🤖 AI Summary
To address the limitations of existing scan statistics for sparse anomaly detection under multi-source heterogeneous coordinate systems—namely, their reliance on strong distributional assumptions and difficulty in calibration—this paper proposes a rank-based high-criticism scoring method. The approach leverages only the relative ordering among independent observations, avoiding parametric modeling entirely, and introduces a novel rank-based high-criticism framework to nonparametrically characterize detectability conditions. We theoretically establish that detection power is uniquely determined by the probability that an anomalous observation exceeds a typical one, and prove asymptotic optimality under both exponential families and convolution models. The method achieves strong robustness and theoretical interpretability: it successfully identifies process anomalies in pharmaceutical quality control data, and simulations demonstrate performance approaching that of the oracle test.
📝 Abstract
Detecting anomalies in large sets of observations is crucial in various applications, such as epidemiological studies, gene expression studies, and systems monitoring. We consider settings where the units of interest result in multiple independent observations from potentially distinct referentials. Scan statistics and related methods are commonly used in such settings, but rely on stringent modeling assumptions for proper calibration. We instead propose a rank-based variant of the higher criticism statistic that only requires independent observations originating from ordered spaces. We show under what conditions the resulting methodology is able to detect the presence of anomalies. These conditions are stated in a general, non-parametric manner, and depend solely on the probabilities of anomalous observations exceeding nominal observations. The analysis requires a refined understanding of the distribution of the ranks under the presence of anomalies, and in particular of the rank-induced dependencies. The methodology is robust against heavy-tailed distributions through the use of ranks. Within the exponential family and a family of convolutional models, we analytically quantify the asymptotic performance of our methodology and the performance of the oracle, and show the difference is small for many common models. Simulations confirm these results. We show the applicability of the methodology through an analysis of quality control data of a pharmaceutical manufacturing process.