# Anticipatory Troubleshooting
## 3 Troubleshooting with Survival Analysis
### Integrating Survival Functions to Improve Diagnosis  
For a given component \(C\) we have two estimates for the likelihood that it is correct: one from the MBD algorithm (\(p(C)\)) and one from its survival curve (\(S_{C}(C_{age})\)). The MBD algorithm's estimate is derived from the currently observed system behavior and knowledge about the system's structure. The survival curve estimate is derived from knowledge about how such components tend to fail over time. We propose to combine these estimates to provide a more accurate and more informed diagnostic report. A simplistic approach to combine these fault likelihood estimates is by using some weighted linear combination, such that the weights are positive and sum up to one. However, we argue that these estimates are fundamentally different: \(S_{C}(C_{age})\) is an estimate given a priori to the actual fault, while \(p(C)\) is computed by the MBD algorithm for the specific fault at hand, taking into consideration the currently observed system behavior. Indeed, MBD algorithms often require information about the prior probability of each component to be faulty when computing their likelihood estimates [9; 16; 17; 18; 19; 20]. These priors are often set to be uniform, although it has been shown that setting such priors more intelligently can significantly improve diagnostic accuracy [21; 22]. Therefore, we propose to integrate the fault likelihood estimation given by the survival curves as part of the priors within the likelihood estimation computation done by the MBD algorithm. We explain how this is done next. Recall that every component \(C\) is associated with a node \(n_{C}\) in the BN, such that the value of this node represents the health state of \(C\). Let \(V_{0}\) be the value of \(n_{C}\) representing that \(C\) is healthy, and let \(V_{1},\ldots,V_{m}\) represent the other, non-healthy possible states of \(C\). Let \(prior(V_{i})\) be the prior probability that \(C\) is in state \(V_{i}\). The sum of these priors, i.e., \(\sum_{i=1}^{m}prior(V_{i})\), expresses the a priori belief that the component is faulty, i.e., the belief before any evidence is taken into account. We propose to integrate the information into the  
Figure 1: An example of exponential survival curves. survival curves with the health nodes' priors, by introducing a revised prior for state \(V_{i}\), denoted by \(priors^{s}(V_{i})\), which is defined as follows:  
\[priors^{s}(V_{i})= \begin{cases}S_{C}(C_{Age})&i=0\\ \left(1-S_{C}(C_{Age})\right)\cdot\frac{prior(V_{i})}{\sum\limits_{j=1}^{m} prior(V_{j})}&\text{otherwise}\end{cases}  \tag{1}\]  
The intuition behind the formula is that the prior probability for being in a healthy state (\(V_{0}\)) can be computed by the survival function given the age of the component. The priors of the other faulty states are normalized by \(\frac{1-S_{C}(C_{Age})}{\sum_{j=1}^{m}prior(V_{j})}\) so that they sum up to one, but still preserve their relative prior probabilities from the BN. As an example of this integration of survival curves and diagnostic model, consider an MBD problem for a car. Figure 2 depicts the BN we use as a system model for this car. Nodes \(lg\), \(B\), and \(R\) correspond to the health variables for the ignition, battery, and radiator, respectively. \(W\) corresponds to the water level variable, and \(C\) corresponds to the observable variable that is true if the car starts properly and false otherwise. The conditional probability tables (CPTs) for all nodes except \(C\) are displayed in Figure 2. For example, the a priori probability that the ignition (\(lg\)) is faulty is 0.01. The value of \(C\) deterministically depends on \(lg\), \(B\), and \(R\): the car can start; i.e., \(C\) is true, if and only if all the components \(lg\), \(B\), and \(R\), are healthy. Modeling such dependency (a logical OR) in a BN is trivial. For simplicity, assume that at most one component--\(lg\), \(B\), or \(R\), can be faulty at a time. This is called the single-fault assumption.