# Causality Analysis with Different Probabilistic Distributions Using Transfer Entropy
## 4 Analysis of Simulation Results
### Causality for the Dataset with Gaussian Noise  
In this subsection, the simulation data impacted by the Gaussian noise is analyzed (see Figure 3). The relationships between five regulation errors of a simulation system are designated using the Transfer Entropy approach based on the Gaussian distribution and \(\alpha\)-stable distribution.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Cauchy** & **\(\alpha\)-Stable** & **Laplace** & **Huber** & **t Location-Scale** \\ \hline \(\epsilon_{1}\) & 3.3440 & **1.4925** & 2.4805 & 1.5098 & 1.5333 \\ \(\epsilon_{2}\) & 3.6252 & 1.3049 & 2.7709 & **1.3007** & 1.4069 \\ \(\epsilon_{3}\) & 3.3966 & **1.4344** & 2.2947 & 1.5182 & 1.5246 \\ \(\epsilon_{4}\) & 3.5771 & **1.9684** & 2.8030 & 1.9850 & 1.9877 \\ \(\epsilon_{5}\) & 3.1062 & 1.9417 & 2.5749 & 1.9473 & **1.8361** \\ \hline \hline \end{tabular}
\end{table}
Table 2: Fit indicators for the dataset with Gaussian noise.  
Figure 11: Histograms with PDF fitting for dataset with Gaussian noise and Cauchy disturbance (control error \(\epsilon_{5}\)).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Cauchy** & **\(\alpha\)-Stable** & **Laplace** & **Huber** & **t Location-Scale** \\ \hline \(\epsilon_{1}\) & 5.1540 & 5.7679 & 5.5035 & 5.6962 & 5.7589 \\ \(\epsilon_{2}\) & 5.0164 & 5.0727 & **4.9170** & 4.9768 & 4.9432 \\ \(\epsilon_{3}\) & 3.1914 & **1.9279** & 2.9657 & 2.1143 & 2.1835 \\ \(\epsilon_{4}\) & 3.9075 & 2.2738 & 3.6509 & 2.3228 & 2.3033 \\ \(\epsilon_{5}\) & 3.5362 & **3.1455** & 3.8385 & 3.4425 & 3.1752 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Fit indicators for the dataset with both Gaussian noise and Cauchy disturbance.  
Results for simulation data impacted by Gaussian noise for the classic TE approach are impossible to present in the form of a table, as most values appear as \(NaN\)-type or \(Inf\)-type values. These are considered to be 0, which indicates no causality. There are two possible reasons for such poor behavior. The first is that the Gaussian distribution does not sufficiently fit the data. The second and more probable reason is that Gaussian disturbance is too weak, and does sufficiently not stimulate the simulation system.  
When using the Transfer Entropy method based on the \(\alpha\)-stable distribution, the results turn out to be satisfactory. The TE coefficients are obtained, and are included in Table 4. Because there is no objective method for applying a threshold value to the obtained entropy values, the highest entropy value in each row is marked in blue and bold. Although the above assumption forces us to show only the direct relationships, it is nonetheless possible to draw the causality graph, which is provided in Figure 12. The analysis of these results allows us to conclude that the change in the probability distribution has a positive effect on the quality of the Transfer Entropy method. Comparing the causality graph with the assumed one presented in Figure 5, it is inconsistent only in the case of the direction of the relationship between the regulation errors \(\epsilon_{1}\) and \(\epsilon_{5}\); in all other cases the causality is determined correctly.