# Granular Description of Uncertain Data for Classification Rules in Three-Way Decision
## 3 Granular Uncertainty Description
### Multi-Dimensional Data with Two Categories  
In high-dimensional datasets, the uncertain region between classes is complicated, different from the intervals in one-dimensional data. Here, we still start with the simple and typical case, namely a high-dimensional data with two categories. The dataset should be normalized into unit space \([0,1]^{n}\) first. Two clusters are generated through FCM, and the partition matrix and membership are expressed as \(\mathbf{U}\) and \(u_{ik}\) according to Section 2. To construct the granule describing the uncertain region, the process is composed of the following two phases as well, namely the determination of a prototype of granule and the optimization of the information granule's size.  
1. Determination of the uncertain granule's prototype According to the process of describing uncertain regions in one-dimensional data, the prototype of an uncertain region is selected according to memberships. Similarly, we define the prototype of an uncertain region in high-dimensional space according to the following formula:  
\[\mathbf{v}=\text{argmin}_{\mathbf{x}}\big{|}A(\mathbf{x})-B(\mathbf{x})\big{|} \tag{17}\]  
where \(\mathbf{v}\) is the determined prototype; \(A(\mathbf{x})\) and \(B(\mathbf{x})\) are memberships of \(\mathbf{x}\) with respect to two FCM clusters; argmin is the operator for seeking for the minimum value.  
(2) Optimization of information granule's size  
Different from one-dimensional data, the shape of information granules in high-dimensional space has several options based on the selection of distance function. Assuming the granule is formed by hypersphere, the distance between the prototype \(\mathbf{v}\) and any point \(\mathbf{x}\) of the uncertain region could be expressed as seen below:  
\[||\mathbf{x}-\mathbf{v}||^{2}=\sum_{l=1}^{n}\left(x_{j}-v_{j}\right)^{2} \tag{18}\]  
Then, an information granule centering on the prototype \(\mathbf{v}\) is constructed with the size parameter \(\rho\) (\(0\leq\rho\leq 1\)). To optimize the granule size for better uncertainty description, the two metrics (coverage and specificity) are still needed in the process of granule construction. These two functions are expressed as follows:  
\[\text{cov}(\rho)=\sum_{\begin{subarray}{c}k=1\\ \frac{1}{2}|\text{i}_{\text{R}}-\text{v}_{\text{R}}||^{2}\leq\rho^{2}\end{subarray}} ^{N}\frac{1}{2}\{(\max(h(A(\mathbf{x}_{k})),h(B(\mathbf{x}_{k})))+\min(h(A(\mathbf{x}_{k} )),h(B(\mathbf{x}_{k})))\} \tag{19}\]  
\[spe(\rho)=(1-\rho)^{\beta} \tag{20}\]  
Considering these two metrics are in conflict for maximization synchronously, the optimal value could be optimized (maximized) as seen below:  
\[\left\{\begin{array}{l}\rho_{opt}=\text{argmax}_{\rho}Q(\rho)\\ Q(\rho)=\text{cov}(\rho)\cdot spe(\rho)\end{array}\right. \tag{21}\]  
Through the above description, the uncertain region between two classes in high-dimensional space is constructed by a hypersphere granule. Assuming this uncertain granule as \(\Omega\), so the classification rules in (4) for testing data could be expressed in detail as seen below.  
\[\left\{\begin{array}{l}Rule\ 1:\ \text{IF}\ \mathbf{x}_{k}\in\Omega,\ \text{THEN }y_{k}\text{ needs ''clerical review''};\\ Rule\ 2:\ \text{IF}\ \mathbf{x}_{k}\
otin\Omega\text{ \emph{and} }A_{1}(\mathbf{x}_{k})>A_{2}(\mathbf{x}_{k}),\ \text{THEN }y_{k}\text{ is in Class 1};\\ Rule\ 3:\ \text{IF}\ \mathbf{x}_{k}\
otin\Omega\text{ \emph{and} }A_{1}(\mathbf{x}_{k})<A_{2}(\mathbf{x}_{k}),\ \text{THEN }y_{k}\text{ is in Class 2}. \end{array}\right. \tag{22}\]