# Machine Learning Approach in Heterogeneous Group of Algorithms for Transport Safety-Critical System
## 4 Results
We performed this particular experiment only once because our goal was just to illustrate the process for a few specific target values. The table lists the final estimate X calculated as an average of 20 \"preliminary\" estimates produced by the neural network as well as the standard deviation sX of these preliminary estimates. Furthermore, the threshold-crossing probabilities \(\Pr\left(X<8.0\right)\) and \(\Pr\left(X>40.0\right)\) and the corresponding 90 percent tolerance interval are shown in the table. The real value of X in the last column (XREAL) is shown just for verification--it was not necessary to the crucial system models. By comparing the individual rows in Table 2 we can see the effect of the average change of X on the threshold-crossing probabilities, which increase or decrease sharply as X enters or leaves each guarded region. For example, as the average X changes from 36.9891 to 39.7402 between rows six and seven, the probability \(\Pr\left(X>40.0\right)\) increases more than tenfold from 0.033197 to 0.394970. Similarly, as X changes from 8.9423 to 10.4178 between rows two and three, \(\Pr\left(X<8.0\right)\) sharply decreases from 0.421507 to 0.032779. Another important fact can be seen in rows four and five--a surprisingly large influence of the standard deviation sX on the threshold-crossing probabilities. Both rows share a very similar value of X yet their threshold-crossing probabilities for both thresholds differ by several orders of magnitude. This is caused by the difference in the deviation sX, since the higher the sX, the higher the probability of exceeding the threshold. In order to cross-check the soundness of our calculations, we have substituted the bounds of our 90 percent tolerance intervals from the sixth column, expecting to get the same result (0.9) for each row. The values that we obtained were indeed very close to 0.9--if we denote the error as \(\epsilon\), our results were \(0.9+\epsilon\), with the maximum absolute value of the error \(\left|\epsilon\text{MAX}\right|<2.6\times 10^{-13}\), which we interpret as the confirmation of sufficient precision in our calculations. As can be seen from our derivations above, the task of predicting the probability of exceeding a predefined threshold is inverse to that of estimating the bounds of an interval within which the target value should reside with a given probability. Both exploit the same mathematical relationship between the bounds and the probability, but they approach it from the opposite sides. This entitled us to verify the correctness of our technique by testing the validity of our 0.9 tolerance intervals, which is much easier to do than to verify the inverse task. Accordingly, we have trained machine learning models on various subsets of our transformed data and repeatedly calculated the bounds for 90 percent tolerance intervals, each time also noting whether or not they contained the actual value of X (which was known to us but hidden from the regression models, Tables 3 and 4). Table 3 lists averaged values for model precision, elapsed calculation time and the success rate of interval estimates for each size of the training set (specified in the NumRec