Final

ex1/ex1/computeCost.m

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+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
+
+% ====================== YOUR CODE HERE ======================
+% Instructions: Compute the cost of a particular choice of theta
+%               You should set J to the cost.
+
+distances = theta(1) + theta(2) * X(:, 2) - y;
+J = sum(distances.^ 2) / (2 * m);
+
+
+
+% =========================================================================
+
+end

ex1/ex1/computeCostMulti.m

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+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 
+
+% ====================== YOUR CODE HERE ======================
+% Instructions: Compute the cost of a particular choice of theta
+%               You should set J to the cost.
+
+distances = X * theta - y;
+
+J = sum(distances .^ 2) / (2 * m);
+
+% =========================================================================
+
+end

ex1/ex1/ex1.m

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+%% 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: Gradient descent ===================
+fprintf('Running Gradient Descent ...\n')
+
+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;
+
+% compute and display initial cost
+computeCost(X, y, theta)
+
+% run gradient descent
+theta = gradientDescent(X, y, theta, alpha, iterations);
+
+% print theta to screen
+fprintf('Theta found by gradient descent: ');
+fprintf('%f %f \n', theta(1), theta(2));
+
+% 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);

ex1/ex1/ex1_multi.m

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+%% 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
+%     gradientDescent.m
+%     computeCost.m
+%     gradientDescentMulti.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
+
+fprintf('Loading data ...\n');
+
+%% Load Data
+data = load('ex1data2.txt');
+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). 
+%
+%               Your task is to first make sure that your functions - 
+%               computeCost and gradientDescent already work with 
+%               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.
+%
+
+fprintf('Running gradient descent ...\n');
+
+% 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');
+
+% Display gradient descent's result
+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);
+