## 3 Imperfect Information Due to Noisy Sensing  
The simplest metric model of measurement error is obtained by assuming the error to be interval-bounded and independent across sensors as well as across time instants of measurements, thus pretending that the error incurred when measuring the same physical quantity by the same sensor at different times is uncorrelated. Sensor-based monitoring under such a model of measurement uncertainty can be realized by an appropriate interval lifting of the STL semantics [12], as standard interval arithmetic (IA) [18] underlying this lifting reflects an analogous independence assumption. This independence assumption, however, is infamously known as the dependency (or alias) problem of interval arithmetic in cases where the independence assumption does not actually apply and IA consequently yields an overly conservative approximation instead [18]. Such overapproximation will obviously also arise when the interval-based monitoring algorithm [12] is applied in cases where the per-sample error of multiple measurements is not fully independent; the overapproximation then shows by reporting inconclusive monitoring verdicts (due to the interval embedding encoded as the inconclusive truth value interval \(\{\bot,\top\}\)) rather than a conclusive truth value  
Dependencies between per-sample measurement errors are, however, the rule and not the exception. As a typical example take the usual decomposition of measurement error into a confounding unknown yet fixed sensor offset that remains constant across successive measurements taken by the same sensor, and a random measurement error that varies uncorrelated between samples at different time instants. The upper bounds of these two values refer directly to the two terms \"trueness\" and \"precision\" used by the pertinent ISO norm 5725 to describe the accuracy of a measurement method. They are consequently found routinely in data sheets of sensor devices, which we consider to be the contracts between component (i.e., sensor) manufacturer and component user (i.e., the monitor designer) in the sense of contract-based design [19], implying that all subsequent logical inferences we pursue are relative to satisfaction of the contract by the actual sensor. Within the ISO parlance, precision identifies the grouping or closeness of multiple readings,i.e., the portion of the total error that varies in an unpredictable way between tests or measurements. In contrast, trueness indicates the closeness of the average test results to a reference or true value in the sense of the deviation or offset of the arithmetic mean of a large number of repeated measurements and the true or accepted reference value. **Definition 3**. : _Let \(S\) be a sensor observing a signal \(\sigma\in\mathit{Var}\) at times \(T\subseteq\mathbb{N}\) with a maximal sensor offset of \(\varepsilon\geq 0\) and a maximal random measurement error of \(\delta\geq 0\). Let \(\tau\) be a (ground-truth) trajectory. Then \(m_{S}:T\to\mathbb{R}\) is a possible \(S\) time series over \(\tau\) iff_  
\[\exists o\in[-\varepsilon,\varepsilon]:\forall t\in T:\exists e\in[-\delta, \delta]:\tau(\sigma)(t)+o+e=m_{S}(t). \tag{1}\]  
_If \(m_{S}\) is an \(S\) time series over \(\tau\), then we symmetrically say that the trajectory \(\tau\) is consistent with \(m_{S}\) and denote this fact by \(m_{S}\vdash\tau\). This notion immediately extends to simultaneous consistency with a set of time series \(m_{S_{1}},\,m_{S_{2}}\), to \(m_{S_{n}}\): we denote the fact that trajectory \(\tau\) satisfies \(m_{S_{i}}\vdash\tau\) for each \(i\in\{1,\dots,n\}\) by \(m_{S_{1}},\dots,m_{S_{n}}\vdash\tau\)._  
Note that the above definition features two additive offsets affecting measurements, the first of which (namely the sensor offset) is uniformly chosen for the whole time series while the second one (the random noise) is chosen independently upon every sample.