Bidirectional GP-ARX Virtual Sensing for Robust Online Imputation in Unreliable Multi-Sensor Systems

Real-time monitoring and control increasingly rely on low-cost IoT sensor networks, yet these networks are often affected by intermittent availability, ranging from stochastic dropouts to prolonged system-wide blackouts. Existing data-driven imputation methods often struggle with such arbitrary missingness patterns in online settings, either relying on batch processing that violates real-time causality or providing limited probabilistic information for uncertainty-aware decision-making. We present a virtual sensor based on a Bidirectional Gaussian Process Autoregression with exogenous inputs (BGP-ARX) that performs online data imputation and returns probabilistic uncertainty estimates. The architecture jointly models (i) the temporal autoregressive dynamics of each channel and (ii) cross-sensor dependencies via a deep-kernel GP hierarchy, enabling information to flow across variables without prescribing fixed missing patterns. A sequential Bayesian updating scheme supports real-time inference and maintains robustness when any or all sensors are absent for certain durations. Beyond point accuracy, BGP-ARX produces predictive distributions whose probabilistic quality can be assessed through calibration-oriented diagnostics. Extensive experiments on the UCI Air Quality dataset and real-world data in the United Kingdom with controlled missingness evaluate accuracy (RMSE/MAE/$R^2$) and probabilistic quality (NLL/CRPS/ECE) across challenging missingness regimes, including long contiguous gaps and simultaneous blackouts. BGP-ARX performs competitively across the evaluated scenarios and shows its clearest advantage under long contiguous outages and settings where cross-sensor information remains informative. On UCI Air Quality, BGP-ARX achieves competitive one-step forecasting accuracy (e.g., CO RMSE = 81.5), substantially outperforming LSTM/NARX (CO RMSE = 102.5/112.4) while matching strong classical baselines such as RF and independent GP. For long single-channel outages, BGP-ARX is most robust: under a 24-hour contiguous CO gap, it reduces RMSE to 156.3 (vs. 188.9 for RF and 178.6 for BRITS, -17\\% and -12\\%, respectively). These results suggest that the BGP-ARX virtual sensor is a practical probabilistic model for augmenting unreliable multi-sensor systems under prolonged and arbitrary missingness.