# On a Framework for Federated Cluster Analysis
## 3 Results
### Experiments
#### 3.2.3 Evaluation on Real-World Data  
In the last two sections it was investigated how the framework behaves on artificial data with well-controlled properties. With this final experiment, we want to evaluate the framework on more real-world-inspired data and demonstrate how it could be used in practice. The data for this experiment were first introduced in [43] and can be accessed through the UCI machine learning repository ([[https://archive.ics.uci.edu/ml/datasets/Bank+Marketing](https://archive.ics.uci.edu/ml/datasets/Bank+Marketing)]([https://archive.ics.uci.edu/ml/datasets/Bank+Marketing](https://archive.ics.uci.edu/ml/datasets/Bank+Marketing)), accessed on 27 March 2022) and are about customers of a bank. In particular, we are interested in the customers' recorded job, age, balance, and education. Based on the job information, we split the data to create a federated setting. Each client has data of only one job group. For example, one client has all records of students, another has all data of retired persons and another has all records of managers, etc. In total, there are 11 job categories such that there are 11 clients in the federated setting. Based on the remaining columns (age, balance, education), we want to form groups of similar customers using (federated) fuzzy \(c\)-means following the framework introduced in Section 2.2.3: determine the number of global clusters, derive a soft partitioning of the data, and validate the clustering. For comparison, we also compute the partitioning on the full, but centralized, dataset as well as the local-only datasets. Before applying the framework, we preprocess the data: we translate education into numerical values (primary: 1, secondary: 2, tertiary: 3), roughly estimating the time spent in school/university, rows with unknown values are dropped, and each column is standardized to have zero mean and standard deviation of 1. In total, we are left with 43,193 rows in the dataset. Each client holds data of only one job group: job group \"management\" has 9216 examples, \"technician\" 7355, \"entrepreneur\" 1411, \"retired\" 2145, \"admin\" 5000, \"services\" 4004, \"blue-collar\" 9278, \"self-employed\" 1450, \"unemployed\" 1274, \"housemaid\" 1195, and \"student\" 775 examples. First, we need to determine the number of clusters by executing the first step of our framework. We set the minimum of clusters \(K_{min}=3\). Each client has at least 3 clusters in its local-only data. We draw that conclusion from calculating the (nonfederated) fuzzy Davies-Bouldin index and applying the elbow method (see Figure 5 for examples). The maximum number of clusters is set to \(K_{max}=9\), because it provides a buffer for the identification of forming clusters. One of the clients (entrepreneur) reports that it has six clusters and we choose \(K_{max}=K_{min}+\max_{i}K^{(l)}=9\) according to our rule of thumb (Section 2.2.3). For each \(K\in[K_{min},K_{max}]\), the partitioning using the federated fuzzy \(c\)-means algorithm with \(k\)-means averaging and the FedFuzzDB index is calculated. The results can be found in Figure 6. The elbow method suggests the number of global clusters to be four, five, or six. Additionally, the figure contains the results of the same analysis in the nonfederated setting. Similarly, the method suggests that there are four or six in the centralized data, showing good consistency. As common in practice, the index only gives a good indication on the number of clusters and the practitioner is left to make the final call. Notably, the nonfederated and federated index values are not the same. Generally, the FedFuzzDB index is slightly lower than the nonfederated index. That implies that the federated method returns better centers (as measured by the Davies-Bouldin index) than its nonfederated counterpart. Second, we want to understand why the federated index is lower than the nonfederated index. Therefore, we compare the federated and nonfederated cluster centers for \(K=4\). We note that three out of the four centers are almost identical.