SPLIT${}^3$: Separating Test Time Training Plasticity for Anomaly Detection and Localization

Multivariate time series (MTS) anomaly diagnosis requires identifying both when anomalies occur and which variables are associated with them. Test Time Training (TTT) continuously adapts a predictive model at test time, but detection and localization can favor different levels of adaptation. We derive an exact recurrence describing how differences in TTT update rates propagate through the evolving fast state and relate the resulting prediction-error differences to the distinct ranking structures of detection and localization. To allow detection and localization to follow task-specific adaptation trajectories within the same trained predictive model, we propose Separated PLasticity In Test Time Training (SPLIT${}^3$), which shares the encoder and learned initialization while maintaining separate fast states with task-specific update rates. Experiments on the SMD, SWaT, and WaDI datasets consistently show that detection favors lower update rates than localization. Across multiple MTS benchmarks, SPLIT${}^3$ achieves strong localization performance while maintaining competitive detection performance.