## 3 Adaptive Thresholds for Robust Fault Detection
### Residual Evaluation
#### 3.3.2 An Equilibrium Point  
In order to improve performance, the system requires high sensitivity with respect to low modeling uncertainty and also requires high adaptability to accommodate environment uncertainty. In fact, these non-functional requirements are difficult to satisfy simultaneously because there is a trade-off relation between them. In this normality model, the size of the normal region defined not only highly depends on the levels of sensitivity and adaptability for the generated residual, but each level of the sensitivity and adaptability is also associated with the false positive rate and the false negative rate, respectively. In our normality model, an increase in the reference point leads to an increase of the region covering the data likely to be in the normal states. However, the probability that the data may actually be anomalous data increases. It means that while the system provides higher adaptability and flexibility, it cannot guarantee high detectability due to an increase in the false negative error rate when the high reference point is used. In contrast, when the system meets high sensitivity as the reference point decreases, it brings about an increase in the anomalous region. Hence, while it is capable of recognizing the anomalous state well and increasing the degree of precision related to modeling uncertainty, it would be a major cause of occurrences of false positive errors. In terms of adaptability, a higher reference point is preferred while a lower reference point is preferred in terms of sensitivity. We determine one reference point as an optimal solution which can represent the best possible compromise between two competing objectives. In other words, a reference point can be represented as a single point, namely an equilibrium point, derived from the trade-off relationship of these objectives. When we find out an equilibrium point that meets the specified conditions where the performance of the sensitivity is maximized without sacrificing the performance of the adaptability, it can be a reasonable and fair solution. After two performance measures are defined to represent the performance of the two objectives, an equilibrium point is formulated as nonlinear programming which can solve optimally. Based on the normality model, two performance measures are calculated by using the defined regions for the anomalous data and transient faults which are derived from each of the combinations of both the reference point and the upper boundary. First, the concept of information uncertainty is adopted to represent the level of sensitivity. The uncertainty denoted as U, of the system means that how uncertain the system is about the presence of the transient faults, not soft faults, in the anomalous region defined by our normality model. It also represents the influence of the false positive error rates with respect to the anomalous region. Therefore, the uncertainty of the system at each reference point for all velocities is given by the conditional entropy which quantifies the amount of information describing the event that transient faults might exist in the suggested anomalous region. When the discrete random variable \(X\) and \(Y\) taking a certain value \(x\) and \(y\), respectively, let \(x\) represent whether the given residual is close to the normal or anomalous state so that \(x=1\) means that the residual is anomalous, and \(x=0\) means that the residual is normal. Let \(y\) represent whether the residual where \(x\) belongs to is a transient fault or a soft fault. \(y=1\) indicates that the residual is a transient fault, and, otherwise, the residual is a soft fault. According to information theory, the conditional entropy of \(Y\) given \(X\) is defined as Equation (2) from which we can obtain the conditional entropy \(H(Y=1|X=1)\) at each reference point for all velocities, which is normalized from 0 to 1. \[U(\rho_{r},\sigma_{u})\equiv H(Y|X)=-\sum_{x}p(x)\sum_{y}p(y|x)log(p(y|x)) \tag{2}\]  
Second, the detectability is defined to represent the adaptability.