## 4 Results
### Scenario Attestation
#### 4.2.4 Constructing the Mass Functions  
Moving forward, we demonstrate the construction of the mass functions for the process at hand. The first observation, as already mentioned, is the linear dependency between interventions and the computed impact matrix. Hence, we proceed to the selection of the \(M^{-}(b^{\prime})\) and upper \(M^{+}(b^{\prime})\) bounds for attestation. For demonstration purposes we select \(M^{-}(b^{\prime})=2\%\) and \(M^{+}(b^{\prime})=6\%\). The interval bounded by these values provides six available interventions. Let \(V\) denote the set of interventions. Then, for each observed variable, we compute the three bounding rectangles that fit these interventions, similarly to the procedure shown in Figure 2(a). Let \(R_{min}(V)\) denote the minimum bounding rectangle that bounds the set of values \(V\) and has centroid \(\mathbf{C}_{min}\). Let \(h,w\) denote the height and width of the rectangle, respectively. Subsequently, the uncertainty bounding rectangle denoted by \(R_{unc}(V)\) contains \(R_{min}(V)\), and therefore has the same centroid \(\mathbf{C}_{min}\). Let \(h_{unc},w_{unc}\) denote the height and width of the uncertainty rectangle, respectively. For the purposes of this experiment, we assume that \(h_{unc}=3h\) and that \(w_{unc}=w\), which means that the width of the uncertainty rectangle does not change. However, the height of this rectangle includes values that deviate at most two times from previous measurements. Obviously, these limits can be adjusted according to the process characteristics and measurement noise. Similarly, we construct the anomaly bounding rectangle \(R_{anom}(V)\), which contains \(R_{min}(V)\), and has the same centroid \(\mathbf{C}_{min}\). For \(h_{anom},w_{anom}\) denoting the height and width of the uncertainty rectangle, respectively, \(h_{anom}=5h\) and \(w_{anom}=w\). An example illustration of the three bounding rectangles obtained for VAc with an intervention on the HAC feed valve is depicted in Figure 10.  
In the next step, the translation to mass function assignments was performed. For this experiment, we assumed that \(m(\Theta)\) has the same value for all of the detectors, and its value is negligible (e.g., \(<\)0.001). Nevertheless, this value can be adjusted to express the level of confidence with respect to the reports originating from a particular detector. Figure 8: Attestation result on two observable variables with intervention on the HAC feed product. Figure 9: Attestation result on two observable variables with intervention on the C\({}_{2}\)H\({}_{4}\) feed product. An example design for the mass functions is provided in Figure 11a. Let \(csl()\) be a function that determines the central splitting line of a given rectangle, and \(dist()\) a function that returns the distance of a given point to a line segment. Subsequently, let \(D_{1}=h/2\), \(D_{2}=3h/2\), and \(D_{3}=5h/2\) denote the distances from the central splitting line to the edge segment of each bounding box, as illustrated in Figure 11b. Let \((M(b),C(y^{s},b))\) be a pair of intervention magnitude and the measured impact that requires attesting. Then, we compute \(d=dist((M(b),C(y^{s},b)),csl(R_{min}))\). According to the value of \(d\) with respect to distances \(D_{1},D_{2}\), and \(D_{3}\), three cases are distinguished. For each case, the mass functions are obtained by projecting the distance \(d\) to each particular line segment that is active:  
* Case 1. \(d\leq D_{1}\). Two hypothesis are active: \(H_{1}=\{NORMAL\}\), and \(H_{2}=\{NORMAL\}\), \(ANOMALY\}\). The rationale for this is that as we move away from the center of \(R_{min}\) the level of confidence in a normal state decreases, and, at the same time, we experience an increase in the level of uncertainty. In this case, the uncertainty is that the detector cannot distinguish between a normal and anomalous state. * Case 2. \(d\in(D_{1},D_{2}]\). Three hypothesis are active: \(H_{1}=\{NORMAL\}\), \(H_{2}=\{NORMAL\}\), \(ANOMALY\}\), and \(H_{3}=\{ANOMALY\}\).