Drift-Aware Online Dynamic Learning for Nonstationary Multivariate Time Series: Application to Sintering Quality Prediction
Authors: Yumeng Zhao, Shengxiang Yang, Xianpeng Wang
Organizations: the National Frontiers Science Center for Industrial Intelligence and Systems Optimization, Northeastern University, Shenyang, China · De Montfort University, Leicester, U.K. · Innovative Artificial Intelligence Cluster, Digital Future Institute, De Montfort University, Leicester, U.K. · Key Laboratory of Data Analytics and Optimization for Smart Industry (Northeastern University), Ministry of Education, Shenyang, China
Accurate prediction of nonstationary multivariate time series remains a critical challenge in complex industrial systems such as iron ore sintering. In practice, pronounced concept drift compounded by significant label verification latency rapidly degrades the performance of offline-trained models. Existing methods based on static architectures or passive update strategies struggle to simultaneously extract multi-scale spatiotemporal features and overcome the stability-plasticity dilemma without immediate supervision. To address these limitations, a Drift-Aware Multi-Scale Dynamic Learning (DA-MSDL) framework is proposed to maintain robust multi-output predictive performance via online adaptive mechanisms on nonstationary data streams. The framework employs a multi-scale bi-branch convolutional network as its backbone to disentangle local fluctuations from long-term trends, thereby enhancing representational capacity for complex dynamic patterns. To circumvent the label latency bottleneck, DA-MSDL leverages Maximum Mean Discrepancy (MMD) for unsupervised drift detection. By quantifying online statistical deviations in feature distributions, DA-MSDL proactively triggers model adaptation prior to inference. Furthermore, a drift-severity-guided hierarchical fine-tuning strategy is developed. Supported by prioritized experience replay from a dynamic memory queue, this approach achieves rapid distribution alignment while effectively mitigating catastrophic forgetting. Long-horizon experiments on real-world industrial sintering data and a public benchmark dataset demonstrate that DA-MSDL consistently outperforms representative baselines under severe concept drift. Exhibiting strong cross-domain generalization and predictive stability, the proposed framework provides an effective online dynamic learning paradigm for quality monitoring in nonstationary environments.
Figures & tables
Fig. 1: Schematic diagram of the sintering process.
Fig. 2: Overall architecture of the DA-MSDL framework. The pipeline integrates Data Preprocessing (Step 1), Drift Detection and Adaptive Fine-tuning (Step 2), and MS-BCNN Feature Extraction (Step 3), supported by the Dynamic Memory Queue and Stable Error Fine-tuning modules.
Describe the raw-material composition and blending conditions before sintering
Process & Thermal Parameters
Ignition temperature; exhaust gas temperature; burn-through point (BTP); main-line and furnace negative pressures; coal-gas flow rate and pressure; air flow rate and pressure
Reflect combustion intensity, heat-transfer behavior, and airflow dynamics
Equipment & Environment
Sintering machine speed; pallet speed; bed height; finished-product weight; production output
Indicate equipment operating status and system-level production conditions
Output Targets
TFe, FeO, SiO 2 , CaO, basicity
Key chemical quality indicators obtained from laboratory assays
TABLE I: Feature categories and representative variables in the sintering dataset.
Fig. 5: MMD-based drift trend during the online detection stage.
Model
NMSE
NMAE
MAPE (%)
DA-MSDL
0.2423±0.1542
0.3052±0.1521
1.09±0.04
OB-ISSID
0.3044±0.2006
0.3809±0.1353
1.31±0.07
Ventingformer
0.2793±0.1989
0.3656±0.1364
1.25±0.07
Transformer
0.4564±0.1766
0.4829±0.1028
1.65±0.17
LSTM
0.3803±0.2204
0.4213±0.1419
1.44±0.08
GRU-PLS
0.4618±0.1707
0.5120±0.0821
1.74±0.08
TABLE II: Overall predictive performance on the shuffled dataset (mean ± standard deviation).
Model
Label
MSE
MAE
MAPE (%)
R2
DA-MSDL
TFe
0.060832 ± 0.006063
0.180206 ± 0.008479
0.320033 ± 0.015002
−0.26131±0.125718
FeO
0.118038 ± 0.006595
0.246699 ± 0.008104
2.682173 ± 0.089205
0.077400 ± 0.051546
SiO 2
0.006607 ± 0.001714
0.056760 ± 0.006537
1.028868 ± 0.117734
0.662256 ± 0.087604
CaO
0.036010 ± 0.003603
0.129945 ± 0.007947
1.092134 ± 0.067053
0.466945 ± 0.053337
Basicity
0.000536 ± 0.000036
0.016871 ± 0.000561
0.784819 ± 0.026099
0.212011 ± 0.053578
Overall Mean
0.768453 ± 0.318736
0.612549 ± 0.145467
1.181605 ± 0.063019
0.231459 ± 0.074357
TABLE III: Predictive performance comparison on the five quality targets (mean ± standard deviation).
Fig. 6: Error scatter distribution illustrating the prediction stability across labels.
Fig. 7: Time-series prediction comparison: true values vs. predicted values for the five quality targets on the test set across different models.
Model
NMSE
NMAE
MAPE (%)
DA-MSDL
0.7693±0.1328
0.5502±0.1954
0.55±0.03
OB-ISSID
1.3479±0.4521
0.6145±0.2545
0.63±0.04
Ventingformer
1.7582±0.9545
0.8336±0.7451
0.85±0.10
Transformer
4.3388±1.5265
1.6894±1.6541
1.71±0.24
LSTM
0.9525±0.4536
0.6247±0.2654
0.64±0.04
GRU-PLS
1.3018±0.5562
0.7286±0.2684
0.74±0.05
TABLE IV: Overall predictive performance on the Water Treatment Plant dataset (mean ± standard deviation).
Fig. 8: Online MAE of MS-BCNN (static) and DA-MSDL (dynamic) on real-world production sequences. Dashed lines mark detected concept drift events where online adaptation is triggered.
Fig. 9: Error difference ( ΔError ) between MS-BCNN and DA-MSDL over time. Persistent positive values indicate consistent gains from dynamic adaptation.
Fig. 10: Component-wise ablation of DA-MSDL measured by overall NMSE. The results show how each component affects prediction performance under nonstationary conditions.
Fig. 11: Recovery-time distribution under different drift severity levels.
Fig. 12: Sensitivity analysis of DA-MSDL with respect to key hyperparameters.