# Causality in Control Systems Based on Data-Driven Oscillation Identification
## 4 Results
### Results for Simulation Data  
Simulation data is decomposed into 7 IMFs for both variants (data length set to 2000 samples for each \(\epsilon_{i}\)). In Figure 10a there are results presented for variable \(\epsilon_{3}\) with Gaussian noise added. Oscillations are barely visible in IMFs \(d_{4}\), \(d_{5}\), \(d_{6}\), and \(d_{7}\). IMFs designated as \(d_{1}\), \(d_{2}\), and \(d_{3}\) are considered as noise. The situation is different in the case of the second variant--the oscillation phenomenon is much more visible for the variant with Cauchy noise and sinusoidal signal (of known frequency) added (Figure 10b). Due to the above, research is conducted on the last four IMF functions. Figure 9: Histograms for time series of the industrial data with PDF fitting. (**a**) Control error \(u(k)\), (**b**) Control error \(w(k)\), (**c**) Control error \(x(k)\), (**d**) Control error \(y(k)\), (**e**) Control error \(z(k)\). Tables 2 and 3 show the values of the Transfer Entropy coefficients for simulation data with Gaussian noise and Gaussian and Cauchy noises respectively. An analysis of the determined TE coefficients values for simulation data with Gaussian noise shows that after the decomposition process, mostly \(NaN\) or \(Inf\) values type appear (considered as 0). According to the calculated results, in this case, it is impossible to determine the causality graph. There can be many reasons for such poor behavior The most probably is that Gaussian disturbance is too weak and does not stimulate the simulation system sufficiently. The oscillations detected in the decomposition process are negligible, as shown in Figure 10a. Therefore, the Transfer Entropy approach is ineffective and the determined values of the coefficients should be completely rejected. Figure 10: Oscillation decomposition sample using MEEMD algorithm for the variable \(e_{3}\) in simulated data. The first subplot shows original data (Data) and the consecutive plots show decomposed frequencies and the residuum in the last subplot. (**a**) Gaussian noise case, (**b**) Gaussian noise, and Cauchy disturbance case. According to the analysis of TE coefficients for oscillations with Gauss and Cauchy noises, the effects of the calculations are completely different and the causality graph can be successfully determined. It is presented in Figure 11. In the case of all IMF functions for individual \(e_{i}\), the largest values of the Transfer Entropy coefficients appear for the same \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{\(T_{row\to column}\)} & \(e_{1}\) & \(e_{2}\) & \(e_{3}\) & \(e_{4}\) & \(e_{5}\) \\ \hline \multirow{3}{*}{\(e_{1}\)} & \(d_{4}\) & NA & 0 & 0 & 0 & 0 \\  & \(d_{5}\) & NA & 0 & 0 & 0 & 0.0535 \\  & \(d_{6}\) & NA & 0 & 0 & 0.0508 & 0.0561 \\  & \(d_{7}\) & NA & 0 & 0 & 0.0779 & 0.0801 \\ \hline \multirow{3}{*}{\(e_{2}\)} & \(d_{4}\) & 0 & NA & 0 & 0 & 0 \\  & \(d_{5}\) & 0 & NA & 0 & 0 & 0 \\  & \(d_{6}\) & 0 & NA & 0 & 0.0420 & 0.0607 \\  & \(d_{7}\) & 0.0651 & NA & 0 & 0.492 & 0.0561 \\ \hline \multirow{3}{*}{\(e_{3}\)} & \(d_{4}\) & 0 & 0 & NA & 0 & 0 \\  & \(d_{5}\) & 0 & 0 & NA & 0 & 0 \\  & \(d_{6}\) & 0 & 0 & NA & 0.0390 & 0.0599 \\  & \(d_{7}\) & 0.0697 & 0 & NA & 0.0528 & 0.0480 \\ \hline \multirow{3}{*}{\(e_{4}\)} & \(d_{4}\) & 0 & 0 & 0 & NA & 0 \\  & \(d_{5}\) & 0 & 0 & 0 & NA & 0 \\  & \(d_{6}\) & 0.0515 & 0 & 0 & NA & 0.0563 \\  & \(d_{7}\) & 0.0807 & 0 & 0 & NA & 0.0597 \\ \hline \multirow{3}{*}{\(e_{5}\)} & \(d_{4}\) & 0 & 0 & 0 & 0 & NA \\  & \(d_{5}\) & 0.0340 & 0 & 0 & 0 & NA \\  & \(d_{6}\) & 0 & 0 & 0 & 0.0626 & NA \\ \cline{1-1}  & \(d_{7}\) & 0.0644 & 0 & 0 & 0.0452 & NA \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Transfer Entropy coefficients for simulated data (Gaussian noise).