## 2 Interval-Based GM
### IFCM Clustering  
In this study, interval-based GM is designed by IFCM clustering to generate information granulation. The IFCM clustering method creates an interval in the output space, taking into account the pattern characteristics of the input and output spaces. The general fuzzy clustering method [43] estimates a cluster by using the distance between its center and the input data in the input space without considering the output space. On the other hand, the IFCM performs clustering by considering the pattern characteristics of the output space, and thus, performs better than the existing clustering method. Next, IFCM determines the cluster centers and the membership matrix using the following steps  
1. Set the number of intervals \(I\) (\(1<I<\) q) and the number of clusters per interval \(C\) (\(2<C<\) n). 2. Initialize the membership matrix with random values between 0 and 1. \[U\big{(}\big{[}u_{ij}\big{]}i=1,\ \ldots,\ C,\ j=1,\ \ldots,\ n\big{)}\] (1)
3. Calculate \(C_{i}\) (i = 1, 2,..., \(C\)), which is the center of each cluster, by the following equation. \[C_{i}\ =\ \frac{\sum_{j\ =\ 1}^{n}u_{ij}^{m}\ x_{j}}{\sum_{j\ =\ 1}^{n}u_{ij}^{m}}\] (2)
4. Compute the partition matrix U considering \(f_{j}\) as follows \[u_{ij}\ =\ \frac{f_{i}}{\sum_{k\ =\ 1}^{C}\big{(}\frac{\big{[}x_{k,x}-c_{i} \big{]}\big{]}}{\big{[}x_{k,x}-c_{j}\big{]}}\big{)}^{\frac{2}{(m\ -\ 1)}}}\] (3) where, \(f_{j}\) represents the degree of inclusion of \(x_{j}\) in the generated cluster. Then, \(f_{j}\ =\ A(y_{i}),\ j\ =\ 1,\ \ldots,\ n\) can be denoted by A to \(y_{j}\). \(u_{ij}\) denotes the membership degrees induced by the \(i\)th cluster and the \(j\)th data. \(x_{ck}\) represents the \(k\)th input data in the \(c\)th cluster. 5. When the equation is satisfied, the above process stops. Otherwise, the process is started again from Step 3. \[J\ =\ \sum_{j\ =\ 1}^{n}\sum_{i\ =\ 1}^{C}u_{ij}^{m}\ \|x_{j}\ -\ C_{i}\|^{2}\] (4)  
In the IFCM, the number of interval and clusters per interval must be set in advance. Figure 1 shows the concept of the IFCM clustering method, where the intervals and clusters are estimated by setting the numbers of sections and clusters to 4 and 3, respectively. As shown in Figure 1, each interval has linguistic meaning and each cluster produced by the intervals is represented by fuzzy if-then rules. Figure 1: Concept of interval-based fuzzy clustering concept. In general, when creating an interval in the output space, the method of dividing the interval without an even overlapping is used. In this study, the performance of GM is evaluated by adding a method to divide the interval flexibly and without any overlapping, based on a stochastic distribution, and a method to divide the interval evenly at overlapping intervals. In general, a method of dividing an interval not being evenly overlapped is used to divide the output space evenly. Thus, we only need to consider the spacing of each interval. We define the general partitioning method as Case 1. The interval division method used in this study is as follows. First, the method of dividing the interval without overlapping flexibly adjusts the interval length from the probability distribution of data in the output. The interval of a portion with a large distribution value is short, while that with a small distribution value is long. We define the method of flexible and non-overlapping partitioning as Case 2. Second, the method of dividing the interval evenly and overlapping a certain range is similar to the general method, but the difference lies in overlapping the ends of each interval. This method allows each section to overlap a certain range to find additional similar features. Figure 2 shows a conceptual diagram of Cases 1, 2, and 3 according to the method of interval division. Figure 2a shows Case 1, which divides the output space uniformly, and Figure 2b shows Case 2, which flexibly divides the output space based on a stochastic distribution. The second interval of Figure 2b is observed to be short, because the distribution value is large.