## 4 Parsimonious Network Based on Fuzzy Inference System (PANFIS)
### Fuzzy Set Extraction and Merging Scenario  
Although PANFIS operates in the high-dimensional space, it is fitted with the fuzzy set extraction scenario, which derives the fuzzy set representation of the non-axis parallel ellipsoidal cluster. This mechanism allows the classical-interpretable of the If-Then rule. The centre of the multivariate Gaussian function can be simply applied to the fuzzy set level, while the radii of the fuzzy set should be carefully determined because the ellipsoidal cluster rotates in any direction. There exist two strategies to elicit the fuzzy set radii of the multivariate Gaussian function: (1) the first method elicits the eigenvalue and eigenvector of the non-diagonal covariance matrix. The disadvantage of the first approach is its prohibitive computational cost function because of the eigenvalue and eigenvector computation, even though it offers accurate approximation; (2) the second method enumerates the distance from the center to the cutting point of the ellipsoids in the main axes. This method is computationally light, although it is rather inaccurate if the ellipsoidal cluster rotates around 45\({}^{\circ}\). In such a circumstance, this method generates a spread that is too small. The present study only focuses on the second method, which is expressed as follows:  
\[\sigma_{i}=\frac{r}{\sqrt{\sum_{ii}}} \tag{12}\]  
where \(r,\sum_{ii}\) are the Mahalanobis distance and the diagonal elements of the covariance matrix, respectively. This mechanism is executed once completing the training process to show the fuzzy rules to operators.  
PANFIS is equipped with the fuzzy set merging strategy, which aims to coalesce highly overlapping fuzzy sets. Although two fuzzy rules are well-separated in the high-dimensional space, the overlapping in fuzzy sets usually results from the projection to the one-dimensional axis. This situation often results in inconsistency of rule semantics because fuzzy rules with similar fuzzy sets generate different rule conclusions. PANFIS utilizes the kernel-based metric principle for a fuzzy set merging scenario, which compares the center and width of Gaussian fuzzy sets in one joint formula [3]. It is expressed as follows:  
\[S_{\text{ker}}(A,B)=e^{-|C_{A}-C_{B}|-|\sigma_{A}-\sigma_{B}|} \tag{13}\]  
This formula holds the following interesting properties:  
\[\begin{split} S_{\text{ker}}(A,B)&=1\Leftrightarrow |C_{A}-C_{B}|+|\sigma_{A}-\sigma_{B}|\Leftrightarrow C_{A}=C_{B}\wedge\sigma_{ A}=\sigma_{B}\\ S_{\text{ker}}(A,B)&<\varepsilon\Leftrightarrow|C _{A}-C_{B}|>\delta\vee|\sigma_{A}-\sigma_{B}|>\delta\end{split} \tag{14}\]  
Two fuzzy sets are merged if the kernel-based metric returns \(S_{\text{ker}}(A,B)>0.8\), as set in [3]. This hyper-parameter controls a tradeoff of the intensity of the fuzzy set merging scenario, where the higher the values, the more frequently the fuzzy set merging scenario is executed, and vice versa. The fuzzy set merging process itself is carried out as follows:  
\[\begin{array}{l}c_{new}=(\max(U)+\min(U))/2\\ \sigma_{new}=(\max(U)-\min(U))/2\end{array} \tag{15}\]  
where \(U=\{c_{A}\pm\sigma_{A},c_{B}\pm\sigma_{B}\}\). This formula is used to perform the exact merging in accordance with their \(\alpha\)-cuts. It is expected that merged fuzzy sets deliver the membership degree around 0.6, which ensures the \(\varepsilon\)-completeness at 0.6. It reflects the inflection points of Gaussian fuzzy set \(c\pm\sigma\).