• Derives a global, data-dependent upper bound for average silhouette width (ASW). • Gives pointwise silhouette-width bounds, and aggregates them to the ASW ceiling. • Computes bounds from pairwise dissimilarities in O(n2 log n) time overall only. • Experiments show bounds aid ASW interpretation; usefulness is datasetdependent. • Extends the framework to an upper bound for the macro-averaged silhouette score. The silhouette coefficient quantifies, for each observation, the balance between within-cluster cohesion and between-cluster separation, taking values in the range [ − 1 , 1 ] . The average silhouette width ( ASW ) is a widely used internal measure of clustering quality, with higher values indicating more cohesive and well-separated clusters. However, the dataset-specific maximum of ASW is typically unknown, and the standard upper limit of 1 is rarely attainable. In this work, we derive for each data point a sharp upper bound on its silhouette width and aggregate these to obtain a canonical upper bound of the ASW . This bound—often substantially below 1—enhances the interpretability of empirical ASW values by providing guidance on how close a given clustering result is to the best possible outcome for that dataset. We evaluate the usefulness of the upper bound on a variety of datasets and conclude that it can meaningfully enrich cluster quality evaluation; however, its practical relevance depends on the specific dataset. Finally, we extend the framework to establish an upper bound of the macro-averaged silhouette.

This paper proposes a novel framework for clustering mixed numerical and categorical data with missing values. It integrates the imputation and clustering steps into a single process, which results in an algorithm named Clustering Mixed Numerical and Categorical Data with Missing Values (k-CMM). The algorithm consists of three phases. The initialization phase splits the input dataset into two parts based on missing values in objects and attributes types. The imputation phase uses the decision-tree-based method to find the set of correlated data objects. The clustering phase uses the mean and kernel-based methods to form cluster centers at numerical and categorical attributes, respectively. The algorithm also uses the squared Euclidean and information-theoretic-based dissimilarity measure to compute the distances between objects and cluster centers. An extensive experimental evaluation was conducted on real-life datasets to compare the clustering quality of k-CMM with state-of-the-art clustering algorithms. The execution time, memory usage, and scalability of k-CMM for various numbers of clusters or data sizes were also evaluated. Experimental results show that k-CMM can efficiently cluster missing mixed datasets as well as outperform other algorithms when the number of missing values increases in the datasets.