## 4 The Concept for the Assessment of Data Set Completeness
### Assumptions and Prerequisites  
Carefully performed data preparation is required before the assessment of data set completeness. This includes identifying relevant data, the data gathering and integration as well as the assessment of other data quality dimensions. It is assumed that possible data quality defects of other data quality dimensions are rectified. Missing synchronicity of sensor and actuator data, for example, leads to a misfit of the time reference. While the observation of one sensor is recorded at time \(t\), the observation of another sensor is recorded at time \(t+1\). In this case, it is not possible to bring both sensors into relation. This needs to be solved before the assessment of data set completeness. Furthermore, it is  
\begin{table}
\begin{tabular}{c l l l} \hline \hline
**Parameters** & & & & \\ \hline \(\overrightarrow{I}\) & Data set including \(n\) variables & \(K_{j}\) & Number of sections of class \(j\) \\ \(n\) & Number of variables & \(\mu_{j}\) & Number of minority \(\sec\)tions in class \(j\) \\ \(\overrightarrow{X}_{i}\) & Variable \(i\) including \(m\) observations & \(\text{DSC}_{j}\) & Data set completeness of class \(j\) \\ \(m\) & Number of observations & \(SV_{i}\) & Section volume of class \(j\) \\ \(d\) & Distance of two vectors & \(SB_{j}\) & Section balance of class \(j\) \\ \(\eta\) & Number of observations in one section & \(Min\) & Minimum value \\ \(\overrightarrow{v}_{j}\) & Class vector of class \(j\) & \(Max\) & Maximum value \\ \(\overrightarrow{s}\) & Variable vector & \(Mean\) & Mean value \\ \(\overrightarrow{v}_{j}\) & Equally distributed class vector of class \(j\) & \(SD\) & Standard deviation \\ \(\overrightarrow{a}_{j}\) & Worst distributed class vector of class \(j\) & \(RSD\) & Relative standard deviation (SD/Mean) \\ \(K\) & Number of sections & \(IQR\) & Interquartile range (3 rd quartile–1 st quartile) \\ \(k_{i}\) & Number of classes of variable \(i\) & & & \\  & Variables & & & \\ \(q\) & Flow & \(i\) & Variable \\ \(s\) & Stroke & \(j\) & Class \\ \(p\) & Pressure upstream (before the valve) & & & \\ \(\Delta p\) & Pressure difference before and after the valve & & & \\ \(p_{s}\) & Signal pressure & & & \\ \hline
**Terms** & & & \\ \hline variable & A variable represents one sensor or actuator of an aPS. The values of a variable are numerical, either continuous or discrete. & \\ \multirow{2}{*}{class} & A class describes a specific value interval of a variable. Classes do not overlap but border to each other and cover the whole value range of a variable. & \\ \multirow{2}{*}{section} & A section represents an area of the \(n\)-dimensional feature space described by the respective classes of the \(n\) variables. & \\ \multirow{2}{*}{feature space} & A section is an n-dimensional bin of the feature space. & \\ \multirow{2}{*}{space} & The feature space is the n-dimensional space, which is created between the variables. & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Nomenclature for the concept of data set completeness.  
assumed that the data set includes all relevant variables required to solve the task at hand, meaning high Schema Completeness [16] is given. Also, the variables of this data set are supposed to be independent and high utilization of the feature space is expected. For dependent sensor data, e.g., the positively correlated stroke \(s\) and the signal pressure of a control valve \(p_{s}\), a fully utilized features space is not possible since a high signal pressure causes a high stroke. Data with high signal pressure and low stroke is unfeasible and not represented in the data set. Based on these assumptions and prerequisites, the data set completeness can be assessed using the following metric.