# Modeling of Fuzzy Systems Based on the Competitive Neural Network
## 2 Theoretical Framework
### Competitive Neural Network
\[d(x,w_{i})=\|x-w_{i}\|=\sqrt{\left(x_{1}-w_{i1}\right)^{2}+...+\left(x_{N}-w_ {iN}\right)^{2}} \tag{4}\]  
Hence, if \(r\) represents the winning process unit, after introduction of the input pattern, the validation that the winning unit possesses the synaptic weight vector most similar to the input vector is expressed through Equation (5). \[d(x,w_{r})\leq\;d(x,w_{k}),\;k=1,\;2,\ldots,M \tag{5}\]  
Figure 1: Architecture of a CNN. Red represents the winning unit and orange the deactivated unit. The aim of the unsupervised learning in the CNN is to autonomously identify groups or classes. To achieve this, the criterion of the minimum squares is calculated using Equation (6). \[E(k)=\sum_{i=1}^{M}a_{ik}\|\mathbf{x}(k)-\mathbf{w}_{i}(k)\|^{2} \tag{6}\]  
In (6), \(E\) denotes the error in iteration \(k\), derived from the distance between the input pattern \(x(k)\) belonging to class M and the synaptic weight vector \(x_{i}(k)\). This is identified as the learning rule, which is denoted in Equation (7). \[\mathbf{w}_{r}(k+1)=\mathbf{w}_{r}(k)+\eta(k)[\mathbf{x}(k)-\mathbf{w}_{r}(k)] \tag{7}\]  
In this scenario, the new synaptic weights vector \(w_{r}(k+1)\) is a linear combination of the vectors \(w_{r}(k)\) and \(x(k)\), with \(\eta(k)\) representing the learning rate, as expressed in Equation (8), where \(k\) is the current iteration and T is the total number of the iterations; therefore, the initial value of \(\eta_{0}\) could be (0, 1). The learning rate will be decrementing; if \(\eta\) reaches 0, the CNN will stop learning. \[\eta(k)=\eta_{0}\bigg{(}1-\frac{k}{T}\bigg{)} \tag{8}\]  
In the state of the art method, we can find a large number of research works, exploiting the advantages of Competitive Neural Networks or Competitive Learning and applying them in different areas. In [9], the author provides a comprehensive overview of competitive learning-based clustering methods. In [11], a framework for different clustering methods like Fuzzy C-Means and K-Means clustering, entropy constrained vector quantization, or topological feature maps and CNN is proposed. In [20], the authors explore the exponential synchronization problem for a class of the CNN. In other applications, the CompNet model based on Competitive Neural Networks and Learnable Gabor Kernels is presented in [21], the model is implemented for Palmprint Recognition; according to the results, the proposal achieved the lowest error rate compared to the most commonly state of the art methods. In [22], a Competitive Neural Network is used to estimate a rice-plant area; the authors demonstrate that these kinds of models are useful for the classification of the satellite data. In hybrid methods, in [23], the CNN is optimized and is applied to solve a complex problem in the field of chemical engineering; the authors proposed a novel neural network optimizer that leverages the advantages of both an improved evolutionary competitive algorithm and gradient-based backpropagation. In [24], the Fireworks Algorithm (FWA) was implemented to optimize the neurons of the CNN and improve the results of the traditional model. In order to improve the CNN, a novel classification algorithm that is based on the integration between competitive learning and the computational power of quantum computing is presented in [25]. These and other researches [26] demonstrate the potential of this area to solve a variety of problems.