From Equations to Predictions: Understanding the Mathematics and Machine Learning of Multiple Linear Regression

Abstract

In this paper, the core concepts of multiple linear regression are explored, with a focus on its mathematical foundations and integration with machine learning principles. The objective is to bridge the gap between theory and practical application, providing readers with a comprehensive understanding of this versatile method and highlighting its synergy with traditional statistical approaches and modern computational methods. The paper begins by applying multiple linear regression to predict wine quality based on physicochemical attributes, using a comprehensive dataset. The least squares method is used to estimate regression coefficients, facilitating the construction of a predictive model. The study also encompasses the testing of assumptions such as homoscedasticity and normality of residuals, along with the assessment of autocorrelation to ensure model robustness. To illustrate the practical implementation of multiple linear regression, a demonstration using PyTorch, a popular deep learning framework, is provided. A linear model is defined, and the significance of gradient descent in optimizing model parameters is elucidated. Additionally, the paper covers topics such as data preprocessing, model evaluation, and insights into interpreting regression results. Furthermore, the performance of linear regression is evaluated in comparison to decision trees, random forests, and support vector regression, showcasing the versatility of this classic technique. By presenting a holistic view of multiple linear regression, emphasizing its mathematical foundations, practical implementation, and integration with machine learning, researchers and practitioners are empowered to leverage the potential of linear regression across various domains.