# Granular Description of Uncertain Data for Classification Rules in Three-Way Decision
## 5 Experiment and Discussion
### Synthetic Two-Dimensional Data  
Considering the two-dimensional data space is a representative of high-dimensional data space, a synthetic dataset is generated to study the proposed method on the high-dimensional case. Similar to the one-dimensional case, we can generate two clusters of Gaussian random data in a two-dimensional space. As shown in Figure 5, each cluster has a number of 500 data points. By uniformizing the original data into [0, 1]\({}^{2}\), the centers of two clusters are determined by FCM as \(\mathbf{v}_{1}=\) (0.3319, 0.5087) and \(\mathbf{v}_{2}=\) (0.7389, 0.5022). Then, according to the formula in (14), the prototype of the uncertain region as \(\mathbf{x}_{0}=\) (0.5354, 0.5055) can be determined.  
Centering on the prototype \(\mathbf{x}_{0}\), different shapes of information granules can be constructed via different distance functions. In this paper, we can simply assume the hypersphere as the basic granule architecture, then the optimal size of granule for uncertainty description in a three-way decision is determined via the optimization of coverage and specificity.  
As the simplest case, we can set the parameter \(\beta\) as 1 for granule construction, then the variances of coverage and specificity for the uncertainty description are shown in Figure 6. It is seen that coverage (a) and specificity (b) are in conflict to become maximum. Their product (c) can be a useful metric to trade off the optimal size \(\rho\) in (18). Therefore, the final size \(\rho\) for constructing the uncertain granule is shown in Figure 6. Different from a one-dimensional data space, \(\beta=1\) is always the optimal solution in granule construction. When a suitable granule is constructed to describe the uncertain region, the performance and overhead of decision-making will be affected. For example, if a large uncertain granule is constructed, namely implying a large percentage of data requires \"clerical review\", then the cost of manual checking will certainty increase, and even the performance on three-way decision might be improved.  
Figure 5: Synthetic two-dimensional data.  
Figure 7a,b shows the values of optimal granule radius \(\rho\) and the percentage of points within the uncertain region (i.e., \(E_{ur}\)) at a different parameter \(\beta\), respectively. It is seen that, when the value of \(\beta\) increases, fewer points within the uncertain region, namely the optimal size of \(\rho\), will decrease. In Figure 7c,d, the classification rate and the total entropy of the uncertain region via (10) are shown at different radius parameter \(\rho\). It is seen that when the value of \(\beta\) increases, the optimal values of \(\rho\) and \(E_{ur}\) correspondingly decrease. That is to say, when the uncertain region size increases, the uncertainty contained by the constructed granule increases but the classification rate decreases. For the well-performed classification rules, both the classification rate and the uncertainty of points for \"clerical review\" should be decided properly. Therefore, by setting different values of \(E_{ur}\) as a constraint condition,  
Figure 6: Performance of different granule sizes.  
Figure 7: Performance of different parameters in uncertain granule construction. (**a**) optimal \(\rho\) vs \(\beta\); (**b**) \(E_{ur}\) vs. \(\beta\); (**c**) classification rate vs. \(\rho\); (**d**) uncertainty \(h(\Omega)\) vs. \(\rho\).