A Gentle Correctness Verification of the Theory of Uncertain Event Prognosis to Compute Failure Time Probability

Abstract

Prognosis of the time of first occurrence of events, particularly failures, is a problem that has been studied and analyzed from different perspectives, being referred to as the “first-hitting time” or “first-passage time” problem, among other names. Within the Prognostics and Health Management community, as well as in other disciplines, the probabilistic benchmark for computing the Time-of-Failure consists of Monte Carlo simulations where the failure event is defined as the moment when the health of the monitored system, typically described by a health indicator, crosses a predetermined threshold that denotes a region known as hazard zone. More recently, this problem was formalized through the introduction of new probability measures based on the concept of uncertain event, which correct widely used expressions for these purposes, and extend the typical threshold criterion for declaring a failure (or other event) to a broader notion of uncertain event likelihood. Following a step-by-step approach supported by code programmed in Python language, this paper verifies the correctness of the aforementioned probability measures, illustrating with a simple example how exactly the same results are obtained using either these new probability measures or the universally accepted benchmark.