Multivariate Linear Regression in Matlab Programming
Multivariate linear regression is an extension of simple linear regression, where the dependent variable yyy depends on two or more independent variables x1,x2,...,xnx_1, x_2, ..., x_nx1?,x2?,...,xn?.
The model is expressed as:
y=β0+β1x1+β2x2+?+βnxn+?y = beta_0 + beta_1 x_1 + beta_2 x_2 + dots + beta_n x_n + epsilony=β0+β1?x1+β2x2?+?+βn?xn?+?
MATLAB provides functions like fitlm and matrix operations to perform multivariate regression efficiently.
Suppose we have the following dataset:
% Independent variables (features)X = [1 2; 2 3; 3 5; 4 7; 5 8]% Dependent variable (target)y = [3; 5; 7; 10; 12];Here, X has two features, and y is the target variable.
Add a column of ones for the intercept:
X_aug = [ones(size(X,1),1) X]; % Augmented matrix with interceptbeta = (X_aug' * X_aug) (X_aug' * y); % Least squares solutiondisp('Regression Coefficients:')disp(beta)beta(1) → Intercept (β0\beta_0β0?)
beta(2:end) → Coefficients for features (β1,β2,...beta_1, beta_2, ...β1?,β2?,...)
fitlm Functionfitlm provides a higher-level interface with statistics:
mdl = fitlm(X, y);disp(mdl)View coefficients, R-squared, p-values, and confidence intervals
predict function can be used for new data:
new_X = [6 9];y_pred = predict(mdl, new_X);disp(['Predicted Value: ', num2str(y_pred)])For two features, you can create a 3D plot:
[X1, X2] = meshgrid(1:0.5:6, 2:0.5:9);Y_fit = beta(1) + beta(2)*X1 + beta(3)*X2;scatter3(X(:,1), X(:,2), y, 'filled')hold onsurf(X1, X2, Y_fit)xlabel('X1')ylabel('X2')zlabel('y')title('Multivariate Linear Regression')grid onhold offR-squared: Measures the goodness of fit
Residual analysis: Check errors for randomness
p-values: Test statistical significance of coefficients
fitlm provides these directly:
R2 = mdl.Rsquared.Ordinary;disp(['R-squared: ', num2str(R2)])Predicting house prices based on size, location, and features
Sales forecasting using multiple market factors
Engineering and scientific data modeling
Machine learning feature-based prediction
Multivariate linear regression in MATLAB allows you to model complex relationships involving multiple independent variables. Using matrix operations or fitlm, you can efficiently compute coefficients, make predictions, and analyze model performance.
Mastering this technique is essential for data analysis, predictive modeling, and engineering applications where multiple factors influence outcomes.
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