10 Chapter 1. Introduction In this dissertation, we focus on Markovian Process-based models for prognostics—a sub-class of data-driven prognostics (Tsui, N. Chen, Q. Zhou, et al., 2015)—due to their ability to model alternative states to failure, crucial for safety-critical systems like bridges (Ranjith, Setunge, Gravina, et al., 2013) and components with slow degradation like sewer mains (Barraud, Bosco, Clemens-Meyer, et al., 2024). We use Markov chains for Multi-State Deterioration modelling, discussed next. Markov chains for Multi-State Deterioration Proposed by the Russian mathematician Andrey Markov, a Markov process models the behaviour of systems—physical or mathematical—based on states and their transitions. It assumes that the future evolution of the system depends only on its current state, not its past (Stewart, 2009). This characteristic forms the basis of Markov chains, where state transitions create a sequence over time in which each future state depends only on the present state, disregarding prior events. Formal definitions of Markov chains are provided in Section II.4.1 (page 95), and details of the Markov chain structures used in this dissertation are found in Section II.4.4 (page 95). Here, we illustrate the concept behind Markov chains with the following example. Figure 1.3 depicts the reliability function of a bearing, where two states are distinguished: nominal (N) and failure (F). The underlying degradation process can be modelled using a two-state Markov chain, with a transition fromNto F, as shown in Figure 1.8(a). In this example, the Markov chain does not allow repairs, i.e., once the state changes to failure, it cannot revert. However, repairs can still be modelled with Markov chains, though this would yield a di!erent example. State probability (t) Bearing age (t) Failure Nominal behaviour Non-Nominal behaviour e.g., Crack Figure 1.7: Multi-State Deterioration of a bearing with three states (example). By introducing an intermediate state, non-nominal behaviour (e.g., due to a crack), we enrich the degradation model, as depicted in Figure 1.7. This can be
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