## 2 Methodology
### ANFIS-Adaptive Neuro-Fuzzy Inference System  
The Adaptive Neuro-Fuzzy Inference System (ANFIS), introduced by [PERSON] [53], is a sophisticated hybrid model that integrates the principles of both artificial neural networks (ANNs) and fuzzy logic systems. This integration leverages the strengths of both approaches, enabling the emulation of human-like intelligent decision-making processes [30; 31; 32; 33; 37; 54]. ANFIS is structured as a multi-layer feedforward network that utilizes the learning capabilities of ANNs and Fuzzy Inference Systems (FISs) to create a powerful tool for nonlinear functions [30]. The system is based on the Takagi-Sugeno fuzzy model, which is structured into five distinct layers: the input fuzzification layer, the product layer, the normalized layer, the defuzzification layer, and the output layer [32]. These layers work in concert to process inputs through fuzzy IF-THEN rules and generate outputs by mapping complex nonlinear relationships [55]. \[\text{Rule 1}:\text{ {if} }x\text{ {is} }X_{1}\text{ {and} }y\text{ {is} }Y_{1}\text{, {then} }f_{1}=k_{1}x+l_{1}y+m_{1} \tag{7}\]  
\[\text{Rule 2}:\text{ {if} }x\text{ {is} }X_{2}\text{ {and} }y\text{ {is} }Y_{2}\text{, {then} }f_{2}=k_{2}x+l_{2}y+m_{2} \tag{8}\]  
where \(x\) and \(y\) are inputs; \(X\) and \(Y\) are fuzzy sets; \(f\) is the output; and \(k\), \(l\), and \(m\) are the parameters defined by the artificial neural network. The typical architecture of ANFIS \begin{table}
\begin{tabular}{l l} \hline \hline
**SPI** & **Category** \\ \hline above and (2.0) & Extremely wet \\ \hline (1.99)–(1.50) & Severely wet \\ \hline (1.49)–(1.00) & Moderately wet \\ \hline (0.99)–(\(-\)0.99) & Near normal \\ \hline (\(-\)1.0)–(\(-\)1.49) & Moderately dry \\ \hline (\(-\)1.5)–(\(-\)1.99) & Severe dry \\ \hline \(-\)2.0 and less & Extremely dry \\ \hline \hline \end{tabular}
\end{table} 
Table 2: SPI categories [23]. consists of five layers. A complete description of ANFIS can be found in [PERSON] [53] and [PERSON] and Sun [56]. The first layer, known as the fuzzification layer, receives inputs and applies a generalized Gaussian membership function to them, determining the degree to which each input belongs to various fuzzy sets. The subsequent layers, including the product and normalization layers, combine these fuzzy inputs using fuzzy logic operators to evaluate the rules of the fuzzy inference system. The defuzzification layer then converts the fuzzy classifications into a crisp output, which is finally aggregated in the output layer. One of the key advantages of ANFIS is its ability to learn and optimize model parameters through adaptive methods such as back-propagation gradient descent and least squares techniques. This adaptability makes ANFIS particularly effective in handling complex, nonlinear problems where traditional linear models are not sufficient. However, determining the optimal parameters for the membership functions can be a challenging and time-consuming process. Furthermore, the model can sometimes become trapped at local optimum points. To address these issues, optimization algorithms such as the GWO have been employed to enhance the model's performance and help it avoid local optima. Despite its complexity, ANFIS has been widely recognized for its effectiveness across various applications, particularly in areas where modeling nonlinear relationships is crucial. The integration of ANN enhances the model's learning and optimization capabilities, making ANFIS a robust tool for predictive modeling in uncertain environments. Overall, ANFIS stands out as a prime example of how combining different computational intelligence techniques can lead to more powerful and versatile modeling approaches, capable of solving a wide range of challenging real-world problems. A comprehensive understanding can be obtained from the foundational works by [PERSON], [53], as well as further explorations and applications discussed by [PERSON] and [PERSON] [57] and [PERSON] and [PERSON] [56].