# Causality Analysis with Different Probabilistic Distributions Using Transfer Entropy
## 5 Conclusions and Further Research  
The presented considerations aim to improve the model-free Transfer Entropy method by altering a key element in the causality detection approach. The original method utilizes the Gaussian probability distribution function and considers the process data as it is. In this paper, we have hypothesized that the identification process depends on the PDF fitting; thus, we propose a modification using common PDFs such as the Cauchy, \(\alpha\)-stable, Laplace, Huber, and t-location scale distributions. It is challenging to compare the TE method with other approaches. Though minor modeling assumptions are made in estimating the transfer entropy in this paper, in general the TE approach itself does not require a model of the analyzed process, while other methods typically assume process models. However, process models are seldom known in the process industry, and any misfitting in modeling can lead to confusion about whether responsibility for any particular issue lies with the model or the approach used for causality detection. Therefore, methods requiring model assumptions \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(T_{\text{equit}\rightarrow\text{coalumn}}\) & \(\epsilon_{1}\) & \(\epsilon_{2}\) & \(\epsilon_{3}\) & \(\epsilon_{4}\) & \(\epsilon_{5}\) \\ \hline \(\epsilon_{1}\) & \(NA\) & 0.1154 & 0.0729 & 0.0599 & 0.0601 \\ \(\epsilon_{2}\) & 0.0561 & \(NA\) & 0.0540 & 0.0491 & 0.0666 \\ \(\epsilon_{3}\) & 0.0568 & 0.0475 & \(NA\) & 0.0699 & 0.0854 \\ \(\epsilon_{4}\) & 0.0772 & 0.0284 & 0.0521 & \(NA\) & 0.0380 \\ \(\epsilon_{5}\) & 0.0643 & 0.0695 & **0.0888** & 0.0623 & \(NA\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Calculated TE coefficients based on the \(\alpha\)-stable distribution for the dataset with both Gaussian noise and Cauchy disturbance. Figure 14: Causality diagram for the dataset with both Gaussian noise and Cauchy disturbance using the Transfer Entropy approach based on the \(\alpha\)-stable distribution. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(T_{\text{equit}\rightarrow\text{coalumn}}\) & \(\epsilon_{1}\) & \(\epsilon_{2}\) & \(\epsilon_{3}\) & \(\epsilon_{4}\) & \(\epsilon_{5}\) \\ \hline \(\epsilon_{1}\) & \(NA\) & 0.1143 & 0.0743 & 0.0608 & 0.0631 \\ \(\epsilon_{2}\) & 0.0589 & \(NA\) & 0.0556 & 0.0490 & 0.0653 \\ \(\epsilon_{3}\) & 0.0568 & 0.0475 & \(NA\) & 0.0699 & 0.0854 \\ \(\epsilon_{4}\) & 0.0285 & 0.0374 & **0.0547** & \(NA\) & 0.0419 \\ \(\epsilon_{5}\) & 0.0685 & 0.0694 & **0.0886** & 0.0646 & \(NA\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Calculated TE coefficients based on Gaussian distribution for the dataset with both Gaussian noise and Cauchy disturbance. Figure 13: Causality diagram for the dataset with both Gaussian noise and Cauchy disturbance using the Transfer Entropy approach based on the Gaussian distribution. are not considered here, as the modeling effort would lead to increased complexity without any positive trade-offs. In this study, we have examined two variants of simulated data, the first including distortion by Gaussian noise and the second including distortion by both Gaussian noise and Cauchy disturbance. In addition, a sinusoidal signal was added to the Cauchy disturbance to simulate known frequency loop oscillations and their propagation. The time series data represent regulation errors (\(\epsilon_{1}\), \(\epsilon_{2}\), \(\epsilon_{3}\), \(\epsilon_{4}\), and \(\epsilon_{5}\)). Due to the high computational complexity of the TE approach, we used short datasets for analysis, with the main focus on using the modified Transfer Entropy algorithm with the collected data in order to generate causality graphs. The complexity of causality analysis is highlighted in the paper, as there is no single solution that can be applied to every case. The study considers two simulation examples. The first example was analyzed using the TE approach based on the \(\alpha\)-stable distribution, and was found to workseffectively.