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# Stanford 机器学习练习 Part 1 Linear Regression

warmUpExercise.m

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

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 = sum((X*theta - y).^2) / (2*m);
% =========================================================================
end

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.
% ====================== 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', 5);
xlabel("x");
ylabel("y");
% ============================================================
end

function [theta, J_history] = gradientDescent(X, y, theta, alpha, num_iters)
%   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.
%
temp = 0;
temp = temp + alpha/m * X' * (y - X * theta);
theta = theta + temp;
% ============================================================
% Save the cost J in every iteration
J_history(iter) = computeCost(X, y, theta);
end
end

ex1.m

featureNormalize.m

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;
m = size(X, 2);
mu = zeros(1, size(X, 2));
mu = mean(X);
sigma = std(X);
for i = 1:m
X_norm(:,i) = (X(:,i).-mu(i))./sigma(i);
end
% ====================== 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.
%

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

end

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

function [theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters)
%   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);

temp = zeros(feature_number,1);

for iter = 1:num_iters
temp = alpha/m * X' * (y - X*theta);
theta = theta + temp;
J_history(iter) = computeCostMulti(X, y, theta);
% ====================== 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.
%
% ============================================================
% Save the cost J in every iteration
J_history(iter) = computeCostMulti(X, y, theta);

end

end

normalEqn.m

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.

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 ----------------------
theta = pinv(X' * X) * X' * y;
% -------------------------------------------------------------
% ============================================================

end

ex1_multi.m

%% 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
%     computeCost.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

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).
%
%               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.
%

% 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');

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);


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