Coursera吴恩达机器学习课程 总结笔记及作业代码——第1,2周

Linear’regression

发现这个教程是最入门的一个教程了，老师讲的很好，也很通俗，每堂课后面还有编程作业，全程用matlab编程，只需要填写核心代码，很适合自学。

1.1 Model representation

起始给出了预测房价的例子。

这个问题属于监督问题，每个样本都给出了准确的答案。
同时属于回归问题，对给定值预测实际输出。

定义 (x(i),y(i)) 为第i个样本，x表示输入值，y表示输出值，上标表示样本。

以下是机器学习运行模型

对于假设h我们可以用一条直线描述，用线性函数预测房价值。
hθ(x)=θ0+θ1∗x

1.2 Cost function

我们取怎样的 θ 值可以使预测值更加准确呢？
想想看，我们应使得每一个预测值和真实值差别不大，可以定义代价函数如下
J(θ0,θ1)=12m∑mi=1(hθ(x(i))−y(i))2
通过使J值取最小来满足需求

下面通过图形方式感受一下代价函数

1.3 Gradient descent

怎样使我们的代价函数取得最小值呢
下面我们采取梯度下降法。

好比我们下山，每次在一点环顾四周，往最陡峭的路向下走，用图形的方式更形象的表示

Gradient descent algorithm
repeat until convergence{
　　 θj=θj−α∂∂θjJ(θ0,θ1) 　　 (for　j=0　and　j=1)
}

注意更新theta值应同时更新，matlab中向量更新即为同时更新，所以应使上式向量化（之后会讲解向量化含义），也可采取下面方式

1.4 Gradient descent for linear regression

repeat until convergence{
　　 θj=θj−α∂∂θjJ(θ0,θ1) 　　 (for　j=0　and　j=1)
}

∂∂θjJ(θ0,θ1)==∂∂θj12m∑i=1m(hθ(x(i)−y(i)))2∂∂θj12m∑i=1m(hθ(θ0+θ1x)−y(i))2

j=0:∂∂θjJ(θ0,θ1)=1m∑mi=1(hθ(x(i)−y(i)))
j=1:∂∂θjJ(θ0,θ1)=1m∑mi=1(hθ(x(i)−y(i)))∗x(i)

2.1 Mul2ple features

如果输入值不止一个，我们的假设函数应修改为
hθ(x)=θ0+θ1x1+θ2x2+⋯+θnxn

为了结构统一，我们设 x0=1
hθ(x)=θ0+θ1x1+θ2x2+⋯+θnxn=θTx

如此一来，便将变量向量化了

New algorithm
repeat until convergence{
　　 θj=θj−α∂∂θjJ(θ)=θj−α1m∑mi=1(hθ(x(i)−y(i)))∗x(i)j 　　 (for　j=0,1,2⋯n)
}

2.2 Feature Scaling

面对输入数据各个特征值范围差距过大的问题，我们可以对输入数据进行标准化。
x(j)i=x(j)i−avg(xi)Si
其中 Si 可以为标准差，也可以为 max(xi)−min(xi)

2.3 Learning’rate

1. 如果 α 太小，则梯度下降法会收敛缓慢
2. 如果 α 太大，则梯度下降法每次迭代可能不下降，最终导致不收敛。

2.4 Features and polynomial regression

除了线性回归外，我们也能采用多项式回归
举例如下假设函数
hθ(x)=θ0+θ1x+θ2x2+θ3x3
我们可以定义为
hθ(x)=θ0+θ1x1+θ2x2+θ3x3=θ0+θ1x1+θ2x21+θ3x31
对于多项式回归，标准化更加重要。

2.5 Normal equa2on

除了梯度下降法，另一种求最小值的方式则是让代价函数导数为0，求 θ 值
J(θ)=12m∑mi=1(hθ(x(i))−y(i))2
∂∂θjJ(θ)=0 　for every j
求得: θ=(XTX)−1XTy

下面这个图比较了两个算法之间的区别

对于 (XTX) 不可逆的情况下，我们可以采取减少特征量和使用正规化方式来改善。

编程作业

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);

ComputeCost.m

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

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

end

gradientDescent.m

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