A robust probabilistic framework for principal component regression: optimizing parameter identification and outlier detection via approximate Bayesian computation

Anomalies and ill-conditioned predictors present considerable obstacles to reliable parameter estimation in regression models. This paper presents an innovative approach that combines principal component regression with approximate Bayesian computation to address these issues. Principal component regression mitigates the effects of ill-conditioned variables by transforming highly correlated predictors into orthogonal components. Meanwhile, approximate Bayesian computation enhances robustness by approximating the posterior distribution of error variance ($\sigma^2$). This flexible framework models uncertainty and noise effectively. The integration of these methods improves both parameter estimation and anomaly detection. By assigning probabilistic scores to potential outliers, the method provides a more accurate and nuanced identification of anomalies. Extensive validation through simulated and real-world datasets demonstrates the favorable performance of the proposed technique over existing robust methods. These findings highlight the potential of approximate Bayesian computation as a powerful tool to improve the robustness and precision of regression analyzes in noisy and complex data environments.