## 4 Fuzzy Neural Network for Predicting Rains and Temperatures
### Training Algorithm  
The model training algorithm is based on obtaining the neural network synaptic weights (and, consequently, the importance of the fuzzy rules of the inference system) through the concepts of using [PERSON]'s pseudo-inverse [91]. Thus, there is no requirement to update the parameters recursively, and, at the same time, there is a reduction in the influence of randomly defined parameters on the model structure. This technique allows weights to be set in one step. To solve possible overfitting problems in FNN training, a resampling regularization technique [92] is used in the objective function of training to define the neurons (or fuzzy rules) that contribute most efficiently to the model. The values of the neural network weights of the hybrid model are qualified for the operationalization of the neuron assignments to implement the function approximation. When synaptic weights are defined analytically, the efficiency of the model is evidenced and favors obtaining fast and accurate solutions. In this paper, this vector is considered by the Moore-Penrose pseudo Inverse [13]:  
\[\mathbf{v}=\mathbf{Z}^{+}\mathbf{y} \tag{10}\]  
where \(y=[y_{1},y_{9},\cdots,y_{n}]^{T}\) is the desired output vector and \(\mathbf{Z}^{+}\) is pseudo-inverse of Moore-Penrose [91] of \(\mathbf{z}\). The \(l\)+1 dimensional input space (\(\mathbf{z}\)), generating a \(n\)\(x\)\(l+1\) feature matrix, is presented as [13]:  
\[\mathbf{z}= \begin{bmatrix}z_{0}&z_{1}(x_{1})&z_{2}(x_{2})&\cdots&z_{1}(x_{1}) \\ z_{0}&z_{1}(x_{1})&z_{2}(x_{2})&\cdots&z_{1}(x_{1})\\ \vdots&\vdots&\vdots&\cdots&\vdots\\ z_{0}&z_{1}(x_{n})&z_{2}(x_{n})&\cdots&z_{1}(x_{n})\end{bmatrix}  \tag{11}\]  
In this view, \(\mathbf{z}\) is the least norm of the least-squares resolution for the weights of the output layer. Basically, the purpose of learning is presented to find the \(\mathbf{v}\) parameter that minimizes the error between the network output and wanted output for all training samples:  
\[\sum_{i=1}^{n}\|z(x_{i})\mathbf{v}-y_{i}\| \tag{12}\]  
Since the number of second layer neurons may contain unnecessary information, it is essential to use analytical techniques to determine the relevance of the collected neurons to the problem analyzed. Statistical evaluations that attempt to discover the correlations between two vectors are performed by regularization techniques, such as Least-Angle Regression (LARS) proposed by [PERSON] et al. [93]. LARS is a regression model for extensive data that can qualify and quantify exactly the regression coefficients as well as a part of candidate regressors to be associate in the final model. When refereeing a set of \(n\) different samples (\(x_{i},y_{i}\)), the cost function of this regression algorithm can be presented to change Equation (12) [93]:  
\[\sum_{i=1}^{n}\|z(x_{i})\mathbf{v}-y_{i}\,\|_{2}+\lambda\|\mathbf{v}\|_{1} \tag{13}\]  
where \(\lambda\) is a regularization parameter, frequently evaluated by cross-validation. The initial expression in Equation (13) corresponds to the Residual Sum of Squares (RSS). The relationship between the training error and the first term of Equation (13) is directly proportional so that the training error should be reduced by decreasing the RSS. The second is a \(L_{1}\) regularization term. First, it develops FNN generalization, bypassing overfitting [94]. Secondary, it can be used to produce sparse models [95]. To explain why LARS can be used as a neuron selection in the model, Equation (13) is rewritten as:  
\[ \begin{array}{ll}\underset{v}{\text{min}}&RSS(v)\\ \text{s.t.}&||v||_{1}\leq\Gamma\end{array}  \tag{14}\]  
where \(\Gamma\) is an upper-bound on the \(L_{1}\)-norm of the weights. A small value of \(\Gamma\) corresponds to a high value of \(\lambda\), and vice versa. The model proposed by [PERSON] [92], called Bolasso (Model Consistent Lasso Estimation through the Bootstrap), uses this methodology to find a combination of neurons that meet a previously defined criterion in a set of bootstrap replications.