## 2 Design of the CGK-Based Fuzzy Granular Model
### Optimal Assignment of IGs
Performance indices play an important role in evaluating model accuracy and clarity, and various methods have been developed to evaluate model performance. Common performance evaluation methods include Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE). RMSE calculates the average of the squares by subtracting the model prediction value from the actual prediction value and then squaring the value to evaluate the performance. MAPE evaluates the performance by subtracting the model prediction value from the actual output value and dividing it by the model prediction value. This performance evaluation method is mainly used when the output value of the model is a numerical value. However, in the case of a granular model composed of information granules, since the model output value is an information granule rather than a numerical value, it is difficult to evaluate the model using a general performance evaluation method. To solve this problem, research on scalability and specificity has been actively conducted as a performance evaluation method for granular models. As the performance index has a higher value, it is a meaningful information granule, and a granular model with excellent performance can be designed. The PI is a non-increasing function of coverage. If the coverage is too high, it may not be allowed. Thus, the coverage can be re-assigned by setting a specific upper limit to generate effective IGs. In order to generate reasonable information granules, two requirements are performed by the performance measure of coverage and specificity of information granules. The coverage [43] indicates whether the target data are included in the formed information granules. In other words, it expresses how much the target total data are accumulated within the range of information granules and the degree of accumulation. The more data accumulated in the information granule, the higher the scalability value. It can prove the legitimacy of the information granule, so it can be a better model in relation to the modeling. Specificity [43] indicates how specific and semantically explainable each information granule is. In other words, information granules should be created as detailed as possible, and each information granule should have an explanatory meaning. When the information granule is in the form of context, the narrower the interval between the upper and lower bounds, the higher the specificity. The general GM generated IGs through cluster centroid and showed a limitation that processed every IG similarly by assigning the same coverage value for every variable. In this study, the coverage of the IGs generated in the output space and the number of IGs in the input space are re-assigning to generate meaningful IGs, which improves GM's prediction performance [43; 44]. Figure 5 shows the process of generating rational IGs by re-assigning the variables of IGs generated in the output space. As shown in Figure 5, the information granule in the form of context has a lower bound and an upper bound. The middle vertex of the triangle is the model value. a1 is the lower bound of the model and a2 is the upper bound of the model. For a1 and a2, \(-\) and \(+\) are the ranges in which to optimize the lower and upper bounds to produce reasonable particles of information. Here, there are upper and lower limits. After random ranges are specified to the upper limits, the upper and lower limits that generate the satisfactory values for the coverage and specificity of IGs are searched using the optimization algorithm GA.  
Fuzzy-Based Granular Model and CGK-Based Fuzzy Granular Model of the Aggregated Structure Based on Rational IGs  
In a hierarchical structure, the CGK-based fuzzy granular model of the aggregated structure uses the linear regression, neural network, and radial basis function neural network, which are generally used as prediction models as sub-models. Moreover, it also uses the CGK-based fuzzy granular model as the upper model.