Mathematical Foundations of Regression Analysis: A Study of Linear and Logistic Models

Regression modelsform the backbone of modern statistical inference and predictive analytics.This paper presents a rigorous mathematical examination of two fundamental approaches: linear regression and logistic regression. Beginning with the formulation of each model, we derive their objective functions—the least squares criterion for linear regression and the log-likelihood for logistic regression. Closed-form solutions for linear regression are contrasted with the iterative optimization required in logistic regression, highlighting the importance of gradient-based methods. Special emphasis is placed on demonstrating how these mathematical principles can be applied to real-life datasets.