# Granular Description of Uncertain Data for Classification Rules in Three-Way Decision
## 5 Experiment and Discussion  
### Synthetic One-Dimensional Data  
Firstly, a simple case is generated, i.e., one-dimensional synthetic data with two classes. This dataset contains a total of 1000 data points equally distributed in two classes with the Normal distribution \(N(-2,1)\) and \(N(2,1.5)\), as shown in Figure 3a. By uniformizing the data into [0, 1], the FCM algorithm is utilized to classify the original data according to Section 2. Then, two fuzzy clusters are generated with prototypes of \(v_{\text{A}}=0.2800\) and \(v_{\text{B}}=0.6699\). When mapping them in the original domain, their values are \(v_{\text{A}}=-2.0736\) and \(v_{\text{B}}=2.0423\) which are consistent with the distribution of the given synthetic data. Moreover, the membership matrix \(\mathbf{U}\) through FCM is calculated. Figure 3b shows the distribution of memberships to two clusters. Then, according to the criterion in (7), the prototype of the uncertain region between two clusters is determined as the median point \(x_{0}=0.4737\). Next, based on the processes of constructing the uncertain granule in a one-dimensional dataset, two size parameters (\(\tau_{1}\) and \(\tau_{2}\)) are calculated using justifiable granulati \begin{table}
\begin{tabular}{c c c c} \hline \hline  & & \multicolumn{2}{c}{**Target**} \\ \cline{3-4}  & & **ClassA** & **ClassB** \\ \hline \multirow{3}{*}{Decision} & ClassA & \(T_{A}\) & \multirow{3}{*}{\(U_{r}\)} \\  & Uncertain & & \\ \cline{1-1}  & ClassB & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Confusion matrix of two-classes classification. Figure 3: Distribution of one-dimensional synthetic data. (**a**) Original distribution; (**b**) FCM clustering and membership distribution. Figure 4 shows the values of \(cov(\tau)\), \(spe(\tau)\) and their product \(Q=spe(\tau)\times cov(\tau)\) at different values of the size parameter \(\tau\). Here, the parameter \(\beta\) is set to 1, and the final size of the uncertain region is determined as \([x_{0}-\tau_{1},x_{0}+\tau_{2}]=[0.1337,0.8137]\). It is known that the optimal uncertain region under \(\beta=1\) is large, which may generate huge cost in (29). The other challenge in a three-way decision is that it is not clear which points are uncertain and how many data points are true uncertain data. Therefore, to address this problem, we can assume a proportion (\(E_{ur}\)) of data as uncertain data, then set constraints to determine the optimal size parameters for constructing the uncertain region. By forming new classification rules with consideration of the constructed uncertain granule, four metrics in (24)-(27) are calculated for performance analysis, as presented below. It is found from the results of Table 2 that, when the value of \(E_{ur}\) increases, the size of the uncertain region increases, implying more uncertain points are taken for \"clerical review\". Values of metrics (\(Acc\), \(R\), \(P\), \(E_{mis}\)) for evaluating classification performance are also improved with the increasing of \(E_{ur}\). Moreover, as the proportion of uncertain data increases to more than 15%, the improvement of evaluation metrics has no big changes. Therefore, with consideration of computing overhead, there should be a trade-off between performance and cost to select suitable parameters in practical three-way decision processes. For example, in this case, system operators can choose \(E_{ur}=15\%\) as the parameters in a decision based on the results of Table 2.