## 3 Methodology
### Pythagorean Fuzzy Rough Approximation Construction
The loss from delaying the decision is less than the loss from rejecting the decision. Similarly, when the object belongs to the abnormal state \(\left(\
eg X_{\alpha}^{\beta}\right)\), the loss from taking a rejection decision is less than from delaying the decision. The loss from delaying the decision is less than the loss from accepting the decision. This is consistent with human perception. Considering the unity of symbols, in Table 3, let \(I\left(X_{\alpha}^{\beta},\theta_{\alpha}^{\beta}(x)\right)\) be the conditional probability when the object \(\vartheta_{\alpha}^{\beta}(x)\) belongs to the state \(X_{\alpha}^{\beta}\) and \(I\left(\
eg X_{\alpha}^{\beta},\vartheta_{\alpha}^{\beta}(x)\right)\) be the conditional probability when the object \(\vartheta_{\alpha}^{\beta}(x)\) belongs to the state \(\
eg X_{\alpha}^{\beta}\), where \(I\left(X_{\alpha}^{\beta},\vartheta_{\alpha}^{\beta}(x)\right)+I\left(\
eg X _{\alpha}^{\beta},\vartheta_{\alpha}^{\beta}(x)\right)=1\). Therefore, the risk loss functions of the object \(\vartheta_{\alpha}^{\beta}(x)\) under the actions \(a_{P}\), \(a_{B}\) and \(a_{N}\) are  
\[R_{\omega}\left(a_{P}\middle|\vartheta_{\alpha}^{\beta}(x)\right) =E_{\omega}(A(\lambda_{PP}))I\left(X_{\alpha}^{\beta},\vartheta_{ \alpha}^{\beta}(x)\right)+E_{\omega}(A(\lambda_{PN}))I\left(\
eg X_{\alpha}^{ \beta},\vartheta_{\alpha}^{\beta}(x)\right) \tag{16}\]  
\[R_{\omega}\left(a_{B}\middle|\vartheta_{\alpha}^{\beta}(x)\right) =E_{\omega}(A(\lambda_{BP}))I\left(X_{\alpha}^{\beta},\vartheta_{ \alpha}^{\beta}(x)\right)+E_{\omega}(A(\lambda_{BN}))I\left(\
eg X_{\alpha}^{ \beta},\vartheta_{\alpha}^{\beta}(x)\right) \tag{17}\]  
\[R_{\omega}\left(a_{N}\middle|\vartheta_{\alpha}^{\beta}(x)\right) =E_{\omega}(A(\lambda_{NP}))I\left(X_{\alpha}^{\beta},\vartheta_{ \alpha}^{\beta}(x)\right)+E_{\omega}(A(\lambda_{NN}))I\left(\
eg X_{\alpha}^{ \beta},\vartheta_{\alpha}^{\beta}(x)\right) \tag{18}\]  
According to the Bayesian minimum risk principle, the following decision rules can be obtained:  
P(1) if \(R_{\omega}\left(a_{P}\middle|\vartheta_{\alpha}^{\beta}(x)\right)\leq R_{ \omega}\left(a_{B}\middle|\vartheta_{\alpha}^{\beta}(x)\right)\) and \(R_{\omega}\left(a_{P}\middle|\vartheta_{\alpha}^{\beta}(x)\right)\leq R_{ \omega}\left(a_{N}\middle|\vartheta_{\alpha}^{\beta}(x)\right)\), then \(x\in POS\left(X_{\alpha}^{\beta}\right)\);  
B(1) if \(R_{\omega}\left(a_{B}\middle|\vartheta_{\alpha}^{\beta}(x)\right)\leq R_{ \omega}\left(a_{P}\middle|\vartheta_{\alpha}^{\beta}(x)\right)\) and \(R_{\omega}\left(a_{B}\middle|\vartheta_{\alpha}^{\beta}(x)\right)\leq R_{ \omega}\left(a_{N}\middle|\vartheta_{\alpha}^{\beta}(x)\right)\), then \(x\in BND\left(X_{\alpha}^{\beta}\right)\);  
N(1) if \(R_{\omega}\left(a_{N}\middle|\vartheta_{\alpha}^{\beta}(x)\right)\leq R_{ \omega}\left(a_{P}\middle|\vartheta_{\alpha}^{\beta}(x)\right)\) and \(R_{\omega}\left(a_{N}\middle|\vartheta_{\alpha}^{\beta}(x)\right)\leq R_{ \omega}\left(a_{B}\middle|\vartheta_{\alpha}^{\beta}(x)\right)\), then \(x\in NEG\left(X_{\alpha}^{\beta}\right)\). Based on Equations (16)-(18), the decision rules P1\(\
eg\)N1 can be further simplified as:  
P(1) if \(I\left(X_{\alpha}^{\beta},\vartheta_{\alpha}^{\beta}(x)\right)\geq\overline{\alpha}\) and \(I\left(X_{\alpha}^{\beta},\vartheta_{\alpha}^{\beta}(x)\right)\geq\overline{\beta}\), then \(x\in POS\left(X_{\alpha}^{\beta}\right)\);  
B(1) if \(I\left(X_{\alpha}^{\beta},\vartheta_{\alpha}^{\beta}(x)\right)\leq\overline{\alpha}\) and \(I\left(X_{\alpha}^{\beta},\vartheta_{\alpha}^{\beta}(x)\right)\geq\overline{\beta}\), then \(x\in BND\left(X_{\alpha}^{\beta}\right)\);  
N(1) if \(I\left(X_{\alpha}^{\beta},\vartheta_{\alpha}^{\beta}(x)\right)\leq\overline{\beta}\) and \(I\left(X_{\alpha}^{\beta},\vartheta_{\alpha}^{\beta}(x)\right)\leq\overline{\gamma}\), then \(x\in NEG\left(X_{\alpha}^{\beta}\right)\).