## 1 Introduction  
Precise and automatic monitoring of the satisfaction of safety constraints imposed on cyber-physical systems is of utmost importance in a variety of settings: traditionally, it facilitates offline or, if supported by the monitoring algorithm, online system debugging as well as, if pursued online in real-time, the demand-driven activation of safety and fallback mechanisms in safety-oriented architectures as soon as a safety-critical system leaves its operational domain or exposes unexpected behavior. An application domain of growing importance is the safety assurance of autonomous systems, such as unmanned aircraft. Such systems are increasingly equipped with decision-making components that carry out complex missions in areas such as transport, mapping and surveillance, and agriculture. In such applications the monitor plays a critical role in assessing system health conditions (such as sensor cross-validation) and regulatory constraints like geo-fencing, which prevents the aircraft from entering protected airspace [1]. More recently, continuous diagnosis in continuous agile development processes like DevOps has caught interest and provides a further field of application [2]. Of special interest here is the provisioning of flexible languages for the specification of monitors, as the pertinent safety constraints vary tremendously across systems and application domains. Answering this quest, Signal Temporal Logic (STL) [3] and similar linear-time temporal logics have been designed for classifying the time-dependent signals originating from continuous-state or hybrid-state dynamical systems according to formal specifications, alongside efficient stream processing languages targeted towards online monitoring [1]. These highly expressive specification languages do, however, induce the follow-up quest for efficient automatic implementation of monitoring algorithms by means of translation from the formal safety or monitoring specifications. There consequently is a rich body of work on synthesis of monitors from logical specifications of temporal or spatio-temporal type (cf. [4] for an overview), with nowadays even robust industrial tools being available [5], as well as hard real-time capable stream-based execution mechanisms for on-line monitoring of even more expressive monitoring languages [1]. Most of the suggested algorithms do, however, not address the problem of epistemic uncertainty due to environmental sensing, with the monitoring algorithms rather taking sensor values and timestamps as is and ignoring their inherent imprecision. Such imprecisions are unavoidable in applications such as autonomous aircraft due to wind and other external influences. A notable exception is provided by robust quantitative interpretations of temporal logic, which can cope with inaccuracy in timestamps [6] as well as in sensor values [7]. The corresponding robust monitoring approaches [8] support a metric, yet not stochastic, error model, and consequently ignore the fact that repeated measurements provide additional evidence, thus ignoring the wisdom and toolset from metrology concerning state estimation [9; 10], consequently providing extremely pessimistic verdicts [11]. Overcoming the latter problem would require equipping the pertinent logics, like Signal Temporal Logic [7], with a truly stochastic (i.e., reporting a likelihood of satisfaction over a stochastic model) rather than a trace-based metric semantics (reporting slackness of the signal values observed across a single trace towards change of truth value of the formula). This remains the subject of our further research. In this article, we do nevertheless show that already in a metric setting of interval-bounded measurement error, as employed in [12], refined algorithms addressing the relation between successive measurements are possible. [PERSON] et al.