## 2 Materials and Methods
### Procedure for the Construction of the Nonlinear Models  
The model generation starts by calculating the Spearman coefficients for the inputs related to the output. Then, the inputs that achieve the specific threshold for \(\rho\) and \(P_{value}\) are selected. Next, it is necessary to train the GP and FNNs models.  
The GP algorithm applies summation (+), subtraction (\(-\)), multiplication (\(\times\)), division (\(\div\)), and exponentiation (\(\lx@sectionsign\)) as operators; numerical coefficients; Intensity (\(x_{1}\)), red light component (\(x_{2}\)), green light component (\(x_{3}\)), blue light component (\(x_{4}\)), white light component (\(x_{5}\)), pulsed frequency (\(x_{6}\)), and duty cycle (\(x_{7}\)) as input variables. Finally, sines and cosines of each variable are used to build Fourier series with which to approximate any function with a finite number of discontinuities [56].  
The mean absolute error (MAE) in Equation (6) is the GP cost function that estimates the error magnitude in models with unknown distribution [57].  
\[MAE(\hat{y},y)=\frac{1}{N}\sum_{i=1}^{N}\lvert\hat{y}_{i}-y_{i}\rvert \tag{6}\]  
where \(y\) is the real output and \(\hat{y}=g(x_{1},\dots,x_{7})\) is the model output.  
Candidate solutions are refined over \(N_{G}\) generations using \(MAE(\hat{y},y)\), and then the best candidate is returned.  
The FNN structure is constructed according to the process described in Section 2.5, and applied to the same vector input as the GP model. For the FNN model, we used the mean square error (MSE) defined in Equation (7). The MSE is the expected error function, because it prioritizes larger reduction errors and generates a higher quantity of derivatives for updating the ANNs weights [52].  
\[MSE(\hat{y},y)=\frac{1}{N}\sum_{i=1}^{N}\left(\hat{y}-y\right)^{2} \tag{7}\]  
where \(N\) is the number of samples, \(y\) is the target, and \(\hat{y}\) as the computed output of the FNNs.  
After generating both models, we carried out a quantitative evaluation with the MAE and MSE metrics in the nonlinear models, and the errors in the construction stage were applied. The precision measures in nonlinear regression models [58; 59] are the standard error of the estimate (SEE) in Equation (8) and the mean absolute percentage error (MAPE) defined in Equation (9). The SEE represents the average distance from a real output value to a predicted one. The MAPE symbolizes the average relative error between the regression model and the right measure.  
\[SEE(\hat{y},y)=\sqrt{\frac{\sum_{i=1}^{N}\left(\hat{y}_{i}-y_{i}\right)^{2}}{ N}} \tag{8}\]\[MAPE(y,y)=\frac{1}{N}\sum_{i=1}^{N}\left|\frac{\hat{y}_{i}-y_{i}}{y_{i}}\right| \tag{9}\]  
Additionally, we used three statistic validation tests to identify the best model. The first was the R\({}^{2}\) statistic measure used in Equation (10); this is commonly used for scoring regression models, as in [60; 61; 62]. R\({}^{2}\) is in the range of [0,1] with 1 for a regression model that fully explains the variability of the output variable [63]. The second is a box plot that compares, in a graphical representation, the media and quartiles of the analyzed groups. In this case, we tested the GP and FNN errors in a single dimension. The third is one-way analysis of variance (ANOVA), which was used to examine equality between two categorical variables (GP or FNN model) of quantitative outcomes with two or more levels of treatments (with metrics of error including MAE, MSE, SEE, and MAPE) [64].  
\[R^{2}(\hat{y},y,\overline{y})=\frac{\sum_{i=1}^{N}\left(y_{i}-\overline{y} \right)^{2}-\sum_{i=1}^{N}\left(y_{i}-\hat{y}_{i}\right)^{2}}{\sum_{i=1}^{N} \left(y_{i}-\hat{y}_{i}\right)^{2}} \tag{10}\]