# Causality in Control Systems Based on Data-Driven Oscillation Identification
## 4 Results
### Results for Industrial Data
#### 4.2.2 The 2 nd Variant  
According to Section 2.3, the median filter is used to shorten the data. Operating with MF where \(n\) divisor equals 10 and determining the Transfer Entropy coefficients for the signal pairs of the whole dataset, only \(NaN\) or \(Inf\) values are obtained. The same situation is repeated for the \(n\) divisor equals 25, so these two variants are completely rejected from the analysis. Correct results in the form of real numbers are obtained for the \(n\) divisors equal to 50 and 100 (except for causality for signal \(u\)--0 for \(Nan\) or \(Inf\)). Tables 8 and 9 show the measure of causal relationships for the oscillations of individual pairs of variables for \(n\) divisor equals 50 and 100, respectively. Due to the lack of consistency and repeatability of the results for the 2 nd variant, it cannot be generalized to the form of only two versions of the causality graph, as in Section 4.2.1. Figure 15 presents causality graphs for oscillations from decomposed detrended signals. Comparing the results obtained as part of the division of datasets according to  
Graphs are similar to each other to a certain extent however, they can not be generalized to a single case. Even worse conclusions are drawn when it comes to the analysis of results presented in Table 9. It can be noticed that the highest TE coefficient value in a row occurs for different pairs of variables within one signal (\(d_{3}\) and \(d_{4}\)). Attention is also drawn to the fact that in the case when the divisor of \(n\) equals 100, for the IMFs marked as \(d_{3}\), for pairs of the variable \(u\) with the variables \(x\), \(y\), and \(z\) there are 0 (\(Nan\) or \(Inf\)). For \(d_{4}\) such a problem does not occur. Nevertheless, causality graphs can be determined and and their form are presented in Figure 15c,d. Graphs are unfortunately so inconsistent from each other that one can not be accepted as a correct result. Due to the occurrence of \(NaN\)'s and \(Inf\)'s adopted as 0, it is impossible to perform the averaging of the entropy coefficients values, as it is possible for the 1 st variant (Tables 6 and 7). As one can see, it is hard to find a single explanation for the above-given industrial example. The process knowledge shows that inferring causality from the highest TE coefficient for oscillations is most likely wrong, and the causality graphs do not reflect the correct relationships. The information contained in the oscillations obtained during the decomposition process for 1 st variant may be insufficient for the successful Transfer Entropy method application or may be lost. The use of the Median Filter has a positive effect on the computational demands. However, further analysis of the oscillations obtained from the minimized datasets does not bring the desired results. These observations clearly show that the analysis of the industrial data differs completely from the theoretical analyzes. Simple relations and approaches developed in the sterile simulated environment do not have to extend towards real applications. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{\(T_{row\to column}\)} & \(u\) & \(w\) & \(x\) & \(y\) & \(z\) \\ \hline \multirow{3}{*}{\(u\)} & \(d_{3}\) & NA & 0.0358 & 0.0395 & **0.1235** & 0.1008 \\  & \(d_{4}\) & NA & 0.0784 & 0.0431 & 0.0662 & **0.0899** \\ \hline \multirow{3}{*}{\(w\)} & \(d_{3}\) & 0.0461 & NA & **0.0997** & 0.0898 & 0.0906 \\  & \(d_{4}\) & **0.0945** & NA & 0.0874 & 0.0675 & 0.0750 \\ \hline \multirow{3}{*}{\(x\)} & \(d_{3}\) & 0 & 0.0541 & NA & 0.1097 & **0.1758** \\  & \(d_{4}\) & 0.0574 & 0.0625 & NA & 0.0679 & **0.1024** \\ \hline \multirow{3}{*}{\(y\)} & \(d_{3}\) & 0 & 0.0741 & 0.0952 & NA & **0.1817** \\  & \(d_{4}\) & **0.1508** & 0.0558 & 0.0587 & NA & 0.0926 \\ \hline \multirow{3}{*}{\(z\)} & \(d_{3}\) & 0 & 0.0946 & 0.1919 & **0.1961** & NA \\  & \(d_{4}\) & 0.0808 & 0.0547 & 0.0888 & **0.0922** & NA \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Transfer Entropy coefficient for 2 nd variant (\(n\) divisor equals 100).