Analytical Modeling of a Dynamic Probability Mixture via Sigmoidal Weighting for Heterogeneous System Degradation

This paper proposes a novel mathematical model based on a dynamic mixture of Erlang and exponential distributions to capture complex degradation processes in heterogeneous systems. The core feature of the model is the use of sigmoidal weighting functions, defined as ω(x) = 1 / (1 + exp(−r(x − a))), where r > 0 and a > 0. This weighting mechanism allows the model to naturally switch dominance between mixture components, mimicking realistic physical stages of defect nucleation, propagation, and accelerated wear—a critical aspect for effective Prognostics and Health Management (PHM) and remaining useful life prediction.

The scientific novelty lies in the development of an analytical method for calculating the normalization constant C of the dynamic mixture. Unlike traditional approaches that rely solely on numerical integration, the proposed method yields a closed-form expression for the normalization constant using the Lerch transcendent function. Specifically, C = (λ/r) * Φ(−exp(ra), 1, λ/r) + 1 − (μ^k / r^k) * Φ(−exp(ra), k, μ/r). This provides a precise analytical representation, significantly increasing calculation speed and stability in most scenarios. For cases where the Lerch function exhibits computational instability in modern CAS, an approximate method based on the Sommerfeld expansion for Fermi–Dirac type integrals is proposed. The error of this method is controlled by subsequent expansion terms, ensuring the precision required for engineering applications.

Model parameters are estimated via Maximum Likelihood Estimation with gradient-based optimization. To ensure reliable convergence, a dedicated initialization procedure based on the method of moments is proposed to establish robust initial values.

The proposed model was tested on synthetic data with known parameters and on real degradation trajectories from the NASA C-MAPSS dataset. The results indicate accurate parameter recovery in simulations and superior adaptability over conventional static mixtures when modeling nonlinear degradation in real heterogeneous systems.