%% Simple linear regression. Generate some linear data, add normally distributed noise, and compute the coefficients by solving the normal equations. 

%% Specify model parameters
a = 1;
b = 2;

%% Generate some data
x = (1:10)';
yTrue = a + b*x;

%% Add normal error to simulate experimental noise
NoiseStd = 0.5; % The standard deviation of the noise
err = NoiseStd*randn(length(x), 1);
yExpt = yTrue + err;

%% Create design matrix
A = [ones(size(x)) x];

%% Solve normal equation using the MATLAB backslash operator
betaHat = A \ yExpt

%% Plot data and fit
plot(x, yExpt, 'ro', 'MarkerSize', 8)
hold on
xFit = (1:0.1:10)';
yFit = [ones(size(xFit)) xFit]*betaHat;
plot(xFit, yFit, 'b-', 'LineWidth', 2)
xlim([0, 11])
ylim([0, 23])
xlabel('x')
ylabel('y')
grid on
legend('Data', 'Fit', 'Location', 'NW')