## 4 Analysis and Discussion of Results
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Distribution** & \multicolumn{2}{c}{**Parameters**} \\ \hline SD+ & gamma & shape & rate \\  & & 1104.25 & 6343.20 \\ \hline Predominant Gauge & beta & shape1 & shape2 \\  & & 3.20 & 4.99 \\ \hline Minimum Curve Radius & gamma & shape & rate \\  & & 222.50 & 234.08 \\ \hline Number of Operational days per year & beta & shape1 & shape2 \\  & & 4.65 & 27.25 \\ \hline Installed Capacity & lognorm & meanlog & sdlog \\  & & \(-\)0.52 & 0.04 \\ \hline Linked Capacity & lognorm & meanlog & sdlog \\  & & \(-\)0.62 & 0.11 \\ \hline Bottleneck & lognorm & meanlog & sdlog \\  & & \(-\)1.36 & 0.03 \\ \hline Linked Capacity Rail Section & beta & shape1 & shape2 \\  & & 81.71 & 16.82 \\ \hline Idleness & lognorm & meanlog & sdlog \\  & & \(-\)0.67 & 0.06 \\ \hline Increasing Ramp Tax & lognorm & meanlog & sdlog \\  & & \(-\)0.73 & 0.17 \\ \hline Auxiliary Power & beta & shape1 & shape2 \\  & & 5.49 & 11.34 \\ \hline \% of Idleness & lognorm & meanlog & sdlog \\  & & \(-\)1.36 & 0.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Distributional fit for residuals obtained in models (1)-(12). Results presented in Figure 12 are of particular significance for mapping the relationship natures among variables--whether cause-effect or feedback--and their respective weights, as depicted in Figures 9-11. In Figure 12, [PERSON] and [PERSON] (2002) [48]'s results for sensitivity analysis are presented for each one of the 12 neural network regressions. The titles for each bar chart correspond to a given dependent variable, while labels for each bar chart correspond to the sorted list of explanatory variables in decreasing importance order. Readers should be aware that the mapped relationship signs between a pair of variables--say, \"a\" and \"b\"--ought to be analyzed in two distinct bar plots: one, where \"a\" is dependent and \"b\", explanatory; the other, where \"b\" is dependent and \"a\", explanatory. Very often, signs are the same, thus suggesting an endogenous or a feedback process between \"a\" and \"b\"--information entropy of the unconditional distribution is higher. However, signs may vary depending on whether \"a\" is dependent and \"b\" explanatory and vice-versa--information entropy of the conditional distribution is higher. This overall picture calls for several measures that can be implemented, not only to improve the performance of the Brazilian rail sections by establishing investment priorities with respect to structural attributes, but also how to use better performance levels for improving short-term, conjunctural management of bottlenecks and idleness, getting the most out of the existing linked capacity. These results significantly contribute to the previous literature on railway performance evaluation and the influencing factors, we further extend the studies of [PERSON] and [PERSON] (2020) [2] by investigating and showing the inner mechanism of the two-way relationships between structural, conjectural attributes and rail performance. Meanwhile, we fill in the gap by being the first to show the insights related to the potential linkage among different attributes, which would be able to generate more objective and accurate policy implications. Robustness check: A robustness check was performed on results presented in Figures 9-12 by increasing the cut-off percentiles for the conditional and unconditional information entropies. Additional percentiles of 0.980; 0.985; and 0.990 were tested with no differences found whatsoever in cause-effect relationships and their relative importance as derived by [PERSON] and [PERSON] (2002) [48]'s sensitivity analysis. Finally, we also check the robustness of our result by using Fuzzy TOPSIS, the results of which are reported in Appendix A.