Stanford coursera Andrew Ng 机器学习课程第二周总结（附Exercise 1）

Exercise 1：Linear Regression---实现一个线性回归

重要公式

1.h(θ)函数

2.J(θ)函数

思考一下，在matlab里面怎么表达？如下：

原理如下：（如果你懂了这道作业题，上面的也就懂了）

下面通过图形方式感受一下代价函数 ：

 

3.θ迭代过程（梯度下降）

First way:批梯度下降：(编程作业使用这个公式，sum转换同理J(θ))

 

Second way：随机梯度下降:

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 ：

 

4.θ的直接求解法（让代价函数导数为0，求θ值）

 

编程作业答案(红色为添加代码)

1.warmUpExercise.m

function A = warmUpExercise()
%WARMUPEXERCISE Example function in octave
%   A = WARMUPEXERCISE() is an example function that returns the 5x5 identity matrix


% ============= 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(5,5);

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

end

2.plotData.m

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);
xlabel('Population of City in 10,000s');
ylabel('Profit in $10,000s');

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

end

3.gradientDescent.m

function [theta, J_history] = gradientDescent(X, y, theta, alpha, num_iters)
%GRADIENTDESCENT Performs gradient descent to learn theta
%   theta = GRADIENTDESENT(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); % theta 就是用上面的向量表示法的 matlab 语言实现

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

end

end

4.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.

J=1/(2*m)*(X*theta-y)'*(X*theta-y);

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

end

5.运行结果

　　　　　　　　　　　　　　　　　　　　　　For population = 35,000, we predict a profit of 4519.767868
　　　　　　　　　　　　　　　　　　　　　　For population = 70,000, we predict a profit of 45342.450129