0∂J(θ)
t e m p 1 : = θ 1 − α ∂ ∂ / θ 1 J ( θ ) temp_1:=\theta_1-\alpha\frac{\partial }{\partial /\theta_1}J(\theta) temp1:=θ1−α∂/θ1∂J(θ)
θ 0 : = t e m p 0 \theta_0:=temp_0 θ0:=temp0
θ 1 : = t e m p 1 \theta_1:=temp_1 θ1:=temp1
其中，
∂ ∂ / θ j J ( θ ) = ∂ ∂ / θ j 1 2 m ∑ i = 1 m ( h θ ( x ( i ) ) − y ( i ) ) 2 \frac{\partial }{\partial /\theta_j}J(\theta)=\frac{\partial }{\partial /\theta_j}\frac{1}{2m}\sum_{i=1}^{m} (h_\theta(x^{(i)})-y^{(i)})^{2} ∂/θj∂J(θ)=∂/θj∂2m1∑i=1m(hθ(x(i))−y(i))2
= 1 2 m ∑ i = 1 m ( θ 0 + θ 1 x ( i ) − y ( i ) ) 2 =\frac{1}{2m}\sum_{i=1}^{m}(\theta_0+\theta_1x^{(i)}-y^{(i)})^2 =2m1∑i=1m(θ0+θ1x(i)−y(i))2
w h e n j = 0 when j=0 whenj=0
∂ ∂ / θ j J ( θ ) = 1 m ∑ i = 1 m ( h θ ( x ( i ) ) − y ( i ) ) \frac{\partial }{\partial /\theta_j}J(\theta)=\frac{1}{m}\sum_{i=1}^{m} (h_\theta(x^{(i)})-y^{(i)}) ∂/θj∂J(θ)=m1∑i=1m(hθ(x(i))−y(i))
w h e n j = 1 when j=1 whenj=1
∂ ∂ / θ j J ( θ ) = 1 m ∑ i = 1 m ( h θ ( x ( i ) ) − y ( i ) ) x ( i ) \frac{\partial }{\partial /\theta_j}J(\theta)=\frac{1}{m}\sum_{i=1}^{m} (h_\theta(x^{(i)})-y^{(i)})x^{(i)} ∂/θj∂J(θ)=m1∑i=1m(hθ(x(i))−y(i))x(i)
自定义函数gradientDescent.m，实现梯度下降循环赋值算法，根据前文分析同时更新 θ 0 , θ 1 \theta_0,\theta_1 θ0,θ1。
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);
theta_s=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 (computeCost) and gradient here.
%
theta(1) = theta(1) - alpha / m * sum(X * theta_s - y);
theta(2) = theta(2) - alpha / m * sum((X * theta_s - y) .* X(:,2)); % 必须同时更新theta(1)和theta(2)，所以不能用X * theta,而要用theta_s存储上次结果。
theta_s=theta;
% ============================================================
% Save the cost J in every iteration
J_history(iter) = computeCost(X, y, theta);
end
end
3，可视化实现
%% 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);
根据最后收敛得到的 θ \theta θ参数，画出线性拟合的直线
得到的数据集的点零散分布在平面内，线性拟合一条直线，使这些点尽可能分布在这条线上或周围。
为了更好得理解 J ( θ ) J(\theta) J(θ),建立水平轴 θ o , θ 1 \theta_o,\theta_1 θo,θ1的空间坐标系，垂直轴为 J ( θ ) J(\theta) J(θ)的值，得到下图左。
代码为：
% 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
以 θ 0 \theta_0 θ0为横轴， θ 1 \theta_1 θ1为纵轴，得到 J ( θ ) J(\theta) J(θ)等高线图，得到下图右。图中红色点为 θ o , θ 1 \theta_o,\theta_1 θo,θ1收敛到最小处时的位置。
二、多变量线性回归
1，回归方程
多变量线性回归方程与单变量线性回归假设的方程类似，区别在于前者的变量较多：
h θ ( x ) = θ 0 + θ 1 x 1 + θ 2 x 2 + θ 3 x 3 + . . . + θ n x n h_{\theta }(x)=\theta _{0}+\theta_{1}x_1+\theta_{2}x_2+\theta_3x_3+...+\theta_nx_n hθ(x)=θ0+θ1x1+θ2x2+θ3x3+...+θnxn
为了后面的运算，在这里我们可以令 x 0 = 1 x_0=1 x0=1,将 θ , x \theta,x θ,x视为向量
可以得到
h θ ( x ) = θ T x h_\theta(x)=\theta^{T}x hθ(x)=θTx
多变量线性回归的代价函数为：
J ( θ ) = 1 2 m ∑ i = 1 m ( h θ ( x ( i ) ) − y ( i ) ) 2 J(\theta )=\frac{1}{2m}\sum_{i=1}^{m} (h_\theta(x^{(i)})-y^{(i)})^{2} J(θ)=2m1∑i=1m(hθ(x(i))−y(i))2
2，缩小特征
为了使梯度下降算法收敛得足够快，需要对缩小特征。
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=X(:,1);
sigma=X(:,2);
mu = (mu-mean(mu))/std(mu);
sigma = (sigma-mean(sigma))/std(sigma);
X_norm=[mu,sigma];
% ============================================================
end
3，梯度下降算法的实现
首先要计算代价函数，根据公式， J ( θ ) = 1 2 m ∑ i = 1 m ( h θ ( x ( i ) ) − y ( i ) ) 2 J(\theta )=\frac{1}{2m}\sum_{i=1}^{m} (h_\theta(x^{(i)})-y^{(i)})^{2} J(θ)=2m1∑i=1m(hθ(x(i))−y(i))2自定义函数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.
J = sum((X*theta-y).^2)/(2*m);
% =========================================================================
end
自定义gradientDescentMulti.m函数实现梯度下降算法
与单变量回归的梯度算法相同，都是每次迭代后需要同时 θ \theta θ值，
{
θ j : = θ j − α ∂ ∂ / θ j J ( θ ) \theta_j:=\theta_j-\alpha\frac{\partial }{\partial /\theta_j}J(\theta) θj:=θj−α∂/θj∂J(θ)
(for j=0 and j=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 -alpha/m*X'*(X*theta-y);
% ============================================================
% Save the cost J in every iteration
J_history(iter) = computeCostMulti(X, y, theta);
end
画出随着迭代次数的增加（ α = 0.01 \alpha=0.01 α=0.01），代价损失下降的关系
可以看到随着迭代更替，代价损失逐渐减少，直至收敛。
再增大学习效率的值 α \alpha α( α = 0.1 \alpha=0.1 α=0.1)，得到图像如下，会发现代价函数收敛速度大大加快。
减小学习效率的值 α \alpha α( α = 0.001 \alpha=0.001 α=0.001)，得到图像如下，会发现代价函数收敛速度变缓。
附上多变量线性回归的完整实现代码
%% 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
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 = 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
% ============================================================
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
% ============================================================
fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...
'(using normal equations):\n $%f\n'], price);
总结
从实验结果看到，正则化与梯度下降法得到的 θ \theta θ不同，下面对两种方法进行比较
