A Comprehensive Framework for Residual Analysis in Regression and Machine Learning

Residual analysis is one of the most crucial methodologies in statistical modeling and machine learning. Generally, it tends to be an important tool in the evaluation of the precision of a model, diagnosing violations of assumptions, and refinement. This paper critically reviews residuals, their mathematical underpinning foundations, and how they feature in model performance evaluation. Key diagnostic methods that have been explored in this paper include heteroscedasticity, non-linearity, autocorrelation, and influential outliers. Further, we have to develop a new case based on the decomposition of residuals and SHAP values for the analysis of unexplained sales trends. This study underlines how residual patterns can indicate hidden deficiencies in the model and how model improvement can obtain better results. We conclude the length in optimizing model performance and future research directions in residual-based diagnostics.