  Predictive monitoring -- making predictions about future states and
monitoring if the predicted states satisfy requirements -- offers a promising
paradigm in supporting the decision making of Cyber-Physical Systems (CPS).
Existing works of predictive monitoring mostly focus on monitoring individual
predictions rather than sequential predictions. We develop a novel approach for
monitoring sequential predictions generated from Bayesian Recurrent Neural
Networks (RNNs) that can capture the inherent uncertainty in CPS, drawing on
insights from our study of real-world CPS datasets. We propose a new logic
named \emph{Signal Temporal Logic with Uncertainty} (STL-U) to monitor a
flowpipe containing an infinite set of uncertain sequences predicted by
Bayesian RNNs. We define STL-U strong and weak satisfaction semantics based on
if all or some sequences contained in a flowpipe satisfy the requirement. We
also develop methods to compute the range of confidence levels under which a
flowpipe is guaranteed to strongly (weakly) satisfy an STL-U formula.
Furthermore, we develop novel criteria that leverage STL-U monitoring results
to calibrate the uncertainty estimation in Bayesian RNNs. Finally, we evaluate
the proposed approach via experiments with real-world datasets and a simulated
smart city case study, which show very encouraging results of STL-U based
predictive monitoring approach outperforming baselines.
