linearRegression.m

%% Initialization
clear ; close all; clc

data = load('examples.txt');
X = data(:, 1); y = data(:, 2);     % x stores the input features and y stores the output feature
m = length(y); % number of training examples

fprintf('First 10 examples from the dataset: \n');
fprintf(' x = %.0f, y = %.0f \n', [X(1:10,:) y(1:10,:)]');

%% Plotting graph for better visualization
fprintf('Plotting Data ...\n')
figure;
plot (X,y,'rx', 'MarkerSize',10);
xlabel('Profit in $10,000s');
ylabel('Popilation of city in 10.000s');

%% Finding min Jtheta using gradient descent
X = [ones(m, 1), data(:,1)];  % Add a column of ones to x (x[0][j]=1)
theta = zeros(2, 1);          % initialize fitting parameters

% Some gradient descent settings
iterations = 1500;            % Gradient descent iterations
alpha = 0.01;                 % Learning rate
fprintf('\nRunning Gradient Descent ...\n')
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) for the examples.txt dataset\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
% Apply feature scaling for huge data like 35,000
predict1 = [1, 3.5] *theta;
fprintf('For population = 35,000, we predict a profit of %f\n',...
    predict1*10000);  % Example1
predict2 = [1, 7] * theta;
fprintf('For population = 70,000, we predict a profit of %f\n',...
    predict2*10000);  % Example2

fprintf('Program paused. Press enter to continue.\n');
pause;

%% PLotting graphs for better visualization
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);