# Causality Analysis with Different Probabilistic Distributions Using Transfer Entropy
## 4 Analysis of Simulation Results  
Analysis using the Transfer Entropy approach begins with the determination of probability distributions adjusted separately for each regulation error. This applies to data  
Figure 4: Control error time series for the data with Gaussian noise and Cauchy disturbance. Figure 5: The causality diagram of the considered simulated benchmark example. Figure 3: Control error time series for the simulation data with Gaussian noise. generated with Gaussian noise as well as to data with both Gaussian noise and Cauchy disturbance. The results for Cauchy, \(\alpha\)-stable, Laplace, Huber, and t-location scale distribution fitting are presented in Figures 6-8 and Figures 9-11, respectively. The data generated with Gaussian noise are fairly normal and close to a normal distribution. A visual evaluation shows that the fit of the probability distributions is satisfactory, especially for the \(\alpha\)-stable distribution. Figure 6: Histograms with PDF fitting for dataset with Gaussian noise: (**a**) control error \(\epsilon_{1}\) and (**b**) control error \(\epsilon_{2}\). In the second case, data are generated with both Gaussian noise and Cauchy disturbance, with an added sinusoidal signal of 30 Hz frequency. For the most part, these data do not exhibit sufficient normality to be regarded as having been drawn from a normal distribution. In particular, this applies to the regulation errors \(\epsilon_{1}\) and \(\epsilon_{2}\). In the first place, this indicates the presence of outliers. According to [36], an outlier is defined as \"_an observation which deviates so much from other observations as to arouse suspicions that it was generated by a different mechanism.\"_ The presence of outliers has a significant effect on data analysis, as they increase the variance of the signal, reduce the power of statistical tests during analysis [37], disturb the normality of the signal and introduce fat tails [38], and introduce bias in regression analyses [39]. Outliers can be caused by erroneous observations or inherent data variability. There are many approaches for detecting these phenomena and removing them from data, i.e., the Interquartile Range method (IQR) [40], generalized Extreme Studentized Deviate (ESD) [41], and Hampel filter [42]. However, the phenomenon observed in the histograms shown here is the result of loop oscillations, and can be omitted in this case. Figure 7: Histograms with PDF fitting for dataset with Gaussian noise: (**a**) control error \(\epsilon_{3}\) and (**b**) control error \(\epsilon_{4}\). Assessment of the fit quality of the probability distributions selected for analysis cannot consist of a solely visual evaluation. To objectively determine which PDF best fits  
Figure 8: Histograms with PDF fitting for dataset with Gaussian noise (control error \(\epsilon_{5}\)). Figure 9: Histograms with PDF fitting for dataset with Gaussian noise and Cauchy disturbance: (a) control error \(\epsilon_{1}\) and (b) control error \(\epsilon_{2}\). the regulation error histogram, a fit indicator is used. The fitting indexes are calculated using the mean square error between the empirical histogram heights and the value of the fitted probabilistic density function as evaluated at the middle of the respective histogram bin. The results for the dataset with Gaussian noise are presented in Table 2 and those for the dataset with Gaussian noise and Cauchy disturbance in Table 3. The lowest values of the fit indicator (which is considered the best fitting) for each \(\epsilon_{i}\) are marked in blue and bold. Figure 10: Histograms with PDF fitting for dataset with Gaussian noise and Cauchy disturbance: (**a**) control error \(\epsilon_{3}\) and (**b**) control error \(\epsilon_{4}\). As shown in Table 2, for the dataset with Gaussian noise the \(\alpha\)-stable distribution appears to have the best fit in the case of regulation errors \(\epsilon_{1}\), \(\epsilon_{3}\), and \(\epsilon_{4}\).