# Causality in Control Systems Based on Data-Driven Oscillation Identification
## 4 Results
### Results for Industrial Data  
In the case of industrial data, due to the length of time series, each signal from the detrended dataset can be decomposed into multiple IMFs. However, it is unacceptable. In the literature, it can be found that only the first five IMFs are selected for analysis, as the remaining ones being the sum of non-significant IMFs. This assumption is too subjective and unjustified to ignore the fact that there are more IMFs obtained for such a long time series. Whereby an application of the MEEMD algorithm is computationally demanding and the following assumptions for industrial data are stated: first analyzed variant is that the whole dataset is split into 50 (and respectively 49) sub-datasets of equal lengths in two ways (\(S_{1}=[s_{11},s_{12},\ldots]\) and \(S_{2}=[s_{21},s_{22},\ldots]\)), as it is shown in the Figure 12. Such overlapping of the obtained sub-datasets will allow for checking the potentially emerging frequency bands that are key in given considerations. The second variant is to shorten the dataset by using Median Filter (MF) due to the mentioned enormous number of IMFs and savings resulting from computation time. This operation is widely described in Section 2.3. Such action is aimed not only at comparing variants of oscillations with different frequencies on the same data but also at confronting the statement that the use of a Median Filter will delete (filter) higher frequencies so they are not visible. For simulation data, the above does not apply because it is possible to generate time series of a required length. Figure 11: Causality diagram for simulated data—Gaussian noise and Cauchy disturbance. Figure 12: Decomposition of signal \(u\) into sub-datasets for the oscillation identification process. As a result, industrial datasets are divided into 50 (and 49) overlapping sub-datasets successively. Next the MEEMD algorithm is applied to decompose individual signals from the control loops. Sample data for 1 st variant is presented in Figure 13. It shows the MEEMD results for a randomly selected pair of overlapping sub-datasets (in this case for signal \(y\)). It can be noticed that there are six IMFs functions (\(d_{1}\) to \(d_{6}\)) for both given sub-dataset lengths in variant 1 st. The first three IMFs are considered as noise. The most interesting ones are the IMFs denoted as \(d_{4}\), \(d_{5}\), and \(d_{6}\) for which actual oscillation occurs. Through analysis of Figure 13 it is confirmed that due to this operation emerging frequency bands are not skipped. According to the multiple numbers of sub-datasets both for 1 st and 2 nd variants, it is impossible to show all the MEEMD algorithm operation results. Figure 13: Oscillation decomposition using MEEMD algorithm for randomly selected pair of overlapping sub-datasets for \(y\) signal. The first subplot shows original data (Data) and the consecutive plots show decomposed frequencies and the residuum in the last subplot. **(a)** Results for 32 of 50 subdatasets (\(d_{132}\)), **(b)** Results for 32 of 49 subdatasets (\(d_{233}\)). The use of the median operator is justified through the fact that industrial control errors time series are in the majority non-Gaussian fat-tailed [28]. Therefore, a robust shift estimator is required. Median is the simplest robust estimator. It holds a breakdown point of 50% [20], meaning that half the points must be outlying observations before the median can be moved outside the inlying range and thus it is used. Due to the large amount of data, it is impossible to illustrate the causality in the form of Transfer Entropy coefficients for each dataset. For both variants, the relationship between every single signal (variable) (\(u\), \(w\), \(x\), \(y\), \(z\)) with the others is determined. It is noted that no universal threshold method for the Transfer Entropy approach has been developed.