# A Novel Adaptive FCM with Cooperative Multi-Population Differential Evolution Optimization
## 4 Proposed Co-Evolutionary Framework
### Evaluation of Clustering Results
The similarities and differences between the form of the wavelet wave and the signal being analyzed are determined by CWT as,

\[C(a,T)=\int\frac{1}{\sqrt{a}}\psi(t)\bigg{(}\frac{t-T}{a}\bigg{)}x(t)dt \tag{25}\]

where \(\psi(t)\) represents the CWT mother wavelet function which is shifted in time by \(T\) and dilated or contracted by a factor and then correlated with the vibration signal represented by \(x(t)\). All feature attributes are scaled and normalized across the column before clustering. The datasets are unlike others used in the paper--high dimensional data with relatively low

Figure 13: Vibration time signatures and corresponding FFT spectrum for two cases. cardinality. The idea is to see if the fault conditions can themselves be or if any combination of operating speed and fault condition can be partitioned. The datasets are described below:

**BearingData1** is three non-overlapping segments with 16 FFT averages, \(n=90\), \(d=16\)

**BearingData2** is three non-overlapping segments with 64 Mexican Hat averaged wavelet coefficients, \(n=90\), \(d=64\)

**BearingData3** is three non-overlapping segments with 64 Coiflet-averaged wavelet coefficients, \(n=90\), \(d=64\)

**BearingData4** is three non-overlapping segments with kurtosis, skewness, RMS, and crest factors for 10 wavelet approximations and 10 wavelet details, \(n=90\), \(d=80\). We also use a low dimensional data with non-overlapping and overlapping regions of the raw data (50% overlap). **BearingData5** is three non-overlapping and two overlapping segments with four statistical features of the raw time signal--mean, skewness, standard deviation, and kurtosis, \(n=150\), \(d=4\). The performance of the proposed method is compared with FCM-AO with random initialization of cluster prototypes using the four cluster validity indices. FCM-AO is implemented for \(c=2\) to \(c=10\). The vector representation used in CDE-FCM is \(d\times 10\). The results are tabulated in Table 9.

The proposed algorithm CDE-FCM almost always outperforms FCM-AO. In cases where the three natural groupings are not found, CDE-FCM finds approximately 6 clusters which are decoded as two subclusters based on speed (high and low) in most cases. The accuracy, precision, and recall rate of both FCM-AO and CDE-FCM are similar meaning they uncover very similar clusters although with FCM-AO it is often difficult to ascertain the optimal number of clusters. A comparative evaluation is performed using sensitivity analysis. In clustering evaluation, a true positive (_TP_) is defined as the decision that assigns two similar data objects in \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(c\) & \(V_{B}\) & \(V_{T}\) & \(V_{K2}\) & \(V_{R}\) \\ \hline \multicolumn{6}{c}{**BearingData1**, \(n=90\)} \\ \hline FCM-AO & 3 & 0.0245 & 1.2238 & 10.3762 & 1.8856 \\  & 5 & 0.0291 & 2.1925 & 22.4657 & 2.5320 \\  & 10 & 0.0132 & 2.5490 & 43.6115 & 6.9894 \\ CDE-FCM & 3 & 0.0239 & 1.2197 & 9.3250 & 1.7534 \\ \hline \multicolumn{6}{c}{**BearingData2**, \(n=90\)} \\ \hline FCM-AO & 3 & 0.4578 & 17.895 & 71.965 & 5.3372 \\  & 5 & 0.5120 & 23.559 & 123.189 & 7.8900 \\  & 10 & 0.3251 & 14.358 & 53.172 & 4.3256 \\ CDE-FCM & 5 & 0.5052 & 19.856 & 112.885 & 7.2505 \\ \hline \multicolumn{6}{c}{**BearingData3**, \(n=90\)} \\ \hline FCM-AO & 3 & 0.7925 & 8.6690 & 45.890 & 3.897 \\  & 5 & 0.8456 & 12.367 & 119.559 & 4.673 \\  & 10 & 1.6251 & 22.145 & 145.335 & 8.212 \\ CDE-FCM & 3 & 0.8280 & 8.5241 & 36.879 & 3.865 \\ \hline \multicolumn{6}{c}{**BearingData4**, \(n=90\)} \\ \hline FCM-AO & 3 & 0.8955 & 8.356 & 67.189 & 2.9836 \\  & 6 & 0.7341 & 5.3536 & 50.338 & 2.5620 \\  & 10 & 1.3802 & 12.7467 & 121.298 & 3.2461 \\ CDE-FCM & 6 & 0.6821 & 5.1130 & 48.827 & 2.5465 \\ \hline \multicolumn{6}{c}{**BearingData5**, \(n=150\)} \\ \hline FCM-AO & 3 & 0.0026 & 1.2876 & 8.9281 & 0.3653 \\  & 6 & 0.0012 & 1.1926 & 6.7172 & 0.3156 \\  & 10 & 0.0104 & 1.8927 & 11.602 & 0.7253 \\ CDE-FCM & 6 & 0.0011 & 1.2110 & 6.5670 & 0.3267 \\ \hline \end{tabular}
\end{table}