## 2 Design of the CGK-Based Fuzzy Granular Model
### CGK-Based Fuzzy Granular Model with the Aggregated Structure  
When extensive multivariate data are processed through a neuro-fuzzy reasoning system and GM, the number of rules increases exponentially as many input variables influence it. The calculation speed and performance of the neuro-fuzzy reasoning system and GM are decreased. A model that is connected hierarchically with two or more systems rather than a single neuro-fuzzy reasoning system or a single GM can be designed to address this problem. This study proposes the CGK-based fuzzy granular model of the aggregated structure that combines the CGK-based fuzzy granular model with the linear regression [37], artificial neural network [38], and radial basis function neural network [39]. The proposed model is constructed of an aggregated structure and uses linear regression, ANN, and a radial basis function neural network as sub-models and the CGK-based fuzzy granular model as an upper model to calculate the final output. The CGK-based fuzzy granular model of the aggregated structure can be designed more efficiently and easily understood than a single system and model when processing data with the same number of input variables [40; 41]. The CGK-based fuzzy granular model of the aggregated structure uses the input variables of the original data as input for the linear regression, artificial neural network, and radial basis function neural network. It uses the model's output after the data are processed in each sub-model as input for the CGK-based fuzzy granular model, which is the upper model. Linear regression analyzes the correlation between input and output variables and is expressed as follows:  
\[y_{i}=\beta_{1}x_{i1}+\beta_{p}x_{ip}+\varepsilon_{i}=x_{i}^{T}\beta+ \varepsilon_{i},\ i=1,\ 2,\ \ldots,\ n \tag{14}\]  
In Equation (14), the parameter of linear regression is expressed as \(p\), and the coefficient of each independent variable is expressed as \(\beta_{i}\). \(x_{i}^{T}\beta\) expresses the inner product of \(x_{i}\) and \(\beta\), and \(T\) denotes transposition. Lastly, \(\varepsilon_{i}\) is an error term and denotes an error variable that indicates the error between independent and dependent variables. The creation of ANN was inspired by the form of synaptic connection, and the model can solve problems by changing the connection strength through training. The general artificial neural network comprises three layers, the input, hidden, and output layers, respectively. The input layer passes the input variables of data to the hidden layer, and the number of input layers is the same as the number of input variables of the data to be processed. The mathematical calculation does not happen in the input layer, and it only plays the role of passing data. The hidden layer is composed of one or more layers. If there are two or more hidden layers, it is called a multilayer ANN. The output layer calculates the final output of the ANN and is processed by the activation function. The radial basis function neural network uses the activation function as a radial basis function of the structure mentioned above of ANN. Unlike the structure of general ANN, the radial basis function neural network only has one hidden layer. The weight can be effectively calculated because there is only one hidden layer. Figure 3 illustrates the CGK-based fuzzy granular model of the aggregated structure. The outputs from the sub-models, i.e., linear regression, artificial neural network, and radial basis function neural network, are combined and used as the input of the upper model, the CGK-based fuzzy granular model. The upper model, the CGK-based fuzzy granular model, generates IGs by selecting the number of contexts and the number of clusters and calculates the final output using them. The conventional GM had the problem of difficulty generating rational IGs when processing extensive multivariate data, and it took much time to process them.