## 2 Materials and Methods
### Using Dynamic Bayesian Networks to Model Complex Engineering Systems  
One of the primary characteristics of Bayesian networks and other causal-based models heavily used in previous research efforts is that evidence concerning the state of one node can lead to an updated estimate of the value of another node within the model through logical inferences. The ability to infer the status of nodes within a dynamic Bayesian network translates into a powerful tool for understanding the current health condition of a system. DBN nodes can represent a wide range of features within a system, from individual sensors to entire subsystems. As inference capabilities are associated with the directed arcs between nodes, i.e., the node relationships, information about the system can act as evidence in one section of the model, which then travels to other parts of the model. Figure 2: Sample static Bayesian network (left) and dynamic Bayesian network (DBN) (right). Both models have a relationship structure of nodes and directed arcs as well as conditional probability tables for those relationships; however, dynamic nodes within the DBN also need an initial distribution for their states. This allows the DBN model to utilize more evidence and provide more insight into the system than otherwise expected. The dynamic aspect of DBNs is the repeated occurrence of evidence for the same node. For this case study, and in most instances, DBNs are characterized as a two-time-step model, in which the current node values are only related to the immediate time step beforehand. This Markov structure, therefore, suggests that for a given Bayesian network with \(X_{n}\) variables, the underlying probability that a certain scenario would occur, \(P\), is based on Equation (1) from [PERSON] et al. (2015):  
\[P(X_{1},X_{2},...,X_{n})=\prod_{j=1}^{n}P(X_{j}|parents(X_{j})) \tag{1}\]  
where \(parents(Xj)\) is the set of nodes with arcs into the variable \(X_{j}\). This may not hold for networks that have more complicated time-based ordering; however, this assumption is often used for the ease of modeling dynamic relationships. Since inference generated within a DBN can run from parent to child nodes and vice versa, information from separate parts of the model can provide information to other aspects; for a CES, this widens the range of information available for use. As a result, DBN models designed to monitor CES health can have nodes for sensors, components or subsystems that a) either provide useful or readily available data for use as system evidence, or b) are aspects of the CES that operators and maintainers would want to know information about. Consider the theoretical DBN shown in Figure 3. Information provided about an operator's behavior or the system's environment may change the expected state of a specific component's or the system's overall health. A DBN model designed to monitor and provide information on the health state of a CES following a major accident should incorporate not only operational system information, but also the conditions of potential accidents. As shown in Figure 4, DBN models representing this CES scenario can be structured using four distinct data regions. Each section of the model has its own node types, data availability, and purpose for managing CES health. These four regions are:  
* **Accident State:** the nodes within this region represent the different accident events the system might encounter that this model covers. Typically, CESs are operating at normal or baseline conditions until one of these events occur; after an accident, the system will be operating under different circumstances. The type of accidents that this model covers may be either external to the system (i.e., an earthquake or a power outage) or internal (sabotage).