## 3 Methodology
### Markovian Deterioration Prediction Model  
The developmental stages of the Markov-based model and the corresponding equations used as part of this study to model and estimate the transition of a system from one condition state to a lower one within a specific time frame is elaborated in this section. In general, according to [28], a Markov chain is a stochastic process allowing for the observation of preset variables at certain points in time, permitting the monitoring of the change in the system states from one time to another. The probability P of a system transitioning from one state m to another state n should conform to the following two conditions:  
\[\left\{ \begin{array}{c}\text{P}_{\text{mn}}\;\geq\;0\text{ ;}\\ \sum\text{P}_{\text{mn}}\;\leq\;1\end{array} \right. \tag{4}\]  
where m is the initial condition state at time t; n is the final condition state at t + 1. On another note, the Markovian property of the conditional transition probability of a system can be given as Equation (5). \[\text{P}\left(\text{ }X_{t+1}\;=\;\text{n}\,|\,X_{0}\;=\;1_{0}\text{, }X_{1}\;=\;1_{1}\text{, }\ldots\text{, }X_{t}\;=\;\text{m }\right)=\text{ P}\left(\text{ }X_{t+1}\;=\;\text{n}\,|\,X_{t}\;=\;\text{m }\right) \tag{5}\]  
where P represents the transition probability at any point in time. In the initial stage of the model development process, historical datasets from 48 hospital building facilities from the province of Alberta were collected and the data points and instances representing the chosen system, which was the oxygen gas piping and supply system, were identified. Eighty percent of the datasets collected were utilized for the design and structuring of the proposed model, while the remaining twenty percent were used for model validation purposes. Two cycles of inspection reports were included in this model as it was suggested by [28] that two is the minimum number of cycles required to derive a reliable and sound Markovian deterioration estimate. After that, uncensored events in the inspection reports were selected as they demonstrate the transition of the system from one state  
Figure 2: Membership functions calculation procedure for different positions of “x”. to another and this is the principal tool to build a robust and relatable Markov deterioration prediction model. Data points were then filtered and grouped according to the age of systems since installation to facilitate the further process of developing the Markovian transition matrices. The filtering process included the removal of any records showing a renewal activity conducted on building components, as well as the exclusion of any progressive record showing an improvement in the condition as this defies the deterioration theory required for developing and executing the model. The initial state, transition probabilities and final states were calculated in the form of the percentages of systems from the overall number of systems included in every combined group in this study. A theoretical example to demonstrate the derivation of the percentages used in the Markovian model is given below. Given a set of 50 system inspection records for a certain year and with an age group of 25 years since installation, 15 of the systems were in an excellent condition state as per the visual inspection reports, 12 were in a good condition state, 10 were in an acceptable condition state, 8 were in a marginal condition state, 3 were in a poor condition state and 2 were in a critical condition state. This will end up in the initial condition state vector shown in Table 1.  
The following step after calculating the initial state percentages, was the calculation of the final percentages of systems in each condition state as per the transition probabilities derived from the fuzzy-based deterioration model. In this study, it was assumed that systems can only experience a single state transition per year.