A Metric and Visualization of Completeness in Multi-Dimensional Data Sets of Sensor and Actuator Data Applied to a Condition Monitoring Use Case  
[PERSON]  
1 Institute of Automation and Information Systems, Technical University of Munich, 85748 Garching, Germany; [EMAIL_ADDRESS]  
time- and labor-intensive, leading to high costs. Consequently, condition-based monitoring provides the potential to save resources and money by maintaining control valves only when required. One approach to detect faults in control valves has been introduced by [PERSON] et al., 2019 [PERSON] et al. (2019). Based on data of stroke, pressure (before the valve) and pressure difference (before and after the valve) a prediction model for the valve's flow is trained. Deviations between predicted and measured flow indicate defects of the valve. In particular, degradation due to corrosion or cavitation as well as contamination due to chemical reactions of the valve's surface and the medium passing the valve are considered. The approach is a so-called hybrid approach providing a data-driven model of the valve behavior for model-based fault detection ([PERSON] et al., 2019). Hybrid approaches are most promising ([PERSON] et al., 2019) since they combine the available knowledge regarding the input and output parameters of the system and the available sensor and actuator data to model the details of the system dependencies. However, the accuracy of such approaches is highly dependent on the quality of available training data. The term data quality refers to the characteristics of the data influencing the usefulness of it. The term 'fitness for use' is also used to describe data quality ([PERSON] et al., 2019). In general, quality is defined by 'the degree to which a set of inherent characteristics of an object fulfills requirements' ([PERSON] et al., 2019). Regarding data quality in aPS for model-based data-driven condition monitoring, it is required that the data can represent all characteristics of the considered dependencies of the sensor and actuator data to model the behavior of the aPS. However, overlaying closed-loop control cycles induce dependencies to the sensor and actuator data of aPSs that do not represent any physical behavior. Furthermore, the data is influenced by the humans' routines operating the machines at specific operating points or performing specific recipes to receive a particular product. Consequently, the data is often limited to specific value ranges, even though the number of observations is huge. In control valves, the operating point is often set at a stroke between 70% and 85%. Therefore, industrial data often contains many observations within this value range and less to no observations in other ranges. The histogram of the valve's stroke in Figure 1a reveals two operating points with many observations, but no observations with a stroke smaller than 60%. In the considered timeframe of 3 weeks of normal operation in an industrial process plant, the valve was never fully closed (stroke 0%). Training a condition-monitoring model that covers the whole scope of function of a control valve (fully closed to fully opened) is not possible based on the available, incomplete data. The data does not fulfill the requirements to train an effective condition-monitoring model. Therefore, data quality is low. When considering a multi-dimensional feature space, e.g., stroke and pressure, the limitations to specific combinations of these variables further impair data quality. The feature space describes the multi-dimensional space between the considered variables. In two-dimensional considerations, a scatter plot can visualize the feature space. The scatter plot in Figure 1b reveals that the observations for a pressure between 18 and 19 bar occur with different values of stroke (marked with a gray box). However, higher values for pressure, which might also occur during production, are underrepresented or almost not existing in the data set.