Local Anomaly Detection

Anomalies with spatial and temporal stamps arise in a number of applications including communication networks, traffic monitoring and video analysis. In these applications anomalies are temporally or spatially localized but otherwise unknown. We propose a novel graph-based statistical notion that unifies the idea of temporal and spatial locality. This notion lends itself to an elegant characterization of optimal decision rules and in turn suggests corresponding empirical rules based on local nearest neighbor distances. We compute a single composite score for the entire spatio-temporal data sample based on the local neighborhood distances. We declare data samples as containing local anomalies based on the composite score. We show that such rules not only asymptotically guarantee desired false alarm control but are also asymptotically optimal. We also show that our composite scoring scheme overcomes the inherent resolution issues of alternative multi-comparison approaches that are based on fusing the outcomes of location-by-location comparisons. We then verify our algorithms on synthetic and real data sets.