Least Square Regression in MATLAB Programming
Least Squares Regression is a fundamental statistical method used to determine the line or curve of best fit through a set of data points. In MATLAB, it forms the basis for data fitting, trend forecasting, and predictive modeling.
The objective is to minimize the sum of squared residuals (the vertical distances between observed data values (y_i) and predicted model values (hat{y}_i)):
[S = sum_{i=1}^{n} (y_i - hat{y}_i)^2]Define the independent variable vector x and dependent variable vector y:
x = [1 2 3 4 5];
y = [2.2 2.8 3.6 4.5 5.1];For a linear model (y = a x + b), you can compute slope (a) and intercept (b) using built-in functions or linear algebra.
% 1 indicates a first-degree (linear) polynomial fit coeff = polyfit(x, y, 1);
a = coeff(1);
% Slope b = coeff(2);
% Intercept fprintf('Slope: %.4f, Intercept: %.4f ', a, b);% Construct design matrix with a column of ones for intercept X = [x(:) ones(length(x), 1)];
Y = y(:);
% Compute least squares solution: theta = (X^T * X)^(-1) * X^T * Y theta = (X' * X) (X' * Y);
a = theta(1);
b = theta(2);
fprintf('Slope: %.4f, Intercept: %.4f ', a, b);Calculate predicted values y_fit and overlay the regression line on top of the raw data points:
y_fit = a * x + b;
figure;
scatter(x, y, 'filled', 'MarkerFaceColor', 'b');
hold on;
plot(x, y_fit, 'r-', 'LineWidth', 2);
title('Linear Least Squares Regression in MATLAB');
xlabel('Independent Variable (x)');
ylabel('Dependent Variable (y)');
legend('Observed Data', 'Fitted Line', 'Location', 'northwest');
grid on;
hold off;Quantify regression accuracy by computing the Coefficient of Determination ((R^2)):
SS_res = sum((y - y_fit).^2);
SS_tot = sum((y - mean(y)).^2);
R_squared = 1 - (SS_res / SS_tot);
fprintf('R-squared (R^2): %.4f ', R_squared);For non-linear relationships (such as exponential curves (y = a cdot e^{bx})), fit models using fit() and fittype():
% Define exponential model structure ft = fittype('a*exp(b*x)');
f = fit(x(:), y(:), ft, 'StartPoint', [1 0.1]);
figure;
plot(f, x, y);
title('Nonlinear Exponential Regression Fit');
xlabel('x');
ylabel('y');
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