*4.4. Results*

In this section, the results were obtained for the following parameters: relief method, distance from Manhattan, *MinPts* = *n* / 2 and ε = 0.094. A 3D visualization was chosen (three principal components). In fact, the 3D results gave a detection accuracy of 96.7% and the 2D results covered an accuracy of less than 93.7%. The results of the BPFO (ball pass frequency outer) simulation showed the fault detected from signal *A*11, for noise levels 0.1*b (t)*, 0.3 *b(t)* and *A*12 for 0.5*b (t)* (Figures 4–6).

**Figure 4.** Effect of amplitude 0.1*b* (*t*): (**a**) *A*10, (**b**) *A*11 and (**c**) *A*51.

(**a**) (**b**) (**c**) 

**Figure 5.** Effect of amplitude 0.3*b* (*t*): (**a**) *A*10, (**b**) *A*11 and (**c**) *A*51.

**Figure 6.** Effect of amplitude 0.5*b* (*t*): (**a**) *A*11, (**b**) *A*12 and (**c**) *A*51.

The follow-up starting after the end of the detection phase. *GV* monitors the growth of the fault with the varied amplitude of signals. The evolution of *GV* was studied for the three noise levels 0.1*b* (*t*), 0.3*b* (*t*) and 0.5*b* (*t*), Figure 7.

Figure 7a represents the Calinski index calculated between the two clusters. The curve values increased with increasing amplitude values. The Calinski index value for the 0.1*b* (*t*) was more significant and the curve was above the others. For a high noise level, the evolution was linear *GV*1 = 0.438*k* + 7.245 *R*<sup>2</sup> = 0.980 , while for the other two noise levels the evolution was exponential (*R*<sup>2</sup> = 0.977 *and R*<sup>2</sup> = 0.742).

Figure 7b represents the Davies–Bouldin index, the curve was the opposite of the Calinski-Harabasz index, which decreased with the increasing amplitude of signals. The results observed here showed a curve of 0.1*b* (*t*), which was above the other curves, and started near to one and ended near-zero. For the three noise levels, the regression was linear. The mathematical model was similar: *GV*2 = −0.0236*k* + 0.954 *R*<sup>2</sup> = 0.999 , *GV*2 = −0.0239*k* + 0.997 *R*<sup>2</sup> = 0.997 and *GV*2 = −0.0252*k*+ 1.064 *R*<sup>2</sup> = 0.994 respectively for 0.1*b* (*t*), 0.3*b* (*t*) and 0.5*b* (*t*).

Figure 7c represents the density of the defected cluster or the second class. The density decreases over the amplitude of signals until it became constantly equal to zero, contrary to the Davies–Bouldin index decrease, to attend near zero at the end of class. The comparison between the curves showed that the density of 0.1*b* (*t*), bigger than the other noise to signal ratios. The evolution was exponential with the mathematical model: *GV*3 = 1169*e*<sup>−</sup>193*<sup>k</sup> R*<sup>2</sup> = 0.975 , *GV*3 = 777*e*<sup>−</sup>*O*.147*<sup>k</sup> R*<sup>2</sup> = 0.950 and *GV*3 = 809*e*<sup>−</sup>0.151*<sup>k</sup> R*<sup>2</sup> = 0.720 , 0.1*b* (*t*), 0.3*b* (*t*) and 0.5*b* (*t*). The correlation was poor for a low noise level.

Figure 7d represents the distance between two clusters, the distance values growing with amplitude. However, the curvy curve had an increasing trajectory form for the three scenarios 0.1*b* (*t*), 0.3*b* (*t*) and 0.5*b* (*t*). Additionally, the distance parameter could observe the trajectory of 0.1*b* (*t*), was above the other curves at the end, but initially, the three curves were conjoined, then started to separate from an amplitude equal to *k* = 31. A linear model mathematic measurement could be done from *k* = 31, *GV*4 = 0.300*k* − 0.714 *R*<sup>2</sup> = 0.963 , *GV*4 = 0.146*k* − 0.434 *R*<sup>2</sup> = 0.769 and *GV*4 = 0.097*k* − 0.297 *R*<sup>2</sup> = 0.698 .

Figure 7e represents the contour of the second cluster, showing the increase of contour with the amplitude of signals. The comparison of the contour with the Calinski index shows, the Calinski index remained increasing with the number of amplitudes. However, the contour values were similar for noise levels at low amplitudes. The contour was relevant for a certain amplitude level, *k* = 31 for low noise levels and *k* = 41 for higher noise levels. The regression models starting from *k* = 31 were *GV*5 = 0.373*e*<sup>−</sup>0.154*<sup>k</sup> R*<sup>2</sup> = 0.948 , *GV*5 = 0.059*e*0.240*<sup>k</sup> R*<sup>2</sup> = 0.986 and *GV*5 = 0.034*e*0.215*<sup>k</sup> R*<sup>2</sup> = 0.924 .

In summary, the Calinski index di fferentiates noise levels for all amplitudes. However, the mathematical regression model was di fferent. For low noise levels, a linear model was interesting, while for high noise levels, the exponential model was preferred. On the contrary, the Calinski index was little influenced by the noise level, thus the linear regression model was relevant and similar. That could show the importance of the Calinski index, which could separate the curves of di fferent scenarios, the value started with zero and grew directly with the amplitude, while the contour parameter increased slowly with the amplitude. The parameters, density and distance, had values close to 0 either for low amplitudes or high amplitudes. The evolutions were only visible for ranges of amplitudes. According to these simulations the Calinski and Davies–Bouldin indexes were preferred.

This numerical investigation made it possible to fix the internal parameters OPTICS, ε = 0.094 (Equation (4)), *MinPts* (= *n*/2) and to optimize the methods involved in the AOC-OPTICS process (relief method, t-SNE and Manhattan distance).

*Processes* **2020**, *8*, 606

**Figure 7.** (**a**) Calinski index, (**b**) Davies–Bouldin index, (**c**) density, (**d**) distance and (**e**) contour.
