## 4 Results  
In this section, a case study showing how the proposed innovative procedure can be implemented for predicting the RUL based on streaming data is introduced. The component under analysis is a critical part of an automatic machine, whose malfunctioning strongly affects the quality of the products. The component is made of four suction cups, which rotate and push the product forward. The time length of a cycle is 0.134 s. The main problem that is related to this component is that, at each cycle, the pressure of the suction cups on the product decreases until one or more of the suction cups becomes detached. It has been noted that if one only suction cup becomes detached, then the quality of the resulting product is still acceptable; however, if two suction cups become detached, then the component is not able to perform its function properly. The measure that best describes the correct functioning of the component under analysis is the pressure. Therefore, the pressure (unit: bar) is collected at a sampling frequency of 10 kHz, under three conditions, named Nominal, Fault 1, and Fault 2, where Fault 1 represents the state in which only one suction cup is missing and Fault 2 represents two suction cups missing. The pressure values in the different conditions are shown in Figure 7. Note that the test during which data was collected was not performed for maintenance purposes. Thus, the choice of the sampling frequency was derived from other considerations. Figure 6: Example of the implementation of the proposed methodology after a fault has been detected. The first step was to identify the segment length for the feature extraction, i.e., the number of data samples, \(k\) in the time window \(t\), from which features are extracted. Subsequently, a decision regarding which features to extract was made. Time-domain features are the most appropriate in streaming applications. Therefore, a brief analysis was made of the most typical time features and the results showed that the most discriminant features are, in this case, the mean, variance, and minimum peak, as shown in Figure 8a. Subsequently, a synthetic value of the extracted features is computed, as follows:  
\[\text{F}=\sqrt{\text{F}_{1}^{2}+\text{F}_{2}^{2}+\text{F}_{3}^{2}} \tag{21}\]  
This value is shown in Figure 8b and it represents the value on which the anomaly detection is based. Because the performance of the proposed procedure mostly depends on the ability to detect anomalies, we first validate anomaly detection and ADP algorithms on the described dataset, treating it as a streaming dataset. In this case, two anomalies were detected, as expected. In Table 1, the time instants at which the algorithm recognized the occurrence of a change in the data structure are summarized. Figure 8: (**a**) Features extracted in the time-domain: F1, F2, F3. (**b**) Synthetic value computed by the extracted features: F.  
Figure 7: (**a**) Signals related to three different conditions. (**b**) Complete dataset containing all conditions. We can conclude the algorithm was able to correctly identify two behavior changes, as shown by the red dots in Figure 9a with latency times of 0.75 and 0.808 s. In addition, as shown in Figure 9b, three clusters were created, which confirms that three different conditions were present in the dataset. Because actual data clusters (labels) are available, the confusion matrix can be used in order to evaluate the performance of clustering (Figure 10). This is a graphic representation that shows the number of observations correctly assigned to a data cloud (true positives) versus the number of misclassifications. The presence of 10 data samples erroneously clustered is due to the latency time for detecting the two anomalous states. The mean value of data samples is recursively updated at each time stamp during the execution of the anomaly detection algorithm. In this case, this value decreases as time passes and the system degrades, i.e., it is a monotonic function.