Andrew Ng coursera上的《机器学习》ex1

Andrew Ng coursera上的《机器学习》ex1

本系列文章是在coursera上学习Andrew Ng的《机器学习》之后，对练习题进行了一些总结。我是初学者，所以肯定存在很多错误，欢迎大家能够给我提意见。

按照课程所给的ex1的文档要求，ex1要求完成以下几个计算过程的代码编写：

exerciseName	description
warmUpExercise.m	Simple example function in Octave/MATLAB
plotData.m	Function to display the dataset
computeCost.m	Function to compute the cost of linear regression
gradientDescent.m	Function to run gradient descent

1. warmUpExercise.m

要求通过写Octave/MATLAB代码返回一个5阶的单位矩阵。

 X = eye(5);

2. plotData.m

要求将二维的训练数据的x和y用图展示出来。

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.

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


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

end

其中传入的参数X,Y分别用下面的代码求出：

data = load('ex1data1.txt');
X = data(:, 1); y = data(:, 2);
m = length(y); % number of training examples

3. computeCost.m

要求：计算出线性回归函数中对应于只有一个特征值（X是二维的）的情况进行计算。

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

其中传入的参数分别为：

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;

其中的X是一个m*2的矩阵，theta是一个2 1的矩阵，所以m个数据集的h(x)= X theta，与之前描述的h(x) = (theta^T ) * X 有一定的区别，需要注意。这些是通过数学的推导得到的结果。

4.gradientDescent.m

利用批量梯度下降来对参数进行最优化求解。梯度下降的公式如下：
theta = theta - α * sum(h(x)-y)*x /m
其中的sum可以用求和符号表示。在Octave中，可以转为矩阵来进行计算。

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*X'*(X*theta-y)/m;






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

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

end

其中传入的参数：

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;