# On a Framework for Federated Cluster Analysis
## 2 Materials and Methods
### Background: Related Work
#### 2.1.4 Federated Clustering  
Due to the similarity in terminology, we start by contrasting clustered federation with federated clustering. Clustered federation is concerned with identifying clusters of clients or model updates that are suitable to be grouped for a focused update of global supervised FL models. It has been proven to be effective when addressing issues caused by non-i.i.d. data among clients [33; 34; 35; 36].  
In contrast, federated clustering is concerned with identifying global clusters in distributed data without sharing the data and, to the best of our knowledge, has not been explored as much. In [19], the \(k\)-means algorithm was extended to the federated setting. Client devices execute the \(k\)-means algorithm and share cluster centers with the central server. Authors propose a global averaging function that calculates a weighted mean of local cluster centers in order to update global cluster centers. The weights are given by the number of local data points assigned to the clusters. Further, the federated fuzzy clustering equivalent of that approach was introduced in [20] and similarly in [21]. In this approach, the clients execute the fuzzy \(c\)-means algorithm and share the results. Then, the fuzzy assignment vectors are used as weights instead of number of data points given by the hard assignments. In their experimental sections, both works focus on scenarios where the data are uniformly distributed among the clients. [PERSON]. [20] found that the federated clustering result was consistent with centralized clustering result, the algorithm converged quickly, and the clustering result was not impacted much by the number of clients participating. Additionally, [PERSON]. [21] observed that even if clients became unavailable during the federation, the algorithm still found good results. However, these findings are limited to scenarios where the data are uniformly distributed among clients. Finally, we acknowledge that [20] also introduced a formulation for vertical FL, which is beyond the scope of this work.  
A different approach on averaging the local cluster centers to obtain global cluster centers was taken in [22] in the context of one-shot learning. The clients performed \(k\)-means clustering. On the central server, the global cluster centers were computed by applying the \(k\)-means algorithm again to the shared local cluster centers. Besides numerical experiments, they also provided proof that the result was similar to an \"oracle clustering\" (e.g., clustering on the assumed centralixed dataset). The federated \(k\)-means algorithm by [19] appears to be the first work in the area of federated clustering. All other papers were published around the same time and appear to be independent of each other.  
In [18], the cluster analysis framework is described in terms of seven steps: selecting training examples for the cluster analysis, selecting variables for the cluster analysis, preprocessing (e.g., standardizing) the data, selecting a (dis)similarity measure, selecting a clustering method, determining the number of clusters, and interpreting and validating the results. Note that the aforementioned works are mostly concerned with the clustering method and (implicitly) with the dissimilarity measure in a federated setting. With this work, we aim to also contribute to determining the number of clusters and cluster validation in a federated setting; however, similar to the other works, we assume the experimental datasets to be preprocessed and prepared for analysis.