A Leave-One-Out Influence Statistic for Density-Based Outlier Detection

2026-07-15

AI Summary

This work addresses the high computational cost of leave-one-out density refitting in unsupervised anomaly detection by proposing a density-based leave-one-out influence score. The score quantifies local perturbation by measuring the difference in density estimates at a fixed grid and bandwidth before and after removing a given observation. For the first time, we derive a closed-form update formula for this score under the linear bin frequency polygon (LBFP) density estimator, significantly improving computational efficiency while preserving interpretability of the underlying density. Theoretically, we establish an asymptotic order separation between normal and anomalous points under this score. Empirical results demonstrate that the proposed method matches or outperforms state-of-the-art baselines across various contamination models, with low computational overhead, and achieves strong detection performance on a 29-dimensional real-world credit card fraud dataset.

Abstract

We propose a density-based leave-one-out influence score for unsupervised outlier detection. The motivation is that outliers are naturally associated with regions of very small probability density, but direct leave-one-out density refitting can be computationally prohibitive. We use the Linear-Blend Frequency Polygon (LBFP) estimator and define a score that compares the full-sample fitted density at an observation with the fitted density obtained after removing that observation, while keeping the grid and bandwidth fixed. The resulting statistic measures a relative density perturbation at the observation's own location. For the LBFP estimator, this score has an exact closed-form update, so the density estimator does not need to be refitted for each observation. This preserves a direct density interpretation while making the method computationally efficient for large samples. We study the score under contamination and show that regular positive-density observations and contamination-driven observations have distinct asymptotic orders. Simulations over a broad range of contamination models illustrate these theoretical regimes, show competitive performance relative to standard benchmarks, and document computing time. A credit-card fraud application with 29 variables illustrates that the method works well on a large real data set.

Problem

• outlier detection
• density estimation
• leave-one-out
• computational efficiency
• unsupervised learning

Innovation

• leave-one-out influence
• density-based outlier detection
• Linear-Blend Frequency Polygon
• computational efficiency
• unsupervised anomaly detection