# A Design of CGK-Based Granular Model Using Hierarchical Structure
## 4 Granular Model Design with a Hierarchical Structure  
### CGK-Based Granular Model Design with a Hierarchical Structure  
As the number of input variables for a fuzzy system and granular model increase, the number of rules increase geometrically. Large rule bases have the problem of reducing the computation efficiency of fuzzy systems and granular models. In addition, they make it difficult to understand the action of granular models, and complicate the adjustment of rules and membership functions. The possibility of generalizing fuzzy systems and granular models with large rule bases is minimized because various prediction-related fields of application provide limited amounts of data. To resolve these problems, rather than using a single fuzzy system and a single granular model, it is possible to design a granular model with a hierarchical structure in which these are mutually connected. Because the granular model is arranged in a hierarchical tree structure, the tree is called the hierarchical structure. The output of the low-level granular models in the hierarchical structure is adopted as the input for the high-level granular models. Granular models with hierarchical structures are computationally more efficient than single granular models with the same number of inputs, and are also designed with a simple structure [50; 51; 52]. Hierarchical structures that can be used in various prediction-related fields of application include incremental, aggregated, and cascaded structures. Figure 11.  
Presents each type of hierarchical structure. In incremental structures, input variables are combined in several stages, while output values are calculated at several levels. As illustrated in Figure 11, the granular model \(GM_{i}{}^{\eta}\) is built with a 3-stage structure. Here, \(i\) is the nth level's granular model. In an incremental granular model, when \(i\) is one, it means that there is one fuzzy inference system on each level. The nth level's \(i\)th granular model's jth input is called \(x_{ij}{}^{\eta}\), while the nth level's ith granular model's kth output is called \(y_{ik}{}^{n}\). When the input variables on each of the levels of an incremental granular model are selected, their ranks are determined according to their degrees of contribution to the final output value. The input variable with the highest degree of contribution is usually used on the lowest level; conversely, the input variable with the lowest degree of contribution is adopted on the highest level. In other words, low-rank input values depend on high-rank input values. In an aggregate structure, the original data's input variables are used on the lowest level, and the output of the low-level granular model that receives each input variable as the input is inputted in the high-level granular models, and the obtained results are combined. For example, the granular model \(GM_{i_{n}}{}^{n}\) is built with two stages as, illustrated in Figure 11. \(i_{n}\) is the index of the granular model on the nth level. The input variables in aggregated granular models are grouped for performing specific decision making. For example, an autonomous robot's search task combines two tasks: searching while avoiding collisions with obstacles, and arriving at the goal. To perform the search task, the granular model adopts input variables related to obstacles. To perform the task of arriving at the goal, input variables related to the robot's current position and movement direction are employed. Aggregated granular models can be modified to design parallel aggregated granular models that directly add up the outputs of low-level granular models to calculate their final output. A cascaded structure is a structure that combines the aforementioned incremental structure with the hierarchical structure, and it is suitable for systems that include both correlated input variables and non-correlated input variables.