## 2 Generation of Information Granules and Design of the Granular Model
### CFCM-Based Granular Fuzzy Model  
As explained in Section 2.1, the granular fuzzy model (GFM) can be designed using CFCM clustering. The premise parameters are determined by the cluster centers estimated by CFCM clustering. The linguistic contexts, which are the consequent parameters, are produced in the output variable, as shown in Algorithm 1. The CFCM-based GFM consistsof four layers. The first layer receives the input data, and the second layer represents the set of activation levels for all clusters related to the linguistic contexts. In the third layer, the CFCM clustering for each context is performed. Hence, when a linguistic context is given, clustering occurs in the input variable corresponding to the context. For each context, the cluster's number is adjusted by the user. The fourth layer is the output layer, and it is implemented in a single granulated form. The output value Y of the CFCM-based GFM is expressed by Equation (7). \[Y=\sum_{\oplus}W_{t}\otimes z_{t} \tag{7}\]  
Here, the addition and multiplication symbols are used to represent information granules. \(W_{t}\) and \(z_{t}\) are the t-th context and t-th firing strength, respectively. The GFM has a single hidden layer formed by clusters obtained by CFCM clustering. The space between hidden and output layers can be expressed linguistically using contexts, unlike the design method of conventional neural networks. The final value of the output layer can be computed as follows  
\[Y=(z_{11}\otimes A_{1}\oplus z_{12}\otimes A_{1}\oplus\ldots\oplus z_{1n1} \otimes A_{1})\oplus\ldots\oplus(z_{c1}\otimes A_{c}\oplus z_{c2}\otimes A_{c} \oplus\ldots\oplus z_{enc}\otimes A_{c}) \tag{8}\]  
The generalized additions and multiplications in Equation (8) are completed using a fuzzy calculation method. Here \(A_{i}\) is the \(i\)-th fuzzy set characterized by context. When the linguistic context is assigned as a triangular fuzzy number, it can be expressed using the following equation:  
\[A_{i}=(a_{i-},\ a_{i},\ a_{i+}) \tag{9}\]  
Here, \(a_{i-},\ a_{i},\ a_{i+}\) denote the lower limit output value, granular model output value, and upper limit output value of the triangular fuzzy set, respectively. The lower prediction value and upper limits of the GFM can be expressed using Equations (10)-(12). \[y_{-}=(z_{11}a_{1-}+z_{12}a_{1-}+\ldots+z_{1n1}a_{1-})+\ldots+(z_{c1}a_{c-}+z_{ c2}a_{c-}+z_{cnc}a_{c-}) \tag{10}\]  
\[y=(z_{11}a_{1}+z_{12}a_{1}+\ldots+z_{1n1}a_{1})+\ldots+(z_{c1}a_{c}+z_{c2}a_{c} +z_{cnc}a_{c}) \tag{11}\]  
\[y_{+}=(z_{11}a_{1+}+z_{12}a_{1+}+\ldots+z_{1n1}a_{1+})+\ldots+(z_{c1}a_{c+}+z_{ c2}a_{c+}+z_{cnc}a_{c+}) \tag{12}\]  
**3. CFCM-Based Granular Model with Hierarchical Tree Structure**  
3.1. Hierarchical Tree Structures  
Increasing the number of input variables in fuzzy-related and granular models also exponentially increases the number of fuzzy rules, resulting in poor computational efficiency and performance for fuzzy-related and granular models. It also makes it difficult to understand the operation of the model and makes it challenging to adjust the parameters of the membership functions and fuzzy rules. Moreover, the generalizability of fuzzy models and granular models can be reduced when processing large multivariate databases. To solve these problems, fuzzy and granular models are designed in an interconnected hierarchical structure [41] rather than in a single monolithic form. Fuzzy and granular models, designed in a hierarchical structure, can be referred to as hierarchical fuzzy and granular models. In this structure, the output of the low-level model is used as the input to the high-level model. A hierarchical fuzzy or granular model designed in this manner is easier to understand and can perform computations more efficiently than a single monolithic fuzzy or granular model with a unified number of inputs. In this paper, we present three types of hierarchical structure, including incremental structure, aggregated structure, and cascaded structure [42; 43; 44]. Figure 1 shows these three hierarchical structures.