Towards a Functional Record Theory: Applications to Anomaly Detection

Record theory provides key probabilistic tools to understand extreme events : phenomena that exceed typical fluctuations and drive critical decisions in science, engineering, and society. It is thus highly relevant for anomaly detection, as anomalies frequently occur at extremes. While classical record theory is well understood in finite-dimensional settings [1,2], the surge of functional and high- dimensional data has created a significant theoretical gap [3,4]. Modern data sources, such as industrial sensor trajectorie, environmental curves, or biomedical signals, are functional objects in infinite-dimensional spaces [4,5,6]. Reducing these data to finite-dimensional vectors often destroys their temporal structures and extreme behaviors, limiting the ability to detect rare functional anomalies, i.e. unusual trajectories or patterns deviating from typical system behavior [4,7]. 

Research on record theory in infinite-dimensional spaces, such as Hilbert and RKHS, remains limited [8]. Classical record models struggle to capture the complex dependencies inherent in high- dimensional data. To overcome this limitation, developing “functional record” approaches inspired by functional data analysis and RKHS-based modeling appears to be a promising direction [8,9]. 

This project aims to establish the mathematical foundations of functional record theory, enabling a deeper probabilistic understanding of extreme patterns in functional data streams. It marks a major departure from the state of the art by extending record theory beyond Euclidean spaces, using tools from functional analysis and RKHS geometry [3,8]. The resulting framework integrates record statistics into functional anomaly detection, a domain where classical record theory falls short [5,10,11,12]. By capturing rare events within the geometry of infinite-dimensional data, it provides interpretable and robust modeling: unlike pointwise approaches, it considers entire trajectories, preserving temporal and geometric structure while leveraging the full informational content. This perspective uncovers rare functional patterns invisible to classical methods [3,7,10,11] and enables the classification of anomalies, from short-term local deviations to persistent or long-term contextual anomalies. Moreover, its solid mathematical formulation simplifies existing models and reduces computational complexity compared to data-driven alternatives, supporting online, unsupervised detection without training. Hence, the project opens a new research direction at the intersection of record theory, extreme value theory, and functional data analysis.

 Applications are broad and impactful. In the industrial sector, functional record theory detects rare operational profiles in sensor networks and cyber-physical systems, and supports intelligent energy monitoring and continuous quality control in complex manufacturing [10,11,12]. In climate science, it offers a novel description of record-breaking temperature or pollution curves [4]. Last but not least, in healthcare, it can identify unusual physiological patterns (ECG curves and brain signals) that pointwise methods overlook [5,12]. These use cases strongly align with IMT’s strategic goals in responsible industry, energy transition, and health engineering.