# A Design of CGK-Based Granular Model Using Hierarchical Structure
## 3 IG Creation and Granular Model Design  
### Creating Rational IG  
Computing and inferences in GrC are centered on IGs, which are considered fundamental concepts and algorithms, rather than being centered on numbers. IGs are a core element in GrC because they play an important role in knowledge representation and  
Figure 3: Comparison of clusters created with fuzzy and GK clustering: (**a**) fuzzy clustering; (**b**) GK clustering. processing [42; 43; 44]. Although IGs created using various types of clustering are relatively limited, they can reflect the general structure of some original data. Original data comprising numbers cannot depict the features and connections in the data, but IGs make this possible. Rational IG creation is focused on using the original data to create meaningful IGs. To create rational IGs, two requirements must be satisfied: coverage and specificity. Figure 4.  
Presents the coverage and specificity in IGs. Coverage refers to whether the target data is included in the formed IG. In other words, it shows how much of the overall target data has accumulated within the IG's range, including the extent of the accumulation. The more data that accumulates in the IG, the higher the coverage value. This can verify the validity of the IG, and the model may be better in terms of modeling functions. incl, which is the degree of inclusion and is specified according to the form in which the IG \(Y_{k}\) is created. When \(Y_{k}\) is in context form, incl has a value close to one when \(y_{k}\) is included in \(Y_{k}=~{}[y_{k}^{-},~{}y_{k}^{+}]\), and it has a value close to zero when it is not included. In other words, coverage can be adopted to count the number that includes the data \(y_{k}\) in the granularized output of the granular model, while an average value can be calculated for all data. Ideally, the coverage has a value that is close to one, and all data is included in the granular model's output. \[Coverage=\frac{1}{N}\sum_{k=1}^{N}incl~{}(y_{k},~{}Y_{k}) \tag{15}\]  
Specificity represents how specifically and semantically the IG \(Y_{k}\) can be described. In general, the specificity of a given IG \(Y_{k}\) must satisfy Equation (16). In other words, the IG must be created with as much detail as possible, and each IG must have a meaning that can be described. When an IG is in context form, the specificity becomes higher as the interval, i.e., the distance between the upper and lower bounds, becomes narrower. If the IG \(Y_{k}\) is reduced to point form, the specificity arrives at a value close to one. \[\text{if }Y_{k}\subset{Y_{k}}^{\prime}\text{ then }specificity(Y_{k})~{}\succ~{}specificity \big{(}{Y_{k}}^{\prime}\big{)},~{}\text{and }specificity(\{y\})~{}=1 \tag{16}\]  
\[Specificity=\frac{1}{N}\sum_{k=1}^{N}exp\big{(}-|{y_{k}}^{+}-{y_{k}}^{-}| \big{)} \tag{17}\]  
A continuous decreasing function of the interval length can be considered instead of the exponential function used in Equation (17). Coverage and specificity can be adopted to evaluate the IG's validity and the granular model's prediction performance. In other words, the granular model can be evaluated by considering the coverage and specificity of the IG, and a method that can simultaneously maximize coverage and specificity should  
Figure 4: Concepts of coverage and specificity for creating rational IGs: (**a**) concept of coverage; (**b**) concept of specificity. be determined. These two properties have a trade-off relationship. This implies that the higher the coverage value, the lower the specificity value. Rational IGs can be represented by Equation (18), and this is called the PI. The PI plays an important role in evaluating the model's accuracy and clarity, and various methods for evaluating model performance have been developed. General performance evaluation methods include root-mean-square error (RMSE) and mean absolute percentage error (MAPE).