PSI - Issue 78

Ebrahim Aminifar et al. / Procedia Structural Integrity 78 (2026) 1466–1473

1472

suggest overlapping or misclassified groupings. This metric provides a standardized and robust basis for comparing the structural validity of the clustering methods applied to the area distribution of churches (Pedregosaet al., 2018). As reported in Table 2, K-Means achieved the highest silhouette coefficient (0.531), closely followed by GMM (0.529) and Agglomerative Clustering (0.513). Quantile binning, while yielding equal-sized clusters, had the lowest silhouette score (0.392), indicating poor intra-cluster cohesion. In addition to numerical scores, cluster interpretability and balance were assessed. Despite K-Means marginally outperforming GMM, the latter was selected as the preferred method due to its ability to model overlapping distributions and to generate more evenly distributed clusters (682 small, 251 medium, 392 large). GMM's probabilistic framework offers a better representation of morphological continuity within the dataset. This is particularly important when modeling historical architecture, where formal variation tends to be gradual rather than abrupt. The clustering results are visualized in Fig. 5, showing sorted area distributions for each algorithm. The GMM clusters exhibit clearly distinguishable boundaries without the overcompression observed in agglomerative clustering or the symmetry imposed by quantile binning. These results confirm the robustness of GMM in capturing the inherent variability of the church sample. In the small-size class, areas range from 5.8 to 150 m². The medium-size class spans 150-440 m², while the large size class, skewed by a minority of enormous churches, extends up to 2050 m². These distributions will be used in future work to guide the generation of nonlinear static analyses and the derivation of typology-specific fragility curves .

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Fig. 5. Area-based clustering of single-nave churches using four different algorithms. (a) K-Means; (b) Gaussian Mixture; (c) Agglomerative; (d) Quantile Binning

Table 2. Clustering Method Comparison

Clustering Method Comparison K-Means (log-transformed) Gaussian Mixture Models (GMM)

Silhouette Score

0.531 0.529 0.513 0.392

Agglomerative Quantile Binning