2026-09-28T11:48:17Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:27827 2026-09-28T06:35:13Z ddc:004 open_access

Statistical Foundations of Local Geometric Fault Detection: The Curvature Channel Provan, Gregory Geometric metrics Diagnosis methods Theory Geometric and topological methods have recently emerged as a promising approach to fault detection in hybrid and cyber-physical systems. Unlike classical residual-based methods, which detect violations of a parametric model, geometric methods monitor changes in the local structure of trajectories through quantities such as intrinsic dimension, smoothness, tangent-cone geometry, and consistency with physical constraints. Despite encouraging empirical results, there is currently no theory characterising when such geometric signatures are detectable, distinguishable, or reliable under noise. This paper develops a statistical theory of local geometric fault detection, with a focus on curvature-based approaches. We study faults detectable by displacement in a curvature signature space, and derive the noise floor of geometric estimators based on finite differences and local neighbourhood averaging. The resulting estimator variance scales as σ_{est} = O(σ/(Δ t²√k)), exposing the fundamental trade-off between sampling rate, noise level, and neighbourhood size. Using this noise model, we derive geometric analogues of the classical detectability and isolability conditions of fault-detection theory, expressed directly in terms of separation in signature space. We further prove an exclusion result showing that a broad class of smooth parametric faults cannot activate curvature-based geometric channels, yielding a form of label-free fault-class identification. These results lead to a practical pre-deployment assessment procedure that determines, from nominal data alone, whether a curvature-based fault detector is expected to achieve a specified false-alarm rate, detection probability, and isolation capability. The theory thereby places local geometric fault detection on the same statistical footing as classical residual-based diagnosis while clarifying its distinctive strengths, limitations, and operating regime. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Gregory Provan 2026 Is Part Of OASIcs, Volume 148, 37th International Conference on Principles of Diagnosis and Resilient Systems (DX 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/OASIcs.DX.2026.13 urn:nbn:de:0030-drops-278279 https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.DX.2026.13 eng https://creativecommons.org/licenses/by/4.0/legalcode