## 2 Materials and Methods
### ANFIS Models  
Experimental data have been used to train the ANFIS model, and parametric analysis has been performed when varying the following membership functions:  
Four membership function shapes: triangular, trapezoidal, generalized bell, and Gaussian; Two fuzzy inference system (FIS) generators: grid partition and sub-clustering;  
Two optimization methods: hybrid and backpropagation.  
Equations (5)-(8) briefly summarize the equations related to the membership functions considered.  
* Triangular membership function (trimf): This is the most basic of various shapes. Three variables are used to determine its three points--a and c for the feet, and b for the higher vertex--as shown in Equations (5)-(8). \[\text{Triangular}(x;\ a;\ b;\ c)=\max(\min\left(\frac{x-a}{b-a}\right),1,\ \left(\frac{c-x}{c-d}\right),0)\] (5)
* Trapezoidal membership function (Trapmf): Its shape is defined by four scalar parameters: \(a\) and \(b\) for the feet; \(c\) and \(d\) for the sides. \[\text{Trapezoidal}(x;\ a;\ b;\ c;\ d)=\max(\min\left(\frac{x-a}{b-a}\right),1, \left(\frac{d-x}{d-c}\right),\ 0\] (6)
* Generalized bell-shaped membership function (Gbellmf): This is a bell-assimilated shape defined by three parameters: the curve width \(a\), an integer \(b\), and the center of the curve. \[\text{Bell}(x;\ a;\ b;\ c)=\frac{1}{1+\left|\frac{x-c}{a}\right|^{2b}}\] (7)
* Gaussian membership function (Gaussmf): Unlike the above ones, the Gaussian is defined by two parameters, the curve center \(c\) and the curve width, as illustrated in Equation (4). \[\text{Gaussian}(x;\ c;\ \sigma)=e^{-\frac{1(\frac{x-c}{2})^{2}}{2}}\] (8) \(c_{1}\) and \(c_{2}\) are acceleration factors; \(r_{1}\) and \(r_{2}\) are two random numbers ranging between 0 and 1; \(w\) is an initial factor; The initial population (swarm size) of size N and dimension D is given by X = [X\({}_{1}\), X\({}_{2}\),..., X\({}_{\text{N}}\)]T, with T' being the transpose operator; Individual particle X\({}_{\text{p}}\), with \(p\) = 1, 2,..., N, is defined as X\({}_{\text{p}}\) = [X\({}_{\text{p},1}\), X\({}_{\text{p},2}\),..., X\({}_{\text{p},\text{D}}\)]; The initial velocity of the population is defined as V = [V\({}_{1}\), V\({}_{2}\),..., V\({}_  
Swarm size population values: 10, 20, 50, 100, 200, and 400; Number of neurons: 5, 6, 7, 8, 9, and 10; Acceleration factors \(c_{1}\): 1.0, 1.5, 2.0, 2.25, 2.5; Acceleration factors \(c_{2}\): 2.0, 2.25, 2.5, 2.75, 3.0.  
The ANN model follows the sigmoid function described in Equation (2). As for the PSO algorithm, set to optimize the ANN, Equations (9) and (10) describe the new positions of each particle in the search space. This new position is defined by the personal experience (Pbest), the overall experience (Gbest), and the actual movement of the particles.  
\[V_{p,q}^{k+1}=w\ x\ V_{p,q}^{k}+c_{1}r_{1}\left(\textit{Pbest}_{p,q}^{k}-X_{p,q}^{k}\right)+c_{2}r_{2}\left(\textit{Gbest}_{q}^{k}-X_{p,q}^{k}\right) \tag{9}\]  
\[X_{p,q}^{k+1}=X_{p,q}^{k}+V_{p,q}^{k+1} \tag{10}\]  
with \(\textit{Pbest}_{p,q}^{k}\) representing the personal best of the \(q^{\text{th}}\) component of \(p^{\text{th}}\) individual, and \(\textit{Gbest}_{q}^{k}\) representing the \(q^{\text{th}}\) component of the best individual of the population up to iteration \(k\).