## 3 Method
### Method Detailed Description
#### 3.2.4 Final Instance Classification  
The final instance classification is a process made of two activities: Decision Reasoning (Activity 7) and State Detection (Activity 8). Activity 7--Decision Reasoning  
Decision Reasoning receives vector \(\overrightarrow{d}^{s+1}\) from both data-driven and model-based ET (activities 4 and 6 respectively). Alternatively, instead of receiving the output of activity 6, the Decision Reasoning process can receive the residues values vector directly from Activity 5, as shown in Figure 2.  
As an output, the Decision Reasoning process delivers to the next activity a harmonized decision vector \(\overrightarrow{d}^{s+1}\), considering both data-driven and model-based preliminary results. This task can be performed considering either a linear combination of model-based and data-driven decision vectors or a revisiting process of the ET results, applying, for instance, ANFIS (Artificial Network Fuzzy Inference System) algorithm to estimate system state. If an algorithm is used to estimate system state directly from the output of Activities 4, 5 and 6, as for instance ANFIS, then Activity 7 and Activity 8 are merged on a single activity that is able to provide system state detection. Activity 8--State Detection  
The State Detection process selects the most probable state from the harmonized decision vector and informs it to the end-user, also estimating the confidence level of this estimate. As highlighted before, data fusion algorithms as ANFIS can also be used on this task, as detailed in the sequence. ANFIS Semantic Aspects  
One of the best strengths of the Fuzzy Inference System (FIS) is its capability to handle subject measurements [6], and this is the main reason to choose it as a manager--its decision is based on its staff. In this method, the physical model, \(\text{ET}_{1}\), and \(\text{ET}_{2}\) are their staff whose opinions are expressed through vector \(\overrightarrow{d}^{s+1}\). The Membership Function (MF) performs the fuzzification step [25]. This technique defines two sets of Gaussian MF, both with the same size, where \(\Omega_{1}=\text{Eliminated}\), Candidate, Winner corresponds to the system-state estimated and \(\Omega_{2}=\text{Negligible}\), Acceptable, Unacceptable corresponds to the indecision degrees. Considering those same vectors from a semantic perspective, it is possible to make the sentences described in Table 4. Different types of membership functions can be used to support the fuzzification process. On Table 4, \(d_{i,j}\) corresponds to the result vector of first instance classification algorithm, where index \(i\) corresponds to the vector origin (physical model, \(\text{ET}_{1}\), and \(\text{ET}_{2}\)), while index \(j\) corresponds to the system state under analysis. If the system has \(s\) states, \(j\) varies from 1 to \(s+1\), where \(s+1\) corresponds to undefined state (indecision grade). Table 4 can be very assertive when one of its elements declares a winner or an eliminated system-state. At the same time, it can be very subjective when declaring a candidate. It is not necessary to relate all system-states among each other since the MFs can provide a certain level of mutual-exclusivity when the sentences are considered assertive. It impacts the number of necessary rules that defines the size of 2 nd, 3 rd, and 4 th layers. So, the set of useful rules for this methodology is the combination of correspondents system-states and correspondents indecision degree, as seen in Figure 4.  
Figure 4: ANFIS applied in this technique. The ANFIS training makes use of Minimum Square (MS) to set its linear parameter in 4 th layer and gradient descent (GD) algorithm to train its non-linear parameters in 1 st layer. Primarily, it is performed a forward phase where non-linear parameters are kept fixed. After the 3 rd layer processes its outputs, the linear parameters are calculated through MS, and the error is calculated when the 5 th layer outputs its value.