## 3 Results and Discussions
### Tennessee Eastman Process
#### 3.2.2 Performance Analysis of Fault Classification for Different Optimized PNN Schemes
Category 9, 10, 15, 16, and 19 faults have \"local optimization\" in CS-BP. Thus, judging the diagnostic performance of different classifiers only by the rate at which each fault category is found is unfair and insufficient. In order to completely analyze the defect diagnostic performance of all classifiers and demonstrate the efficacy of the proposed model, the study chose the accuracy rate and F1-score as evaluation indices. The confusion matrix is a crucial metric for assessing the  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Category**} & \multicolumn{2}{c}{**MSSA-PNN**} & \multicolumn{2}{c}{**LDA**} & \multicolumn{2}{c}{**QDA**} & \multicolumn{2}{c}{**KNN**} & \multicolumn{2}{c}{**SVM**} & \multicolumn{2}{c}{**MaxEnt**} & \multicolumn{2}{c}{**CS-BP**} \\ \cline{2-13}  & **N** & F & **N** & **F** & **N** & **F** & **N** & **F** & **N** & **F** & **N** & **F** & **N** & **F** \\ \hline
1 & 1.00 & 1.00 & 1.00 & 0.99 & 1.  
performance of classification models. Table 9 shows that it has four values: true positive (TP), true negative (TN), false positive (FP), and false-negative (FN). TP denotes the number of correctly predicted positive samples; TN denotes the number of correctly predicted negative samples; FP denotes the number of predicted positive samples that are actually negative; FN denotes the number of predicted negative samples that are actually positive. Precision refers to the ratio between the number of samples that are correctly predicted as positive labels and the total number of samples that are predicted as positive labels, as shown in Equation (21):  
\[\text{precision }=\frac{\text{TP}}{\text{TP}+\text{FP}} \tag{21}\]  
The recall is the ratio of the number of samples that were correctly predicted to have positive labels to the number of samples that were labeled positive, as shown in Equation (22):  
\[\text{recall }=\frac{\text{TP}}{\text{TP}+\text{FN}} \tag{22}\]  
The two key evaluation indicators required in this paper can also be obtained from Table 9, and the calculation formula is as follows:  
\[\text{accuracy }=\frac{\text{TP}+\text{TN}}{\text{TN}+\text{TP}+\text{FN}+ \text{FP}} \tag{23}\]  
\[\text{F1-score }=\left(1+\beta^{2}\right)\frac{\text{precision }\times\text{ recall}}{\left(\beta^{2}\cdot\text{ precision }\right)+\text{ recall}} \tag{24}\]  
In this paper, \(\beta=1\), indicating that precision and recall are considered with the same weight. The accuracy of MSSA-PNN and other classifiers is displayed in Table 10. MSSA-PNN's diagnostic accuracy is 88%, which is greater than LDA's 69%, QDA's 81%, KNN's 70%, SVM's 75%, MaxEnt's 52%, and CS-BP's 84%. In addition, MSSA-PNN has a considerable advantage over other classifiers in eight fault categories: category 3, category 9, category 10, category 11, category 15, category 16, and category 19. Although it falls short in a few defect categories, the disparity is negligible. Therefore, MSSA-PNN provides more accurate fault diagnosis performance. The F1-score of MSSA-PNN and other classifiers are displayed in Table 7.0 From comparing Tables 8 and 11, it is evident that certain classifiers may have a greater rate of fault diagnosis than MSSA-PNN. However, the F1-score is lower than the MSSA-PNN score. For instance, in the SVM's categories 8, 9, 14, etc., the fault diagnosis rates are more significant than those of MSSA-PNN, but their F1-score is significantly lower than that of MSSA-PNN. It shows that SVM's category 8 and other fault categories display over-fitting. Consequently, the F1-score and the accuracy rate can be used together to give a complete picture of the classifier's ability to find faults. Meanwhile, the average F1-score of MSSA-PNN is 91%, which is higher than that of other classification models, while LDA is at 74%, QDA is at 83%, KNN is at 79%, SVM is at 84%, MaxEnt is at 46%, and CS-BP is at 88%.