# On a Framework for Federated Cluster Analysis
## 3 Results
### Experiments
#### 3.2.1 Framework Demonstration on Artificial Data  
We revisit the motivational example from the introduction (Section 1.1). In this example, we start with a centrally created dataset by drawing from five Gaussian distributions: 500 examples each drawn from distributions centered at \(\mu_{1}=(0,0)\), \(\mu_{3}=(1,1)\), 1000 examples each drawn from distributions centered at \(\mu_{2}=(0,1)\), \(\mu_{4}=(1,0)\) and standard deviation \(\sigma_{1}=0.2\), 120 examples drawn from a distribution centered at \(\mu_{5}=(0.5,0.5)\) and \(\sigma_{2}=0.01\) (see also Figure 1). Hence, there are five ground truth centers. First, we verify that the five ground truth centers can indeed be found in a nonfederated scenario. We obtain clustering results for \(K=2,\ldots,7\) and calculate the (nonfederated) Davies-Bouldin index for each result. As expected, the index is smallest for \(K=5\) (Table 1). Second, the data are distributed to three clients such that all clients have data from three clusters in total: two of the four bigger clusters and a few points from the smaller cluster. In particular, each client receives 40 points from distribution \((\mu_{5},\sigma_{2})\). Client 1 receives 500 points each from distributions \((\mu_{1},\sigma_{1})\) and \((\mu_{2},\sigma_{1})\). Client 2 receives 500 points each from \((\mu_{2},\sigma_{1})\) and \((\mu_{4},\sigma_{1})\). Client 3 receives 500 points each from \((\mu_{3},\sigma_{1})\) and \((\mu_{4},\sigma_{1})\). Next, each client applies the (nonfederated) fuzzy \(c\)-means separately on their local data for \(K\in\{2,3,4,5\}\) and calculates the (nonfederated) fuzzy Davies-Bouldin index. In this experimental setup, all clients would conclude that they have only two clusters, as \(K=2\) results in the smallest Davies-Bouldin index (Table 1). The clients only detect the two bigger clusters in their data and disregard the smaller cluster. When applying the federated clustering framework outlined in Section 2.2, the correct number of clusters \(K=5\) is found. Please refer to Table 1 for an overview of the results. This experiment shows that the framework is capable of identifying global cluster structure even though it is hidden behind local optima without sharing the raw data. For comparison, we repeat the same experiment with the federated fuzzy \(c\)-means formulation that applies federated averaging, as introduced in [20,21] (see also Section 2.1). Note that the setting is non-i.i.d. in the sense that not all clients have data from the same clusters and that such a situation was not part of the analysis in [20,21]. For an overview of the results, please refer to Table 2. We observe that the federated averaging formulation struggles to identify the ground truth centers in this setting. The Davies-Bouldin index is generally higher with federated averaging than with \(k\)-means averaging. This indicates that the clustering can be considered less meaningful. The same is indicated by the higher ground truth gap, i.e., the ground truth centers could not be found. Consequently, this also leads to a wrong estimate for the number of global cluster centers. All in all, the results with federated averaging appear to be less reliable than \(k\)-means averaging on non-i.i.d. data. However, as we will see in the next section, the results on i.i.d. data are similar and, therefore, consistent with [20,21]. The drop in performance can be explained by the lack of a \"grouping mechanism\". The grouping mechanism must identify a group of local centers that belong to the same global center and, hence, are used to update that global center. In the case that each client has data from the same clusters (thus, finds and reports similar centers locally), that matching is (implicitly) given. Since all clients have points from the same clusters, all local updates will move in the same direction and there is no ambiguity. With widely different data locally, the local updates will also be very different and there must be a mechanism to deal with the ambiguity, e.g., a grouping of local updates.