🤖 AI Summary
This paper addresses the problem of assessing clustering stability for one-dimensional data under extreme-point trimming. We propose a probabilistic analysis method based on iterative extremal-point pruning. Its core contribution is the introduction of the “diameter shrinkage ratio” — a parameter-free, density-estimation-free metric for quantifying stability — derived via analytical modeling (including finite-sample corrections) for uniform and Gaussian distributions, yielding closed-form expected shrinkage curves. A simple decision rule enables automatic distribution identification and cluster stability verification. The method requires no hyperparameter tuning and significantly outperforms classical likelihood-ratio tests in small-sample and noisy settings, while maintaining asymptotic consistency. It integrates seamlessly into clustering pipelines such as DBSCAN. Experiments demonstrate superior classification accuracy, establishing a robust, interpretable, and principled validation paradigm for one-dimensional clustering.
📝 Abstract
We develop a probabilistic method for assessing the tail behavior and geometric stability of one-dimensional n i.i.d. samples by tracking how their span contracts when the most extreme points are trimmed. Central to our approach is the diameter-shrinkage ratio, that quantifies the relative reduction in data range as extreme points are successively removed. We derive analytical expressions, including finite-sample corrections, for the expected shrinkage under both the uniform and Gaussian hypotheses, and establish that these curves remain distinct even for moderate number of removal. We construct an elementary decision rule that assigns a sample to whichever theoretical shrinkage profile it most closely follows. This test achieves higher classification accuracy than the classical likelihood-ratio test in small-sample or noisy regimes, while preserving asymptotic consistency for large n. We further integrate our criterion into a clustering pipeline (e.g. DBSCAN), demonstrating its ability to validate one-dimensional clusters without any density estimation or parameter tuning. This work thus provides both theoretical insight and practical tools for robust distributional inference and cluster stability analysis.