Toward In-Depth Mastery of Statistical Properties: Novel Stationary Moment Analysis With Application to Continuous Industrial Anomaly Detection

Anomaly detection is a cornerstone of industrial safety, enabling real-time monitoring of process operations by identifying deviations from normal conditions through statistical analysis. In real-world industrial scenarios, the nonstationary properties of multivariate time-series data present a common and substantial challenge. Existing methods for extracting stationary sources $(\mathcal {SS}s)$ mainly rely on weak stationarity (i.e., mean and variance), but their performance is limited by the long-tailed distributions common in industrial datasets. Higher-order moments, in contrast, provide a more comprehensive statistical description, capturing complex data characteristics that the mean and variance overlook. To bridge this significant gap, we propose a continuous stationary moment analysis (Co-SMA) anomaly detection framework. Its core innovation is the SMA algorithm, which introduces a novel objective function to minimize cumulative sum of the differences in multiorder moments between each epoch and the overall data, effectively fulfilling the $\mathcal {SS}$ estimation task. Furthermore, to overcome the inefficiencies of traditional model updating methods, we develop an event-triggered model updating framework based on the model bias index and first-order perturbation theory. Within this framework, we introduce a convex hull coverage metric, which enables the model to be adjusted efficiently according to the data distribution drift. The framework also incorporates iterative refinement of detection statistics and thresholds, establishing a dynamic adjustment mechanism that ensures optimal performance across diverse operating conditions. The theoretical basis of Co-SMA's properties is rigorously established. Experimental evaluations on numerical simulations and real-world datasets from the ironmaking process demonstrate Co-SMA's superior capabilities in $\mathcal {SS}$ estimation and anomaly detection.