PSI - Issue 81

Dmytro Tymoshchuk et al. / Procedia Structural Integrity 81 (2026) 35–40

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a

b

c

d

Fig. 1. Weights of the base models calculated based on 1/MSE for loading frequencies: (a) 0.3 Hz, (b) 0.5 Hz, (c) 3 Hz, and (d) 5 Hz.

The distribution of weights across different frequencies confirms the stability of the dominant role of tree-based and neural network models, whereas less robust algorithms gradually lost significance. These findings validate the effectiveness of the ensemble approach, which leverages the complementary strengths of diverse models to achieve superior overall performance. 3.2. Performance Evaluation of the Voting Ensemble Model The accuracy of the ensemble Voting model was evaluated using four metrics. The obtained results confirm the high predictive precision of the developed model both on the test datasets and during independent cycle forecasting. According to the testing results (Table 2), the average MAE did not exceed 0.022, indicating minimal deviations of the predicted curves from the experimental ones. The MSE values remained within the range of 0.0003 – 0.0008, suggesting the absence of significant outliers. The coefficient of determination ( R 2 ) exceeded 0.998, reaching 0.9996 for the 0.3 Hz frequency, which implies an almost complete explanation of the experimental data variance. Meanwhile, the low MAPE values (0.005 – 0.008) further confirm the model’s high accuracy in relative terms.

Table 2. Prediction errors (test dataset) Prediction errors

Frequency, Hz

0.3

0.5

3

5

0.0008 0.0216 0.9996 0.0068

0.0003 0.0149 0.9993 0.0053

0.0004 0.0165 0.9984 0.0079

0.0003 0.0135 0.9982 0.0075

Mean squared error (MSE) Mean absolute error (MAE) Coefficient of determination ( R 2 )

Mean absolute percentage error (MAPE)

During additional testing on cycles 251 and 300, which were outside the training dataset (Table 3), the model also showed stable performance. Although the error values slightly increased compared to the test dataset, the overall level of agreement remained high. For instance, for the 251st cycle, the MAE ranged from 0.0134 to 0.0203 depending on the frequency, while the R 2 exceeded 0.998. For the 300th cycle, a more noticeable increase in errors was observed, particularly at 5 Hz, where MAE reached 0.0788 and MAPE was 0.0591. Nevertheless, even in this case, the coefficient of determination ( R 2 ) remained at 0.9653, confirming that the ensemble model maintained strong predictive capability.