# Granular Description of Uncertain Data for Classification Rules in Three-Way Decision
## 3 Granular Uncertainty Description
### One-Dimensional Data with Two Categories  
As we know, one-dimensional data with two categories is the simplest and primary data formation in pattern recognition. Assuming a one-dimensional dataset \(\{\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{N}\}\) uniformized into [0, 1], it is divided into two clusters (ClusterA and ClusterB) via FCM, shown in Figure 2, where uncertain data points requiring \"clerical review\" are included in the square region. As the above assumption, there are two parameters required in the process of granular descriptor construction, such as the prototype \(\mathbf{x}_{0}\) and the size (\(\tau_{1}\) and \(\tau_{2}\)) of the granule. (1) Selection of the prototype \(x_{\rm o}\)  
The function of the granular descriptor's prototype is to locate the uncertain region so we can select the prototype as the median point having the highest uncertainty. As we know, the membership could reflect the uncertainty of data points to some extent, so they could be referred to in order to select the prototype. First, by referring to (1) and setting the dataset's dimension \(n=1\) and the number of clusters \(c=2\), a point's memberships to two clusters are calculated in detail as seen below. \[A(x) = 1/\sum_{i=A,B}\left(\frac{\left|x-v_{\rm A}\right|}{\left|x-v_{i} \right|}\right)^{2}=\frac{\left|x-v_{\rm B}\right|^{2}}{\left|x-v_{\rm A} \right|^{2}+\left|x-v_{\rm B}\right|^{2}} \tag{5}\] \[B(x) = 1/\sum_{i=A,B}\left(\frac{\left|x-v_{\rm B}\right|}{\left|x-v_{i} \right|}\right)^{2}=\frac{\left|x-v_{\rm A}\right|^{2}}{\left|x-v_{\rm A} \right|^{2}+\left|x-v_{\rm B}\right|^{2}} \tag{6}\]  
where \(A(x)\) and \(B(x)\) are the calculation of membership with respect to (w.r.t) two clusters and \(v_{\rm A}\) and \(v_{\rm B}\) are the prototypes of two fuzzy clusters. Then, according to Figure 2, the median point \(x_{\rm o}\) most possibly could be decided as the following criterion:  
\[A(x_{\rm o})\approx B(x_{\rm o}) \tag{7}\]  
(2) Determination of the uncertain region size  
It is seen from Figure 2 that the uncertain region is determined by two parameters, namely \(\tau_{1}\) and \(\tau_{2}\). These two parameters determine the range of data points in each cluster required for \"clerical review\". Therefore, with respect to one-dimensional data, the descriptor of the uncertain region around \(x_{\rm o}\) is expressed by the following interval:  
\[[x_{\rm o}-\tau_{1},x_{\rm o}+\tau_{2}] \tag{8}\]  
In the real-world decision-making process, we expect to obtain high accuracy with low computation overhead, so it is required to correctly classify which data points require \"clerical review\". To realize these objectives, selection of the uncertain region's size is first required to contain most of the data with high uncertainty degree. Then, this region should also guarantee high specificity of uncertain data for high-performed decision-making. To realize these requirements, the justifiable granulating algorithm [29] is applied here to realize the optimization of \(\tau_{1}\) and \(\tau_{2}\) separately. For determining the value of \(\tau_{1}\), we define coverage and specificity functions first [30]. To describe the coverage of uncertain data, a function considering the data points' uncertainty degree is defined as seen below. \[\mathrm{cov}(\tau_{1})=\sum_{x_{k}\in[x_{0}-\tau_{1},\ x_{0}]}\frac{1}{2} \{(\max(h(A(x_{k})),h(B(x_{k})))+\min(h(A(x_{k})),h(B(x_{k})))\} \tag{9}\]  
where \(x_{k}\) is a data point belonging to the region \([x_{0}-t_{1},\ x_{0}]\) and \(h(. )\) is denoted as an entropy function representing the uncertainty degree. To describe the uncertainty of data in the boundary of two classes, the function \(h(. )\) is required to satisfy the following conditions. Firstly, in the left region \([0,\,1/2]\), \(h(. )\) is a monotone increasing function; conversely, it is a  
Figure 2: Diagram of uncertain data in one-dimensional data space. monotone decreasing function in the region of [1/2, 1]. Secondly, it should satisfy \(h(0)=h(1)=0\), \(h(1/2)=1\).