# Deformation of Air Bubbles Near a Plunging Jet Using a Machine Learning Approach
## 3 Results and Discussion
However, Model 3 proved to be less accurate than the other two models: the absence of the number of Eotvos among the input variables significantly reduced its effectiveness. Moreover, it should be noted that the SVR algorithm has consistently shown the same level of accuracy across all models and input variables. This result highlights its robustness, as there is an indication that the algorithm is significantly sensitive to input parameters. Figure 6 shows a comparison between the predicted and measured values of \(D_{\alpha q+1}\), relative to Model 1. In addition, it shows the relative error as a function of the measured \(D_{\alpha q+1}\). Overestimation errors were observed more frequently than underestimation errors in all Model 1 variants. Furthermore, the maximum relative errors were observed for bubbles with equivalent diameters less than 2 mm. These errors showed a tendency to decrease as the equivalent diameter grew and generally did not exceed \(\pm 25\%\). The exception was represented by the SVR algorithm. If the bubble diameter was less than 2 mm, SVR generally tended to overestimate the diameter itself and the error could also exceed 30%, while, if the diameter was greater than 4 mm, SVR tended to underestimate the diameter, with an error which did not exceed 15%. If the equivalent bubble diameter was between 2 and 4 mm, the SVR-based model variant provided results comparable to those of the other variants. Figure 6: _Cont._  
The comparison between the predicted and measured values of \(D_{\alpha q+1}\) shown in Figure 7 relates to Model 2. As in the previous case, the same figure also shows the relative error as a function of the experimental \(D_{\alpha q+1}\). The error distributions were quite similar to those of Model 1, while the maximum relative errors were observed when \(D_{\alpha q}<2.5\) mm. In this case, both K-Star and SVR led to relative errors, in some cases greater than 30% in the forecasts for smaller bubbles. Figure 6: Equivalent diameter prediction “Model 1”: in the left column, predicted versus experimental values, in the right column, residuals versus experimental values. Figure 7: _Cont._  
An alternative effective representation of the models' errors in predicting \(D_{eq+1}\), in terms of residuals (i.e., absolute errors), is provided by the notched box plots in Figure 8. These plots show the range of errors with respect to the experimental values. The lower end of each box plot denotes the lower quartile Q1 (25 th percentile), the upper end denotes the upper quartile Q3 (75 th percentile), the band inside the box represents the median of the data. The whiskers extend from the bottom of the box to the smallest non-outlier and from the top of the box to the highest non-outlier in the dataset. The box plots further confirm what was previously argued in terms of the accuracy of the different models. It can also be noted that the variants of the models based on the Bagging and RF algorithms have their residual distributions characterized by greater symmetry than the others. Figure 9 shows the \(D_{eq}\) time series of two of the analyzed bubbles. Again, it can be observed that all models' variants provided predictions in good agreement with the experimental data. In addition, it is interesting to note that higher errors were observed in correspondence with higher changes in the experimental values between two consecutive time steps. Finally, it may be worth highlighting that the algorithm that provided the most accurate predictions for a single bubble may be different from the algorithm that was the most accurate in the entire dataset. Figure 8: Equivalent diameter prediction: box plots of the residuals. Figure 7: Equivalent diameter prediction “Model 2”: in the left column, predicted versus experimental values, in the right column, residuals versus experimental values. The aspect ratio (AR) results are summarized in Table 2. The accuracy of the prediction models was significantly lower than that of the analogous forecast models of equivalent diameter.