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# ML-Linear Regression_Programming ex1  # Task #1 Linear regression with one variable

## 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
%     computeCost.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);%分别取第一列和第二列 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; %迭代次数 alpha = 0.01; %rate 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);  ## 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) ylabel('profit in$10,000s')
xlabel('Population of City in 10,000s')

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

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
J=0;
% You need to return the following variables correctly

% ====================== 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);%注意用.^  数组中的每个数带入都需要平方

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

end


J =sum((X * theta-y).^2) / (2*m);
%X*theta = θ0*x0 + θ1*x1 + θ2*x2


The X is a n×2 matrix and theta is a 2×1 matrix 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.
%
%theta = theta - alpha*(1/m)*(X'*(X*theta-y));
theta1=theta(1,1)-alpha/m*sum(X*theta-y).*1;
theta2=theta(2,1)-alpha/m*sum((X*theta-y) .* X(:,2));
theta(1,1)=theta1;
theta(2,1)=theta2;

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

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

end

end



Solution 1

theta1=theta(1,1)-alpha/m*sum(X*theta-y).*1;
theta2=theta(2,1)-alpha/m*sum((X*theta-y) .* X(:,2));
theta(1,1)=theta1;
theta(2,1)=theta2;


Solution2

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


$$\sum_{m}^{1} x_{i}y_{i} = X^{T}Y$$

## 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.
%
theta = inv(X'*X)*X'*y;
% ---------------------- Sample Solution ----------------------

% -------------------------------------------------------------

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

end



# Task #2 Linear regression with Multi variable

## 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 = (([1,1650,3]*theta)-mu)/sigma; % You should change this

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

fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ...
'(using gradient descent):\n $%f\n'], price); fprintf('Program paused. Press enter to continue.\n'); pause; %% ================ Part 3: Normal Equations ================ fprintf('Solving with normal equations...\n'); % ====================== YOUR CODE HERE ====================== % Instructions: The following code computes the closed form % solution for linear regression using the normal % equations. You should complete the code in % normalEqn.m % % After doing so, you should complete this code % to predict the price of a 1650 sq-ft, 3 br house. % %% Load Data data = csvread('ex1data2.txt'); X = data(:, 1:2); y = data(:, 3); m = length(y); % Add intercept term to X X = [ones(m, 1) X]; % Calculate the parameters from the normal equation theta = normalEqn(X, y); % Display normal equation's result fprintf('Theta computed from the normal equations: \n'); fprintf(' %f \n', theta); fprintf('\n'); % Estimate the price of a 1650 sq-ft, 3 br house % ====================== YOUR CODE HERE ====================== price = [1,1650,3]*theta; % You should change this % ============================================================ fprintf(['Predicted price of a 1650 sq-ft, 3 br house ' ... '(using normal equations):\n$%f\n'], price);



## 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)/m/2

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

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);

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

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

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

end

end



## 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;
mu = zeros(1, size(X, 2));%X数组列数（1是行数，2是列数）
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 = mean(X);
sigma = std(X);
for i = 1:size(X,2)  %X的feature数量（即列数）
X_norm(:,i) = (X_norm(:,i) - mu(i))/sigma(i)
end

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

end



The aim of Normalization is to make gradient descent faster
$$X\;normalize= \frac{X\;origin – X\;average}{standard\;deviation}$$

# Review

1. Divide process into multiple functions and files could make programming more explicit. Architecture is important.

2. When we meet equation like gradient descent, using matrix instead of SUM could simplify programming.

3. Please remember ‘.*’ and ‘,^’ in MATLAB.

4. How to write function : input X/ output Y

   function [y1,...,yN] = myfun(x1,...,xM)

1. Go back and review Linear Algebra

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