Multivariate Linear Regression in Matlab Programming

Multivariate linear regression extends simple linear regression by modeling the relationship between a single dependent target variable (y) and multiple independent predictor variables ((x_1, x_2, dots, x_n)).

The general mathematical model is expressed as:

• (beta_0): Intercept term
• (beta_1, beta_2, dots, beta_n): Feature slope coefficients
• (epsilon): Residual error term

In MATLAB, you can perform multivariate linear regression using either raw matrix operations (Ordinary Least Squares) or built-in functions like fitlm().

Step 1: Preparing the Dataset

Define your predictor matrix (X) (where each column represents a feature and each row an observation) and target vector (y):

% Independent variables (2 features: X1 and X2) X = [1 2; 2 3; 3 5; 4 7; 5 8];
% Dependent target variable y = [3; 5; 7; 10; 12];

Step 2: Matrix Approach (Ordinary Least Squares)

To solve for coefficients using matrix algebra, augment (X) with a column of ones to represent the intercept (beta_0), then solve (boldsymbol{beta} = (X^T X)^{-1} X^T y):

% Add column of ones for the intercept X_aug = [ones(size(X,1), 1) X];
% Calculate coefficients using normal equations (or backslash operator) beta = (X_aug' * X_aug)  (X_aug' * y);
disp('Calculated Coefficients (Intercept, Beta1, Beta2):');
disp(beta);

Step 3: Using fitlm() for Higher-Level Statistical Analysis

The fitlm() function automatically fits the model and calculates key statistical metrics including (R^2), p-values, and standard errors:

% Fit linear regression model mdl = fitlm(X, y);
disp(mdl);
% Predict response for new observation [X1=6, X2=9] new_X = [6 9];
y_pred = predict(mdl, new_X);
disp(['Predicted Value: ', num2str(y_pred)]);

Step 4: Visualizing a 2-Feature Regression Surface

When working with two predictor features, you can visualize the regression plane in 3D using scatter3() and surf():

% Create 2D grid for surface plotting [X1_grid, X2_grid] = meshgrid(1:0.5:6, 2:0.5:9);
Y_fit = beta(1) + beta(2)*X1_grid + beta(3)*X2_grid;
% Plot data points and fitted surface 
figure;
scatter3(X(:,1), X(:,2), y, 'filled', 'MarkerFaceColor', 'r');
hold on;
surf(X1_grid, X2_grid, Y_fit, 'FaceAlpha', 0.5);
xlabel('Feature X1');
ylabel('Feature X2');
zlabel('Target Y');
title('Multivariate Linear Regression Surface');
grid on;
hold off;

Step 5: Evaluating Model Fit

Inspect model performance using stats extracted directly from the fitted model object:

% Extract Ordinary R-squared value R2 = mdl.Rsquared.Ordinary;
disp(['R-squared: ', num2str(R2)]);

Practical Applications

• Real Estate Valuation: Estimating home prices based on square footage, bedroom count, and age.
• Financial Forecasting: Modeling stock or revenue returns using multiple economic indicators.
• Engineering Systems: Analyzing multi-factor sensor measurements and system degradation.