A Multi-Scale Time-Series Anomaly Detection Approach for Modeling the Time-Lagged Effects of Exogenous Variables
Yuanzhao Shang1, Xin Liu1,2,*, Fengbiao Zan1
1 School of Intelligence Science and Engineering, Qinghai Minzu University, Xining, China
2 Qinghai Provincial Key Laboratory of IoT, Xining, China
* Corresponding Author: Xin Liu. Email: 
Computers, Materials & Continua https://doi.org/10.32604/cmc.2026.084688
Received 27 April 2026; Accepted 26 June 2026; Published online 16 July 2026
Abstract
Multivariate time-series anomaly detection is widely used to identify abnormal operating patterns in complex monitoring systems. Exogenous or contextual inputs can provide useful information for anomaly detection, but their effects on endogenous variables may appear with temporal delays. Direct synchronous modeling is therefore insufficient for capturing delayed response patterns. This study proposes EMS-uDTWAD, a multi-scale anomaly detection framework that combines explicit lag alignment with uncertainty-aware normal-pattern matching. First, the time-series data are divided into fine-grained and coarse-grained windows. At the coarse-grained scale, a lag alignment module estimates delayed responses from exogenous or contextual inputs to endogenous variables and constructs lag-enhanced joint windows. These lag-enhanced windows are then matched against a normal reference library using uncertainty-aware dynamic time warping (uDTW). Thus, uDTW is used for global normal-pattern matching rather than for directly estimating inter-channel time lags. At the fine-grained scale, a graph embedding module models local structural dependencies and cross-channel correlations through a learned adjacency representation. Finally, the anomaly scores from the two branches are fused to obtain the final detection result. Experiments are conducted on two real-world datasets, MSDS and SWaT, and one synthetic Linear Dataset. MSDS is used as a proxy auxiliary-context setting, SWaT is used as a real industrial operational-context setting, and the Linear Dataset provides native exogenous variables. Across these settings, EMS-uDTWAD improves the F1 score over the strongest compared baseline by 2.0 percentage points on MSDS, 1.0 percentage point on the Linear Dataset, and 0.8 percentage points on SWaT. The ablation results further show that the lag-enhanced representation is more useful than direct synchronous concatenation when delayed contextual responses are present.
Keywords
Time series anomaly detection; exogenous and contextual inputs; time-lag modeling; uncertainty-aware dynamic time warping
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