# On a Framework for Federated Cluster Analysis
### Motivational Example  
Previous works in federated clustering mostly assume application scenarios to be given. However, it is not necessarily obvious that sophisticated federated cluster analysis algorithms are indeed required and, for example, exchanging locally optimal cluster analysis results is not sufficient. This motivational example is designed to close the gap. In the following example, locally optimal results obtained by nonfederated fuzzy \(c\)-means do not reveal the global cluster structure. We illustrate this example by outlining a potential practical application. Imagine a multinational company with several local markets selling similar consumer goods in all markets. Each local market has data about their customers (e.g., age, place of residency, sold good, etc.) and applies (fuzzy) clustering algorithms to generate customer segments. The cluster analysis insights are utilized to steer marketing activities. The company wishes to derive global clusters to better understand their global customer base and identify unlocked potential in the local markets. Due to strict privacy regulations, the company is not allowed to gather all data in a central database (e.g., European customer data are not allowed to be transferred to most countries outside of Europe). The company could ask each local market to share their local cluster centers, but this approach disregards that clusters might only become apparent when the data are combined. Such a situation exists, as we will show. For the purpose of this example, we spare the details of the datasetcreation, because a detailed explanation can be found in Section 3.2.1. It is enough to know that there are five global clusters in the dataset, because of how it was created. Four of the clusters have relatively high cardinality, and the fifth cluster has fewer points. We verify that the correct number of clusters is detected in the centralized dataset. To achieve this, the (nonfederated) fuzzy \(c\)-means algorithm (Section 2.1.2) is applied with multiple (potential) number of clusters \(K\), and the result is evaluated with the fuzzy Davies-Bouldin index (Section 2.1.3). The best fuzzy Davies-Bouldin score is achieved with \(K=5\), and we can conclude that the correct number of clusters can be found in the centralized case. This example is designed to prove that global structure in federated data can hide behind locally optimal solutions. To create a federated setting, the data are distributed among three clients in a certain way: each client receives data from two of the four bigger clusters and a few points from the smaller cluster (see Figure 1 for a visualization). Next, we calculate the number of local clusters for all clients using the same method as before (nonfederated fuzzy \(c\)-means in combination with nonfederated Davies-Bouldin). For each client now, the best number of clusters is 2 (as it gives the best DB index score) and none identify the smaller cluster present in their data. That calls for a federated cluster analysis (federated clustering method and federated cluster validation metric) that is able to detect the global structure. In the multinational company example, the smaller cluster could represent a group of customers that each local market falsely assigned to their bigger clusters. As a consequence, the company targets those customers with inappropriate marketing activities. Exploiting insights from the global cluster analysis could lead to new, more targeted marketing strategies and unlock (previously hidden) potential. This work is concerned with introducing the federated fuzzy Davies-Bouldin (FedFuzzDB) index as a federated cluster validation metric. It can be leveraged to identify all five clusters in the data without the need for sharing the raw data, as we will see in Section 3.2.1. To the best of our knowledge, there exists no other federated cluster validation metric in the literature. Another, equally important, challenge is to correctly identify the global cluster structure given the number of clusters.