## 5 Application and Discussion of the Proposed Metric and Visualization  
The proposed metric is evaluated based on the use case of condition monitoring of control valves. In this paper, a model-based data-driven approach is chosen. Expert knowledge is used to identify a valve's relevant input and output parameters (model-based) and data is used to train the dependencies between these parameters (data-driven). In this way, available expert knowledge is utilized, but detailed physical modeling involving fluid mechanics is avoided. In literature, also other approaches are proposed. Signal-based approaches, for example, observe the characteristics of individual signals to detect deviation of normal behavior. Leakage of valves is a common problem that is detected based on the characteristics of the acoustic [48; 49; 50] or pressure [51] signal. However, the system dynamics are not considered in signal-based approaches, making them vulnerable to regular changes and movements, raising false alarms. Also, classification algorithms are used, which require a labeled training data set, including fault-free and faulty data. Several signals and combinations of signals serve to identify the patterns, which separate fault-free and faulty data. Acoustic and vibration data [52; 53; 54] or parameters such as pressure, temperature and stroke [55] provide features for classification. Furthermore, novelty detection algorithms are applied to identify changes in valve behavior. Neural networks are the most common algorithms for novelty detection [56; 57]. Besides, also one-class support vector machines are applied [58]. These methods cause high computational costs for the identification of the best hyper-parameters. In contrast, model-based data-driven approaches reduce complexity in the training process due to the input of experts regarding the input and output parameters and their relevant features. Therefore, this approach is promising for a broad application in the industry. The approach in this paper aims at predicting the flow of the valve \(q\) considering the valve's stroke \(s\), pressure \(p\) as well as the pressure difference \(\Delta p\) as input variables [2]. The residuals of the predicted and the actual flow serve as a feature for anomaly detection. High residuals reveal a change of the dependencies between the variables and indicate an anomaly. The relevance of the variables stroke, pressure and pressure difference is explained easily: a higher stroke increases the cross-section of the valve and a higher flow can pass the valve. Analogous to this, a higher pressure or pressure difference leads to a higher flow. A data-driven model trains these dependencies to predict the flow \(q\) of the valve given the values for stroke \(s\), pressure \(p\) and pressure difference \(\Delta p\). Figure 8: Workflow for the assessment of data set completeness in sensor and actuator data of aPSs. The available data set was gathered at test runs. Experts designed a procedure that covers a wide range of the scope of function for the considered control valve. This enables data-driven models to learn the valve behavior. The test run was repeated 10 times to include the normal volatility of the data in the data-driven model. In total, \(27,000\) observations with a sampling rate of 0.5 s were considered in the following. It was expected that the completeness of the data set is high. For reasons of anonymization, the variables are normalized between 0 and 1. A train-test-split was performed, isolating 30% of the data for test purposes. Prior to evaluating the proposed metric and visualization for data set completeness, the descriptive statistics of the training data set are discussed. The stroke \(s\) is equally distributed between the minimum and maximum (cf. Figure 9). The distribution shows a mean of 0.50 and a standard deviation of 0.29. The pressure \(p\) and the pressure difference \(\Delta p\) clearly indicate four different operating points, which were caused by the design of the test runs (cf. histograms of Figure 9).