 ktl014
5/28/2017 - 8:04 PM

## CourseEra (Gradient Descent,Cost Function) ~ Multivariate Linear Regression Code starts with ex1

CourseEra (Gradient Descent,Cost Function) ~ Multivariate Linear Regression Code starts with ex1

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); % Plot the data
ylabel('Profit in $10,000s'); % Set the y-axis label xlabel('Population of City in 10,000s'); % set the x-axis label % ============================================================ end  function [theta, J_history] = gradientDescent(X, y, theta, alpha, num_iters) %GRADIENTDESCENT Performs gradient descent to learn theta % theta = GRADIENTDESCENT(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. % h = X*theta; error = h - y; theta_change = alpha*(1/m)*X'*error; theta = theta - theta_change; % ============================================================ % Save the cost J in every iteration J_history(iter) = computeCost(X, y, theta); end end  6.1101,17.592 5.5277,9.1302 8.5186,13.662 7.0032,11.854 5.8598,6.8233 8.3829,11.886 7.4764,4.3483 8.5781,12 6.4862,6.5987 5.0546,3.8166 5.7107,3.2522 14.164,15.505 5.734,3.1551 8.4084,7.2258 5.6407,0.71618 5.3794,3.5129 6.3654,5.3048 5.1301,0.56077 6.4296,3.6518 7.0708,5.3893 6.1891,3.1386 20.27,21.767 5.4901,4.263 6.3261,5.1875 5.5649,3.0825 18.945,22.638 12.828,13.501 10.957,7.0467 13.176,14.692 22.203,24.147 5.2524,-1.22 6.5894,5.9966 9.2482,12.134 5.8918,1.8495 8.2111,6.5426 7.9334,4.5623 8.0959,4.1164 5.6063,3.3928 12.836,10.117 6.3534,5.4974 5.4069,0.55657 6.8825,3.9115 11.708,5.3854 5.7737,2.4406 7.8247,6.7318 7.0931,1.0463 5.0702,5.1337 5.8014,1.844 11.7,8.0043 5.5416,1.0179 7.5402,6.7504 5.3077,1.8396 7.4239,4.2885 7.6031,4.9981 6.3328,1.4233 6.3589,-1.4211 6.2742,2.4756 5.6397,4.6042 9.3102,3.9624 9.4536,5.4141 8.8254,5.1694 5.1793,-0.74279 21.279,17.929 14.908,12.054 18.959,17.054 7.2182,4.8852 8.2951,5.7442 10.236,7.7754 5.4994,1.0173 20.341,20.992 10.136,6.6799 7.3345,4.0259 6.0062,1.2784 7.2259,3.3411 5.0269,-2.6807 6.5479,0.29678 7.5386,3.8845 5.0365,5.7014 10.274,6.7526 5.1077,2.0576 5.7292,0.47953 5.1884,0.20421 6.3557,0.67861 9.7687,7.5435 6.5159,5.3436 8.5172,4.2415 9.1802,6.7981 6.002,0.92695 5.5204,0.152 5.0594,2.8214 5.7077,1.8451 7.6366,4.2959 5.8707,7.2029 5.3054,1.9869 8.2934,0.14454 13.394,9.0551 5.4369,0.61705  %% 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: 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;
alpha = 0.01;

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);

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)*sum(((X*theta)-y).^2);

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

end