## 5 Simulation Results  
We compared five approaches, namely IFCME, IFCMC, IPFCME, IIFPFCME, and IIFPFCMC with IDS1-2 datasets and IFCME, IFCMC, IIFPFCME, and IIFPFCMC with IDS1-6 datasets that have several datasets that include outlier interference. We also presented two real datasets concerning exchange rates for validation of the proposed methods. For a detailed description of the datasets used in this paper, please refer to [34; 35]. For all datasets, we used the following computational protocols: stop requirement \(\varepsilon=0.00001\), and iteration limit \(L=300\). All experiments were run on a computer with Intel(R) Xeon(R) CPU E3-1231 v3 @ 3.40 GH, 32 GB memory, Microsoft Windows 10, and MATLAB 2018b. The performance using the root-mean-square error (RMSE) to computer the true cluster center and corresponding cluster center is defined as:  
\[\text{RMSE}=\sqrt{\frac{\sum\limits_{i=1}^{c}(g_{i}-\widetilde{g}_{i})^{2}}{c}} \tag{29}\]  
where \(g_{i}\) is the true cluster center and \(\widetilde{g}_{i}\) is the corresponding cluster center. The first dataset (DS1) consists of five clusters in 2D space which satisfy the standard normal distributions, whose centers are (1, 1), (1, 9), (9, 1), (5, 5), and (9, 9). For the first four clusters, the number of data points per cluster was 30, and their respective covariance matrices were \( \begin{pmatrix}1&0\\ 0&1\end{pmatrix} \). For the last cluster centered at (9, 9), there were 120 data points, and their covariance matrix was \( \begin{pmatrix}2&0\\ 0&2\end{pmatrix} \). These data are shown in Figure 4a. In order to build the interval data (symbolic) sets, this was defined as \(([21-\gamma_{1}/2,z1+\gamma_{1}/2],[2z-\gamma_{2}/2,z2+\gamma_{2}/2])\), from which \(\gamma_{1}\) and \(\gamma_{2}\) were drawn randomly from [0.2, 1], as shown in Figure 4b. The results of IFCME and IFCMC with \(m=2\) for IDS1 are shown in Figure 5, where * is the true cluster center; the same color denotes the same clustering (group), and the symbol data with a fill-in color on each color (clustering) is the cluster center of results. The results of IIFPFCME and IIFPFCMC with parameters \(\eta=0.01\) and \(\eta=0.1\) for IDS1 are shown in Figures 6 and 7, respectively. The results of IPFCME for IDS1 are shown in Figure 8. There are three indexes: the RMSE, the average usage time of ten times, and the average of ten iterations as a comparison on Tables 1-6. The results of the comparison are shown in Table 1. From Table 1, the proposed IIFPFCME and IIFPFCMC on performance and average times all are better than IFCME, IFCMC, and IPFCME. From Table 1, the proposed IIFPFCME appears to be better than the proposed IIFPFCMC under IDS1. IDS1 is a symbolic dataset without outliers. From Figure 5, the clustering results are not good, as the center of the five clusters cannot be found correctly. Hence, we used  
Figure 8: The results of the IPFCMC parameter with (a) \(m=1.5\), \(n=2\), \(a=20\), \(b=50\) and (b) \(m=1.5\), \(n=2\), \(a=50\), \(b=50\) for IDS1. Figure 6: The results of the IIFPFCME parameter with (a) \(\eta=0.01\) and (b) \(\eta=0.1\) for IDS1. Figure 7: The results of the IIFPFCMC parameter with (a) \(\eta=0.01\) and (b) \(\eta=0.1\) for IDS1.