# A Design of CGK-Based Granular Model Using Hierarchical Structure
## 4 Granular Model Design with a Hierarchical Structure
### CGK-Based Granular Model Design with an Aggregated Structure  
This paper presents a design for a granular model that adopts an aggregated structure. When an aggregated structure is built, rather than using low-level and high-level granular models, the low level comprises LR (LR) models, neural network models, and radial basis function networks, and each prediction model's output is adopts as the input for the high-level fuzzy granular model to calculate the final output. An LR model [53] models the linear correlations between input and output variables. Figure 12 shows the concept of linear regression. Simple linear regression models are based on explanatory variables, while multiple linear regression models are based on two or more explanatory variables. Linear regression models estimate unknown parameters in the data. A linear regression model can be expressed as:  
\[y_{i}\ =\ \beta_{1}x_{i1}\ +\ \ldots\ +\ \beta_{p}x_{ip}\ +\ e_{i}\ =\ {x_{i}}^{T}\beta\ +\ e_{i},\ \ i\ =\ 1,\ 2,\ \ldots,\ n \tag{34}\]  
where \(\beta_{i}\) and p denote each independent variable's coefficient and the number of parameters estimated by the linear regression model, respectively. \(T\) indicates transposition, while \({x_{i}}^{T}\beta\) represents the inner product of \(x_{i}\) and \(\beta\). Furthermore, \(e_{i}\) is the error term, and it represents the error variables. This refers to the error between the dependent and independent variables. Neural networks [54] are algorithms that are created with inspiration from biological neural networks in cognitive science and machine learning. These models can solve problems by altering the strength of the connections between synapses via the learning performed by the nodes that constitute the neural network by combining synapses. Figure 13.  
Presents the structure of a simple neural network. A simple neural network consists of an input layer, hidden layer, and output layer. The input layer inputs the data's input variables into the neural network, and the number of input variables must equal the number of input layer nodes. Usually, no calculation is performed in the input layer, and the layer simply performs the role of passing the values on. The hidden layer is between the input and output layers. If there are two or more hidden layers, it is called a multi-layer neural network. The output layer calculates the neural network's output. To achieve this, it adopts an activation function that is suitable for the problem to be solved. Figure 12: Linear regression model concept. A radial basis function network [55; 56] is a neural network that adopts a radial basis function, instead of a sigmoid function, as the activation function in a conventional neural network structure. Figure 14. presents the structure of a radial basis function network. A radial basis function network has a simple structure because there is only one hidden layer and the form of the output is linear; therefore, weight value calculations can be performed efficiently. The outputs from a linear regression model, neural network, and radial basis function network are combined and adopted as the input of a high-level fuzzy granular model. The high-level fuzzy granular model determines the number of contexts created in the output space and the number of clusters created in the input spaces, and then creates the IGs. Accordingly, the final output of the hierarchical structure is calculated. Existing fuzzy granular models are limited in creating meaningful IGs in the input and output spaces when processing large-scale data, and they have the problem of long computation times. In contrast, the fuzzy granular model with a hierarchical structure proposed in this study has the advantages of being able to create meaningful IGs from large-scale data and reduce processing times by taking the data created by combining the output from the low-level linear regression model, neural network, and radial basis function network, as well as adopting it as the input of the high-level fuzzy granular model.