# Granular Description of Uncertain Data for Classification Rules in Three-Way Decision
## 4 Evaluation  
In the classification assessment, confusion matrix [33] is usually used for evaluation. For assessing the performance of the new classification rules with consideration of uncertain regions, we modify the standard confusion matrix as seen below.  
In Table 1, two classes are assumed to be discussed in the modified confusion matrix; \(T_{A}\) and \(T_{B}\) represent true events of two classes in the decision-making, namely truly classified data; \(F_{A}\) and \(F_{B}\) represent false events, namely misclassified data; \(U_{r}\) represents the uncertain data for \"clerical review\". Based on this table, several evaluation metrics can be given out for performance assessment in a three-way classification, such as accuracy, precision, recall and so on [34]. Four commonly used metrics are defined as seen below.  
\[A_{cc}=\frac{N_{T_{A}}+N_{T_{B}}}{N_{T_{A}}+N_{T_{B}}+N_{F_{A}}+N_{F_{B}}} \tag{24}\]  
\[P(ClassA)=\frac{N_{N_{T_{A}}}}{N_{T_{A}}+N_{F_{B}}},\ P(ClassB)=\frac{N_{T_{B}}}{N _{T_{B}}+N_{F_{A}}} \tag{25}\]  
\[R(ClassA)= \frac{N_{T_{A}}}{N_{T_{A}}+N_{F_{A}}},R(ClassB)=\frac{N_{T_{B}}}{N _{T_{B}}+N_{F_{B}}} \tag{26}\]  
\[E_{mis}=\frac{N_{F_{A}}+N_{F_{B}}}{N_{T_{A}}+N_{T_{B}}+N_{F_{A}}+N_{F_{B}}} \tag{27}\]  
where \(N(\text{*})\) is a function counting the number of data points; \(A_{cc}\) represents the accuracy metric, calculated as the percentage of correctly classified points in all classified data; \(P(.)\) and \(R(.)\) are the values of precision and recall in a given class; \(E_{mis}\) represents the percentage of misclassified points in all classified data. On the other hand, the constructed uncertain region will be different by using different parameters of coverage and specificity functions or different distance functions. Therefore, to assess the influence of the constructed uncertain granule in classification, we also define a new metric based on confusion matrix, namely the rate of uncertain points for \"clerical review\", defined below:  
\[E_{ur}=\frac{N_{U_{r}}}{N_{total}}\times 100\% \tag{28}\]  
where \(E_{ur}\) is the new metric; \(N_{Ur}\) is the number of points in uncertain region; \(N_{total}\) is the number of points in the whole dataset. Combining all the above metrics, we could implement the performance evaluation of new classification rules with consideration of uncertain regions and comparison with traditional classification rules.  
While the purpose of reconstructed classification rules is to reduce the cost and to improve the efficiency, we additionally define extra cost coefficients to evaluate classification rules, such as \(\alpha\)_w.r.t._ false cases and \(\gamma\)_w.r.t._ uncertain cases. Since true cases are expected, no extra cost will be added. Then, based on these cost coefficients, we can define a metric named as the total cost to evaluate the performance of classification rules on a given testing set. The new metric is defined below.  
\[\text{Cost}=\alpha\cdot E_{mis}+\gamma\cdot E_{ur} \tag{29}\]  
where the cost function is expressed by considering the metrics of both \(E_{mis}\) and \(E_{ur}\). In a given coefficient pair (\(\alpha\), \(\gamma\)), we could construct the optimal uncertain granules and form classification rules with high performance on both efficiency and effectiveness.