Bayesian Fractional Weibull Regression for Reliability Prognostics and Predictive Maintenance

This paper introduces a novel Bayesian Fractional Weibull (BFW) regression framework, which generalizes the classical Weibull accelerated failure time model using fractional calculus. The proposed methodology addresses key challenges in big data reliability engineering and predictive maintenance by incorporating a fractional order parameter that provides adaptive flexibility when classical Weibull assumptions are violated. We obtain fractional score equations using Caputo derivatives and provide theoretical consistency by proving that classical maximum likelihood estimation emerges as a special case when the fractional order approaches unity. A Bayesian implementation enables full uncertainty quantification and robust inference, particularly in reliability applications characterized by limited data and complicated failure mechanisms. Comprehensive numerical experiments demonstrate the efficacy of the framework: synthetic data validate theoretical properties under both well-specified and misspecified scenarios, while a real-world case study using the NASA C-MAPSS turbofan engine dataset—a standard benchmark in recent reliability literature—shows substantial improvements in predictive performance. The BFW model achieves a 21.7% improvement in predictive performance. This framework combines the theoretical rigor of fractional calculus with the practical advantages of Bayesian inference, directly addressing the need for interpretable and robust methods in big data reliability analytics.