98 Part II: Multi-state deterioration modelling - Qij(t, ς) can be parametrised by hazard rates ϖ(t; ς), derived from the probability density function f(t; ς) and the survival function S(t; ς), where ς represents the hyper-parameters. The IHTMC’s temporal evolution is described by the Forward Kolmogorov equation: φPij(t, ϱ) φt = k→S Pik(t, ϱ)Qkj(t) (4.2) Here Pij(t, ϱ) is the transition probability matrix (see Definition 6). Using Eq. 4.2, the master equation of Markov chains is derived, expressing the probability flow between states by incorporating inflow and outflow terms: φpk(t) φt = i→S,i↔ =k pi(t)Qik(t) →pk(t) j→S,j↔ =k Qkj(t) (4.3) Here, pk(t) is the probability of being in state k ↓ S at time t. The term )i→S,i↔ =k Qkj(t) captures the rates of transition from state k to all other states j, excluding self-transitions. Continuous-Time Markov Chain A Continuous-Time Markov Chain (CTMC) is deemed homogeneous because its hazard rates ϖ(t; ς) are constant over time, exhibiting the memoryless property. Formally, a Continuous-Time Markov Chain (CTMC) is defined as follows. Definition 10 (Continuous-Time Markov Chain). AContinuous-Time Markov Chain is defined by the tuple M=↔S, p(0), Qij(ς)↗, see S and p (0) in Definition 6. Here: - Qij(ς) : S↘S ↑R represents the time-independent transition rate matrix. Unlike the inhomogeneous case, this matrix function, parametrised only by ς, is constant over time. - Since the transition rates ϖ(t; ς) are constant, Qij does not depend on time but may still be parametrised by ς which a"ect the transition rates. WhenT consists of discrete, equally spaced intervals, we transition to Discrete-Time Markov Chains, which are discussed below. Discrete-Time Markov Chain When discretising t into discrete, equally spaced intervals of length ∆t, denoted as n, we transition from continuous to Discrete-Time Markov Chain. A connection to CTMCs can be established through the matrix of exponents: Pij(t)=exp(Qijt) (4.4)
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