This paper comprises a description of a data-driven approach to the real-time monitoring of a physical system. Specifically, a hierarchical anomaly detection algorithm that can identify both instantaneous pointwise anomalies and gradual trajectory anomalies is proposed. To detect anomalies, we first construct a multioutput Gaussian process regression (MOGPR) model that can predict, probabilistically, the outputs of the target system. Using the constructed prediction model, we then propose the statistical decision-making strategies to determine the abnormal operations of the target system by comparing its measured and the predicted responses. For pointwise anomaly detection, we regard a single measurement as abnormal if the difference between the measurement and the prediction exceeds the threshold based on an extreme value theory. For the trajectory anomaly detection, we consider a sequence of measurements abnormal if the Mahalanobis distance between the measured and predicted trajectories is highly improbable. The proposed monitoring strategy does both the pointwise and the trajectory anomaly detection in a single framework. The proposed strategy was applied to detecting abnormal operations of gas regulators. Validating with the actual gas regulator data demonstrated that it could identify the anomalies robustly and accurately.