# On a Framework for Federated Cluster Analysis
## 3 Results
### Experiments
#### 3.2.2 Evaluation on Benchmark Data
Similarly, the knowledge gap statistics are  
Figure 3: Correct number of cluster guesses on all \(100\) G2 sets per dimension. The values are averaged over all values of \(\sigma\) and runs. We observe few correct guesses in the two dimensional case, a peak for \(D=8\), then a decline that stabilizes after \(D=64\). The overall trend is similar for all number of clients. The figure shows results of the \(k\)-means averaging, but they are similar with federated averaging. Figure 2: Correct number of cluster guesses on all \(100\) G2 sets per standard deviation \(\sigma\). The values are averaged over all dimensions and runs. We observe a decline of correct number of cluster guesses with increasing \(\sigma\). The figure shows results of the \(k\)-means averaging, but they are similar with federated averaging. consistent, but not the same (Tables 6 and 7). Hence, in some cases, FedFCM converges to different centers. As we will explain, the differences mostly occur due to sparsity. It is important to note that those are the cases where FedFCM did not find a good clustering (high knowledge gap and high FedFuzzDB) in either the central case or in the federated settings. Overall, we conclude that the clustering results on i.i.d. data are similar in the central case and with both federated clustering methods. \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline
**Knowledge Gap (\(K=2\))** & **Central** & **Two Clients** & **Five Clients** & **Ten Clients** \\ \hline
25\% Quantile & 0.2850 & 0.2724 & 0.2700 & 0.2587 \\ \hline
50\% Quantile & 0.8904 & 0.8897 & 0.8900 & 0.8898 \\ \hline
75\% Quantile & 5.9544 & 5.9272 & 5.8335 & 5.4764 \\ \hline Minimum & 0.0286 & 0.0278 & 0.0265 & 0.0253 \\ \hline Maximum & 77.6472 & 77.540 & 77.7092 & 76.0134 \\ \hline \end{tabular}
\end{table} 
Table 6: Statistics of the knowledge gap on the 100 G2 test sets with \(k\)-means averaging. \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline
**Fuzzy Davies–** & **Central** & **Two Clients** & **Five Clients** & **Ten Clients** \\ \hline
25\% Quantile & 0.6762 & 0.6771 & 0.6766 & 0.6766 \\ \hline
50\% Quantile & 0.8640 & 0.8638 & 0.8627 & 0.8627 \\ \hline
75\% Quantile & 1.3092 & 1.3073 & 1.2961 & 1.2944 \\ \hline Minimum & 0.5460 & 0.5459 & 0.5459 & 0.5458 \\ \hline Maximum & 56,784.5518 & 57.6910 & 23.6987 & 20.0192 \\ \hline \end{tabular}
\end{table} 
Table 4: Statistics of the (federated) fuzzy Davies–Bouldin index on 100 G2 test sets with \(k\)-means averaging. \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline
**Fuzzy Davies–** & **Central** & **Two Clients** & **Five Clients** & **Ten Clients** \\ \hline
25\% Quantile & 0.6762 & 0.6767 & 0.6766 & 0.6766 \\ \hline
50\% Quantile & 0.8640 & 0.8627 & 0.8623 & 0.8620 \\ \hline
75\% Quantile & 1.3092 & 1.2997 & 1.2990 & 1.2934 \\ \hline Minimum & 0.5460 & 0.5460 & 0.5459 & 0.5459 \\ \hline Maximum & 56,784.55 & 31,307.76 & 54,809.42 & 9021.3485 \\ \hline \end{tabular}
\end{table} 
Table 5: Statistics of the (federated) fuzzy Davies–Bouldin index on 100 G2 test sets with federated averaging. With this in mind, we enter the discussion of the results and offer an explanation for some of the observations. First, we want to understand why the cluster number detection rate is lower for higher values of \(\sigma\). Recall that the Davies-Bouldin index is the ratio of \"cluster spread\" and \"center closeness\". Hence, the index is high for clusters that naturally have a high spread, i.e., high \(\sigma\) as depicted in Figure 4 (while keeping the center distances fixed, as in our experiments). In such cases, the index could be reduced by introducing a new cluster, because the gain through the lower spread is relatively big. This behavior is intensified by the poor performance of fuzzy \(c\)-means on sparse data. In such cases, (local) fuzzy \(c\)-means centers (regardless of \(K\)) tend to converge to the center of mass of the whole dataset [30].