MARCO TEÓRICO Y CONCEPTUAL

BNs are in their first formulation a static model. But since then, there have been a number of efforts for extending the networks’ definition in order to make them able to represent time, for situations where, for example, the probability of a variable changing its state depends on the time or if the parameters of the system changes with time. In literature there are several definitions of Dynamic Bayesian Networks or Temporal Bayesian Networks or, in general, BNs that incorporate temporal features. The authors give similar names to different con- cepts and sometimes the same name for different definitions. There are two approaches that can be distinguished: in [36], the authors classify them considering whether they represent the time as points or instances or if they divide it in time intervals. In some approaches, Dynamic networks are seen as a general case of Temporal networks where the term dynamic refers to any change the system is subjected to as the change in state [37].

In the PhD thesis [38], the author considers an approach where the events considered occur at an instant in time but the network is studied evolving at different time slice. The author then modifies the definition to allow growing complexity in the networks but they become more difficult to apply to a large scale problem.

2.3

Converting FTs into BNs

In [39] the authors make a comparison between BNs and FTA techniques for dependability problems. They show how a FT can be mapped into a BN and that any analyses performed with the FT methods by means of the minimal cut sets procedure can be carried out in a BN. Furthermore, some new analysis are permitted in a BN, such as the calculation of the posterior probability of a subset of components given the fault. Therefore any FT corresponds to a BN and any techniques applied to a FT can be performed in a BN, but the latter allows some more modelling solutions [40]. These arguments will be presented with a simple example.

2.3.1 FT Conversion Methods

The operations of the algorithm to obtain a BN from a FT are given below. It is assumed that the FT will have only AND and OR gates, the resulting BN will by binary, the variables will represent states of the components of a system and the two values they can assume will be labelled with FALSE ( ¯V) for the working state, and with TRUE (V ) for the failing state. However the algorithm can be generalised to any FT.

Regarding the qualitative part of the BN:

1. any basic system component of the FT corresponds to a root node in the BN;

2. any gate of the FT corresponds to a node in the BN, in particular the gate whose output is the top event in the FT will be labelled as fault node;

3. the nodes in the BN has to be connected as the gates in the FT.

Figure 2.14 shows how the structure of a simple FT is converted into the structure of a BN. Regarding the probability, the quantitative part of the BN:

1. to any root node in the BN it is assigned the same prior probability of its corresponding basic event in the FT;

2. to any node in the BN corresponding to a AND gate in the FT it is associated a CPT such that the node is TRUE with probability 1 if and only if all parents are TRUE; 3. to any node in the BN corresponding to a OR gate in the FT it is associated a CPT

such that the node is TRUE with probability 1 if and only if at least one of its parents is TRUE.

Figure 2.14– Basic events B1, B2, B3 and B4 correspond to the homonym nodes; gates G2 and G3 correspond to nodes G2 and G3 and the top event corresponds to the node FAULT.

Figure 2.15– CPT corresponding to an AND gate.

For an AND gate, such as G2 in figure 2.14, the CPT will result as in figure 2.15. For an OR gate, such as G3 in figure 2.14, the CPT will be that in figure 2.16

Figure 2.16– CPT corresponding to an OR gate.

The conversion method can be extended to FTs with other gates and the CPTs in the corresponding networks will follow the logic tables of the gates. For example, for the NOT gate in figure 2.17, the CPT will be as in figure 2.18.

Figure 2.17–Configuration corresponding to a NOT gate.

Figure 2.18– CPT corresponding to a NOT gate.

The unavailability of the top event in a FT corresponds to the prior probability of the node labelled as fault in the BN. The unavailability of a sub-system in a FT corresponds to the prior probability of the corresponding nodes in the BN. In a FT, these computations are obtained by means of the minimal cut sets, in a BN they can be obtained as P (H | e) where H represents the fault (or the variables of the sub-system) and the evidence is the empty set, e = ∅. The posterior probability can be also computed in a BN and this can be considered for a single component, for a subset of components or for all components except the ones to which evidence has been assigned. When the fault is given as evidence, the posterior probability of each component gives the criticality of each of them and the posterior probability of a sub-system gives the criticality of the sub-system in causing the system failure.

2.3.2 Bayesian networks from Fault Trees with repeated events

When in a FT a basic event appears more than once, it is said that it has repeated events. FTs with repeated events can be mapped into BNs simply creating a single node for every basic event and linking with more than one link the nodes that correspond to the repeated events. In figure 2.19 the FT has the basic event A appearing twice as output of the gate G2 and G3.

Figure 2.19– Example of FT with repeated events.

The BN corresponding to the FT in figure 2.19 is shown in figure 2.20. Node A appears only once but is linked with two links to both node G2 and G3.

Figure 2.20– BN corresponding to figure 2.19.

When a FT has many repeated events, its corresponding BN can assume a graphical structure where the links intersect themselves in a way that can make the visual understanding of the network more complicated. The same procedure can be applied for a FT with repeated branches. The corresponding BN will result in having nodes with more than one link as in figures 2.21 and 2.22.

Figure 2.21– Example of FT with repeated branches.

Figure 2.22– Example of BN corresponding to the FT in figure 2.21.

representations of the BNs. In the following section, the conversion procedure from FTs to BNs is shown with the example of a pressure tank system.