## 5 Results and Analysis of the Optimization Algorithm  
### Results of the Probabilistic Neural Optimization  
According to the sample screening results, the constituent eigenvectors of the eight characteristic signals are used as the input vectors of the probabilistic neural network.  
In Figure 11, the red dot represents the result of the probabilistic neural network for each input vector classification prediction, while the blue circle represents the actual data result. When the red dot coincides with the blue circle, the classification can be regarded as correct. It can be seen from the figure that when under the normal state, the exhaust pipe leakage state and the intermittent flameout of engine state, some sample points are misclassified, while under the other states, the classification result is completely correct. By statistically correcting the classification prediction results, the classification results of the probabilistic neural network are obtained. The value of \(\sigma\) can affect the accuracy of the classification. When the smoothing factor \(\sigma=0.5\), the classification accuracy of the whole test sample is as high as 92.8%, and the result is ideal. On the one hand, the high accuracy rate proves the excellent classification characteristics of the probabilistic neural network. On the other hand, it also reflects that the screening of characteristic signals and the establishment of fault data samples are accurate.  
The process of PSO optimization probabilistic neural network is based on training samples, through the idea of cross-validation to find the optimal smoothing factor. The cross-validation ideal means to group the original data, one part as a test set and the other part as a verification set, and using the training set to train the classifier firstly, then using the verification set to test the trained model to obtain the classification accuracy as a performance indicator for evaluating classifiers. The basic mathematical relationship of the objective function in a probabilistic neural network can be expressed as Equation (13).  
\[value=f(\sigma) \tag{13}\]  
In the objective function, \(\sigma\) is the input variable, and _value_ represents fitness. The actual meaning of _value_ is the average classification accuracy under cross-validation. The purpose of optimization is to find the maximum fitness value in the optimization process and the corresponding \(\sigma\).  
In the PNN network, the optimization process is relatively simple because only one parameter \(\sigma\) needs to be changed. Set the value of \(\sigma\) to [0.01, 1]. The maximum evolution number of the population in the particle swarm is 50, and the population number is 20. The iterative optimization process and the final classification result are shown in Figure 12. After solving, the probability of a probabilistic neural network is the highest when \(\sigma\) = 0.1803 and finally reaches 96.4%. The error rate that still exists may be caused by the unclassifiable samples.  
Figure 11: Classification results of PNN.  
Figure 12: PNN iterative process and results after PSO optimization. (**a**) Iterative optimization process, (**b**) Final classification results.