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Heidarnejad, et al.

Table 5. Tuning parameters of the gene expression unequivocally demonstrate the supremacy of the MLP
programming model model in comparison to the remaining three models.

Parameter Value/description Table 7 presents the statistical indices of the residual
Population size 110 values for the output of each of the four MLMs
employed in this study during both the training and
Number of genes 3 testing phases. As observed, the MLP model exhibits
Gene head length 10 the lowest deviation in both phases, indicating superior
Gene tail length 11 performance in terms of consistency and accuracy. On
Mutation rate 0.044 the other hand, the MARS model shows the highest
Inversion rate 0.1 error in estimating C among the four models. The mean
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Gene transposition rate 0.1 prediction errors for the MLP model during the training
One point recombination rate 0.3 and testing phases are −0.002 and 0.002, respectively –
these are the lowest values among all evaluated models.
Two-point recombination rate 0.3 Conversely, the MARS model demonstrates the highest
Gene recombination rate 0.1 deviation in this metric. The total residual error for the
Fitness function Root mean square error MLP model is −0.14 in the training phase and 0.086 in
the testing phase, which are also the smallest among the
Table 6. Multivariate adaptive regression splines models, further confirming its accuracy and robustness.
model’s basis functions and their corresponding Overall, analysis of the residual error values clearly
coefficients indicates that the MLP model provides higher prediction
Basis function Coefficient Equation accuracy for C compared to the other models.
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A graphical representation of the residual error
h (x) 0.0025329 Max (0,0.00765854- H d ) variations during the training and testing phases for all
1
P four MLMs is provided in Figure 5. In this figure, the
residual error values for the MLP model are highlighted
h (x) −0.0012013 Max (0, B -1) in red. As can be seen, the MLP model’s errors are
2
w closest to the horizontal axis, indicating the highest
1
accuracy (i.e., the smallest deviation) among all models.
h (x) 0.0020736 Max (0, w 2 -0.003) Figure 6 presents a comparative evaluation of model
3
w
1 performance in the training and testing phases, assessed
using the DDR index. As elucidated in the index
²h (X)
coupled with a bell-shaped curve in its vicinity, signify
C=0.00409658+ ∑ 3 i=1 m m (XVI) description, elevated values along the vertical axis,
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enhanced performance. Based on these criteria, the MLP
Figure 4 presents a comparative analysis of the model exhibits the best performance in the training and
performance of the four distinct models, evaluated testing phases. The values of C d(DDRmax) for the training
through the distribution of data points along a line with and testing stages are 8.07 and 11.32, respectively.
a 1:1 slope, during both the training and testing phases. Subsequently, the SVM, GEP, and MARS models
Enhanced model performance is indicated by reduced sequentially secure the second to fourth positions in
proximity to this reference line. In the testing phase, terms of performance ranking.
the output data of the GEP and MARS models exhibit Predicting C of different weirs is important due to
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a notable deviation from the ideal line. In contrast, various hydraulic and environmental reasons. The C
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the MLP and the SVM models demonstrate data governs the flow over the weirs; therefore, it determines
distribution closer to the reference line, suggesting their the efficiency of flow, which can be handled over the
superior performance. The MLP model demonstrates a weir. The flood water discharges or diversion over the
relatively higher efficacy than the SVM model due to its weir can be efficiently and optimally controlled through
data’s closer alignment with the 1:1 reference line. In accurate prediction of the weir C . Labyrinth weirs are
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general, it can be inferred that among the four models capable of increasing discharges without raising the
under consideration, the MLP model exhibits greater headwater level. Therefore, these are fine hydraulic
consistency with the actual measured C values. In the structures for water control in reservoirs and irrigation
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data distribution plot of the testing phase, the outcomes systems. In addition, accurate C predictions prevent
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Volume 22 Issue 6 (2025) 82 doi: 10.36922/AJWEP025120081