# A Novel Adaptive FCM with Cooperative Multi-Population Differential Evolution Optimization
## 4 Proposed Co-Evolutionary Framework
### Evaluation of Clustering Results
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & \(c\) & \(V_{B}\) & \(V_{T}\) & \(V_{K2}\) & \(V_{R}\) \\ \hline FCM-AO & 8 & 8.775 & 0.3721 & 72.456 & 5.978 \\  & 10 & 13.942 & 0.9885 & 125.814 & 8.775 \\  & 11 & 22.857 & 0.9925 & 216.380 & 12.394 \\  & 12 & 23.617 & 1.0225 & 229.773 & 12.945 \\ \hline ADEFC & 10 & 14.332 & 0.9885 & 120.035 & 8.674 \\ \hline PSO-V & 8 & 8.456 & 0.3389 & 70.990 & 5.825 \\  & 10 & 13.880 & 0.9727 & 118.336 & 8.650 \\  & 11 & 21.352 & 0.9880 & 207.350 & 12.218 \\  & 12 & 21.445 & 1.0225 & 219.872 & 12.945 \\ \hline GAKFCM & 8 & 8.356 & 0.3392 & 7.1855 & 5.978 \\  & 11 & 21.335 & 0.9891 & 208.392 & 12.218 \\ \hline EwFCM & 8 & 8.470 & 0.3391 & 70.990 & 5.824 \\  & 10 & 13.442 & 0.9738 & 115.275 & 8.458 \\  & 11 & 21.832 & 0.9890 & 208.360 & 12.394 \\  & 12 & 21.975 & 1.0225 & 220.825 & 12.945 \\ \hline CDE-FCM & 12 & 21.329 & 1.0225 & 217.375 & 12.885 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Performance of algorithms on wine quality data. or machine failure. Bearing fault diagnosis has been a subject of great importance in machine condition monitoring, predictive maintenance, and machine failure prevention and analysis [64]. Bearings conditions are presented in [65]. Techniques used in fault severity evaluation in rolling bearings are reviewed and discussed in [66]. The review is mainly focused on data-driven approaches such as signal processing for extracting the fault signatures associated with the fault degradation, and the approaches that are used to identify degradation patterns. Modern predictive maintenance techniques are increasingly adopting data analysis techniques such as pattern recognition and machine learning for bearing fault diagnosis. In this case study using the proposed cluster initialization method, wavelet analysis is used to process vibration signals from three bearing cases--no fault, inner race fault, and ball fault, under varying rotating speeds. A schematic of the experimental setup is shown in Figure 11. The faults are introduced as a single rough surface spot simulating pitting wear and are created by using a small grinder as shown in Figure 12. The rotor is run at 10 different speeds (from 500 rpm to 1400 rpm) in increments of 100 rpm. At each speed level, vibration signals are acquired using a PCB accelerometer (model PCB 302A) which is mounted on the outboard test bearing as indicated in Figure 10. The sampling rate is 10,000 samples/s. A radial load of about 2000 lb was kept constant during all tests. A small unbalance mass is added to the rotor to introduce sustained periodic vibration excitation. Three sets of samples are collected, each for 0.5 s. The analysis is performed for each of the three sets, followed by sets obtained using a 50% overlap between the non-overlapping sets resulting in five sample sets (3 non-overlapping sets and 2 overlapping sets) for a single operating condition--speed and type of fault (10 different speeds and 3 fault conditions including no fault). The raw time series is analyzed for statistical features such as mean, variance, standard deviation, skewness, and kurtosis. The times series are also subject to fast Fourier transform (FFT) and continuous wavelet transform (CWT) techniques. Figure 13 shows two sample raw vibration time series and their corresponding frequency spectrum obtained using FFT. After experimenting with several wavelet transform functions, the 'Mexican hat' and the Coiflet wavelet functions were used. The CWT implementation was

Figure 11: Experimental setup. Figure 12: (a) Ball bearing used in the test, (b) ball fault, (c) inner race fault. performed using MATLAB's wavelet toolbox. MATLAB's CWT can be considered as a filter that scales a mother wavelet function along the time axis. At each scale, the CWT function will superimpose the scaled mother wavelet wave form over a segment of the signal under analysis.