## 2 Generation of Information Granules and Design of the Granular Model
### Performance Evaluation of CFCM-Based MGFM  
The GFM (granular fuzzy model) is constructed by the information granules generated using linguistic concepts and information. The information granule can be realized as a specific fuzzy set rather than a single numerical entity. Conventional numerical data are evaluated by the standard deviation as well as the mean or median value. The information granules must be generated rationally so that they can optimally contain the data's information. Additionally, the GFM is evaluated based on specific concepts of coverage and specificity. The first coverage criterion concerns whether experimental evidence of sufficiently high dimension is accumulated to form the information granule that supports the existence of the data. The second specificity criterion is to maintain the high specificity of the information granule generated due to this result. The production of rational information granules focuses on generating meaningful information granules using the original data. To generate rational information granules, the coverage and specificity of the information granules should be satisfied [45]. Figure 5 illustrates a conceptual diagram of the coverage and specificity required to generate rational information granules. Coverage is a criterion to indicate whether data are included in the information granules generated by the GFM, and it shows the degree to which the output values are included in the range of the triangular information granules. When \(Y_{k}\) is a triangular context, it has a value close to 1 if \(y_{k}\) is included in \(Y_{k}=\left[y_{k}^{-},\,y_{k}^{+}\right]\) and a value close to 0 if \(y_{k}\) is not included. In other words, the number of target data included in the output of the GFM can be calculated by the coverage, and then the average value for all data is calculated. When all data are included in the output of the GFM, the coverage has a value close to 1. \[Coverage=\frac{1}{N}\sum_{k=1}^{N}incl(y_{k},\,Y_{k}) \tag{13}\]  
Specificity indicates the precision of the constructed information granules as Equation (14). Shorter information granules indicate higher specificity, meaning that the resulting information granules have a well-defined meaning. The narrower the interval between the upper and lower limit values, the higher the specificity value. When the output \(Y_{k}\) of the GFM decreases to a narrow width, the specificity has a value close to 1. \[Specificity=\frac{1}{N}\sum_{k=1}^{N}exp\big{(}-\left|y_{k}^{+}-y_{k}^{-} \right|\big{)} \tag{14}\]  
The GFM is evaluated by the coverage and specificity of information granules, and we need to find a method that maximizes coverage and specificity simultaneously. Two characteristics of information granules need trade-offs. It means that when the coverage  
Figure 5: The concept of coverage and specificity for rational information granule generation. has a high value, the specificity has a low value. A rational information granule can be expressed by Equation (15), and this equation is referred to as the performance index (PI). \[Performance\ index=coverage(\epsilon)\cdot specificity(\epsilon) \tag{15}\]  
The performance index plays an important role in evaluating the accuracy and clarity of models. Several studies have been conducted on various methods of evaluating the performance of models. Common performance evaluation methods include root mean square error (RMSE) and mean absolute percentage error (MAPE). These methods are mainly used when the output of the model is a numerical value. However, the output of GFM is expressed by fuzzy numbers in a linguistic form. In this sense, the performance evaluation method of GFM is challenging. Thus, we use the performance index as a new performance evaluation method for GFM. The higher the value of the performance index, the more meaningful the information granule, making it possible to design the GFM with excellent performance. Furthermore, we can verify the relationship between the coverage and specificity by the performance index calculated from the GFM.