## 3 Experiment Results and Analysis
### Clustering Results Similarity Analysis  
External indexes have been commonly used to compare clustering algorithms by calculating how many pairs of data points are partitioned consistently between two clustering solutions. For both solutions to be consistent, a pair of points must be assigned to either the same cluster or a distinct cluster. This offers an estimation of point-level similarity but no direct information regarding cluster-level similarity. In most cases, information about the cluster-level similarity of the algorithm is of more interest than the estimation of point-level similarity. [PERSON] et al. [57] proposed Centroid index (CI) measures cluster-level differences of two solutions, and centroid similarity index (CSI) measures point-level differences of two solutions. The CI index has clear intuitive interpretation on how many clusters are differently located in the two solutions by an integer value. A higher CI value indicates a lower similarity between the two solutions when the number of cluster classes is determined. Considering the effect of the number of clustering, this paper employs a normalized adjusted CI index (ACI) to compare solutions with the following formula:  
\[ACI=\frac{CI-abs(K_{1}-k_{2})}{min(K_{1},k_{2})-1} \tag{7}\]  
where \(K_{1},k_{2}\) represents the number of clusters in two solutions; \(abs()\) represents the absolute value function; \(min()\) represents the minimum function. Tables 9 and 10 are the comparison tables for the novel FPSO-GAK algorithm with other algorithms.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \multirow{2}{*}{**City**} & \multirow{2}{*}{**Algs**} & \multirow{2}{*}{**GAK**} & \multicolumn{2}{c}{**PSOK**} & \multicolumn{2}{c}{**PSOK**} & \multicolumn{2}{c}{**PSOWK**} & \multicolumn{2}{c}{**PSO-GAK**} & \multicolumn{2}{c}{**K-Means++**} \\ \cline{2-13}  & & **Mean** & **Std** & **Mean** & **Std** & **Mean** & **Std** & **Mean** & **Std** & **Mean** & **Std** & **Mean** & **Std** \\ \hline Aracaju & 0.266 & 0.060 & 0.186 & 0.027 & 0.190 & 0.028 & 0.198 & 0.035 & 0.204 & 0.022 & 0.126 & 0.031 \\ Beijing & 0.182 & 0.075 & 0.111 & 0.027 & 0.200 & 0.062 & 0.190 & 0.054 & 0.161 & 0.042 & 0.096 & 0.022 \\ Chongqing & 0.307 & 0.042 & 0.151 & 0.022 & 0.208 & 0.029 & 0.198 & 0.025 & 0.207 & 0.026 & 0.131 & 0.037 \\ Roma & 0.364 & 0.039 & 0.162 & 0.030 & 0.249 & 0.023 & 0.243 & 0.027 & 0.272 & 0.025 & 0.181 & 0.024 \\ San Francisco & 0.353 & 0.044 & 0.204 & 0.041 & 0.205 & 0.040 & 0.222 & 0.037 & 0.231 & 0.040 & 0.165 & 0.039 \\ \hline \end{tabular}
\end{table}
Table 9: ACI index for novel FPSO-GAK vs. other algorithms.  
Table 9 shows that the ACI indices of FPSO-GAK and other algorithms are greater than 0.15, indicating a substantial difference in their clustering-level similarity. Obtaining the above outcomes under identical data input conditions reveals that the equivalent mathematical models of the FPSO-GAK algorithm differ significantly from those of other algorithms. However, the value of the ACI index is generally less than 0.25, indicating that the difference between equivalent mathematical models is not fundamental. The CSI indices in Table 10 are more concerned with point-level similarity, and their values indicate similar model analysis information.