Depth and Local Depth in Clustering: Algorithms and Applications with Minimal Assumptions

Abstract

Clustering is a fundamental problem in data analysis, but classical methods are often constrained by assumptions about cluster shape and require user-specified parameters. This dissertation advances clustering methodology by integrating statistical depth and local depth concepts, yielding a unified and flexible framework for a wide range of clustering challenges. The first contribution introduces depth-based local center clustering (DLCC), which can address both convex and non-convex cluster shapes, accommodate balanced or unbalanced cluster sizes, and perform well with both well-separated and overlapping clusters. The second part develops a novel local depth measure, -integrated local depth ( -ILD), to robustly quantify local centrality. Its properties are analyzed in detail. The third contribution presents an automatic version of DLCC that leverages -ILD for local center selection and employs adaptive merging strategies, achieving strong performance on both synthetic and high-dimensional real datasets without the need for parameter tuning. Taken together, these contributions provide a practical and theoretically grounded approach for applying data depth in clustering tasks, opening new directions for clustering complex data with minimal prior assumptions and knowledge about data structure. iv