Application of crested porcupine optimized-back propagation neural network under feature dimensionality reduction for rolling bearing fault diagnosis

2.3.1. Initialization phase

In this phase, the CPO operates similarly to other population-based metaheuristic algorithms. CPO begins by randomly initializing a set of candidate solutions, with the initialization formula given by:

where, ${\overrightarrow{X}}_i$ is the $i$ alternative scheme in the solution space, $\overrightarrow{L}$ and $\overrightarrow{U}$ are the maximums and minimums in space scope, respectively, and $\overrightarrow{r}$ is stochastic number from 0 to 1.

2.3.2. Cyclic population reduction

In this phase, the algorithm simulates only the defensive mechanisms activated by threatened crested porcupines. This phase occurs when the population size decreases to a certain threshold, after which the algorithm increases the population size to enhance diversity and accelerate convergence, continuing this process until the maximum number of epochs is reached. The mathematical formula is:

where, $T$ is a parameter of the frequency of circulation, $t$ is the circulation frequency evaluation, $T_{max}$ is $t$ upper limit, $\%$ is remainder, $N$ is the newly population and $N_{min}$ is the min number of newly population.

2.3.3. Exploration phase

(1) First and second defense strategies. The CPO algorithm explores the solution by simulating the first two defense mechanisms of the crested porcupine. This phase can be considered a global search in the algorithm, aimed at discovering potentially optimal regions within the solution space. The equation for visual perception is Eq. (5), and the equation for acoustic defense is Eq. (6):

where, $\overrightarrow{X_{CP}^t}$ is the optimal solution of $t$, predator at the $t$ moment position is indicated by $\overrightarrow{{y_i}^t}$, ${\tau }_1$ is the stochastic number of normal distribution, ${\tau }_2$ and ${\tau }_3$ are random values in the interval [0, 1]. $r1$ and $r2$ are two stochastic number from $\left[1,N\right]$.

(2) Third and fourth defense strategies. The CPO algorithm refines the identified optimal region by simulating the last two defense mechanisms of the crown porcupine. This phase can be seen as a local search in the algorithm, aiming to find a more precise solution within the identified region. The equation for olfactory recognition is Eq. (7), and the equation for physical attack defense is Eq. (8):

where, $r3$ is stochastic number from $\left[1,N\right]$. $\delta$ is the parameter of the control search route. ${\tau }_3$ and ${\tau }_4$ are random values in the interval $\left[\mathrm{0,1}\right]$. $S_i^t$ is the olfactory diffusion factor. $\alpha$ is the convergence speed factor, and $\overrightarrow{{F_i}^t}$ represents $i$ predator for CP strength.

To more intuitively demonstrate the implementation steps of CPO, the pseudocode of CPO is shown in Table 2.

The defense mechanism integrated within the CPO algorithm inherently mitigates the shortcomings of BPNN. The initial and secondary defense mechanisms prioritize global exploration, thus reducing the propensity of BPNN to converge on local minima. Simultaneously, the third and fourth defense mechanisms promote regional development, expediting the convergence to the best solution by identifying prospective areas. The cyclic population reduction process preserves diversity while balancing exploration and exploitation, thus diminishing the susceptibility of BPNN to initial weight selection.

4.2.1. t-SNE visualization

Fig. 6 displays the t-SNE visualization of each algorithmic model. A comparison of the t-SNE plots reveals that the MCB-Net model (Fig. 6(g)) exhibits superior clustering results compared to other models. Specifically, the t-SNE plot of MCB-Net shows clearer category separation and less overlap, which indicates that the model has higher accuracy and efficiency in feature extraction and classification. In addition, the MCB-Net model is able to handle complex nonlinear relationships more efficiently by optimizing the weights and thresholds of the BP neural Network, which improves the prediction accuracy of the model. As shown in the t-SNE plot, data points of different categories form more dispersed and independent clusters in the low-dimensional space.

4.2.2. Accuracy analysis

Table 6 shows the accuracy and standard deviation of different methods. Fig. 7 plots the accuracy box plot of different methods under 20 times. As shown in Table 6, MCB-Net achieved an average accuracy of 98.20 %, the highest among all algorithms. The result was 3.95 % higher than the worst-performing LEA and 0.40 % higher than the best-performing S-CNN. Fig. 7 also shows that MCB-Net has the highest accuracy on the vertical axis, with the accuracy of the intermediate data concentrated in the high range. Compared with other algorithms, MIV-LEA-BP has the lowest box position and large fluctuations. The boxes of PSO and GA are also much lower than MCB-Net. Although HO and S-CNN have high performance, the accuracy of the middle segment data of MCB-Net is more concentrated. In addition, the standard deviation of MCB-Net is 1.52 %, second only to S-CNN’s 0.74 % among all models, but its 95 % confidence interval is [97.49, 98.91], which almost covers the upper limit of all other algorithms. The results demonstrate that even with fluctuations, the overall performance of MCB-Net remains significantly higher than that of most algorithms.

Fig. 7Box plots of different methods

To more clearly demonstrate the results of each method, Fig. 8 shows the confusion matrix of the corresponding model. MCB-Net shows significantly better classification performance than other methods. S-CNN misclassified categories 4, 7, 9, and 10, particularly category 10, where 8.33 % were misclassified as category 6. The SVM correctly classified category 6 only 86.11 %, with a high percentage of misclassifications into category 8 (8.33 %). Misclassifications were also prominent in categories 7, 8, and 9, resulting in the worst overall performance. For example, the optimized algorithm MIV-LEA-BP misclassified 13.89 % of class 3 into class 9, 5.56 % of class 7 into class 6, and 8.33 % into class 9. In contrast, MCB-Net, excluding classes 3, 4, and 9, had a 100 % accuracy rate for all other classes, significantly higher than other methods. The good performance is mainly due to the precise measurement of feature importance by MIV, combined with the efficient optimization of BP neural network parameters by the CPO optimization strategy, which enables the model to more accurately capture the discriminative features of samples, thereby significantly improving the classification accuracy of samples in each category and effectively reducing the degree of confusion between categories.

Fig. 8Accuracy analysis

a) S-CNN

b) MIV-LEA-BP

c) SVM

d) MIV-HO-BP

e) MIV-GA-BP

f) MIV-PSO-BP

g) MCB-Net

4.2.3. Error analysis

This research quantitatively evaluates the merits and demerits of the four algorithms by conducting 20 epochs for each and use MAE and MSE as comparative assessment measures. Table 7 and 8 present the MAE and MSE for each approach. Excluding MCB-Net, the MAE indicator reveals that the average error values of the remaining methods are clustered between 0.037 and 0.043. Notably, MIV-PSO-BP exhibits the lowest average error at 0.037432; however, the discrepancy between the maximum and minimum values is considerable, suggesting that its overall performance stability is moderate. The average MAE of MCB-Net is 0.045083, marginally above that of certain approaches; however, the disparity between its highest and minimum values is merely 0.001884, indicating substantial stability and consistency. The fluctuation ranges of other approaches, specifically MIV-HO-BP and S-CNN, are 0.002398 and 0.000472, respectively, signifying a degree of uncertainty in both under varying operating conditions.

In terms of MSE indicators, the advantage of MCB-Net is more obvious, with an average MSE of only 0.009752, which is significantly lower than other methods. Specifically, MCB-Net is 33 % higher than MIV-LEA-BP and 15 % higher than MIV-PSO-BP and MIV-GA-BP. In addition, the gap between the maximum and minimum values of MCB-Net is only 0.000391, further demonstrating that it can maintain highly consistent prediction performance. To more clearly demonstrate the differences between the methods, Fig. 9 plots the MAE and MSE comparisons for each method. The figure clearly shows that MCB-Net has the most significant overall advantage in MSE. In contrast, the other algorithms have higher mean errors or larger fluctuations in MSE, indicating greater instability in their prediction results.

4.2.4. Volatility analysis

Algorithm volatility is commonly used to evaluate the stability and predictability of its performance. This study conducted 20 independent epochs for each algorithm to analyze its convergence behavior. Fig. 10 illustrates the volatility of each algorithm during epochs. The horizontal axis represents the number of epochs, while the vertical axis shows the SD of the convergence curve, reflecting the degree of convergence fluctuation within a single run. Higher values on the vertical axis indicate lower convergence stability during that epochs. It should be noted that since the performance of SVM is extremely stable, volatility is not plotted. From the figure, we can see that the MCB-Net curve shows much better stability than other algorithms during the 20 epochs. Specifically, MIV-PSO-BP and MIV-LEA-BP experience large fluctuations during epochs, and the S-CNN model exhibits large local interference. However, the overall fluctuation amplitude of the CPO curve is gentle and regular, indicating that when MCB-Net adjusts the BP neural network parameters through the CPO optimization strategy, combined with MIV's precise measurement of feature importance, the model can more stably approach the optimal solution during training epochs.

Fig. 9MAE and MSE comparison of four algorithms (CWUR)

a) MAE

b) MSE

Fig. 10Volatility across algorithms

4.3.1. Accuracy analysis

Table 10 and Fig. 11 show the performance of different methods. The accuracy of MCB-Net reached 97.26 %, the highest among all methods, an improvement of 1.06 % compared to SVM and 1 % compared to S-CNN’s 96.93 %. Compared with other optimization algorithms, the accuracy of BPNN optimized by CPO is improved by 2.03 %. Further combining the SD, we can see that MCB-Net's SD is only 1.05%, indicating that its results are less volatile across multiple experiments and exhibit greater stability. Although S-CNN’s SD is 0.83 %, showing better stability, its accuracy is still lower than MCB-Net, indicating that CPO achieves a better balance between stability and accuracy. The box plot results further show that the distribution range of MCB-Net is more compact and its overall position is higher, which means that it cannot only remain stable in different experiments but also achieve better performance.

Fig. 11Box plot of different methods (MFTP)

Fig. 12Accuracy analysis

a) S-CNN

b) MIV-LEA-BP

c) SVM

d) MIV-HO-BP

e) MIV-GA-BP

f) MIV-PSO-BP

g) MCB-Net

Fig. 12 shows the confusion matrix of different methods. As can be seen from the figure, except for MCB-Net and MIV-PSO-BP, the other algorithms all made misclassifications in category 1. The LEA-optimized model had the largest classification errors, with significant misclassifications in categories 1, 6, 7, 8, and 9. The SVM performed second worst, primarily in categories 4, 7, and 8. Meanwhile, the MCB-Net achieved accuracy rates above 95 % for all categories except categories 2 and 9. Other algorithms, such as S-CNN and SVM, exhibit relatively high values in some off-diagonal locations. Judging from the confusion matrix, the MCB-Net algorithm demonstrates superior classification performance, thanks to CPO's fine-tuning of model parameters during the optimization process, which better captures data features and thus excels in multi-class classification tasks.

4.3.2. Error analysis

In this dataset, the same 20 experimental epochs were conducted, and the selection of evaluation index parameters aligned with the CWUR dataset. The results for MAE and MSE are presented in Table 11 and 12, as well as Fig. 13. As can be seen from the table, the MCB-Net algorithm shows both balance and superiority. The maximum MAE is 0.099026, which is significantly lower than 0.120079 of MIV-LEA-BP and 0.100611 of SVM. It effectively avoids large deviations in the prediction process, thereby reducing the adverse effects of extreme errors on the overall performance. Although the minimum value is 0.098726, which is slightly higher than some optimization algorithms, the difference between the maximum and minimum values is only about 0.0003, highlighting its robustness in terms of control errors. The MSE indicator shows that compared with other methods, the average MSE is only slightly higher than S-CNN's 0.037496, and the overall performance is superior. In addition, the difference between the maximum and minimum MSE values is only about 0.0010, which is much smaller than 0.0037 of MIV-LEA-BP and 0.0023 of MIV-HO-BP. The visualization in Fig. 13 more directly demonstrates the performance of MCB-Net. With the exception of LEA, which has a significantly higher MAE, the other algorithms are largely comparable. A comparison of MSE reveals that MCB-Net achieves the best performance, further validating its ability to achieve high accuracy and stability in complex prediction tasks.

4.3.3. Volatility analysis

Fig. 14 shows the SD of the convergence curves of each algorithm after 20 epochs. Overall, the MCB-Net algorithm has a very small standard deviation fluctuation during the entire iteration process, showing high stability. In contrast, the standard deviation curve of the PSO algorithm fluctuates greatly from the 10th to the 15th iteration, indicating that its optimization effect is unstable at this stage.

Fig. 13MAE and MSE comparison of four algorithms (MFPT)

a) MAE

b) MSE

Fig. 14Volatility of the MFPT bearing dataset

The LEA algorithm also experienced similar dramatic fluctuations between the 5th and 10th epochs, indicating that its optimization process at this stage was greatly disturbed. Although the S-CNN and HO algorithms showed a certain downward trend in the middle and late stages, the overall fluctuation range was still higher than that of the MCB-Net algorithm, and the convergence speed was relatively slow, failing to effectively suppress the uncertainty in the training process. For example, after the 15th iteration, the HO standard deviation curve shows a significant rebound. MCB-Net effectively reduces the randomness and uncertainty in the optimization process through optimized control strategies and improved update mechanisms, enabling the algorithm to converge more smoothly to the optimal solution.

4.4.1. Accuracy analysis

Table 14 shows the comparative experimental results of various methods on the SEU dataset. From the experimental results of the SEU dataset presented in the table, the MCB-Net method achieved the highest accuracy of 97.78 %, with a standard deviation of 0.98 %, showing good stability. Although MIV-HO-BP is close in accuracy, reaching 96.82 %, its standard deviation is 1.10 %, indicating that MCB-Net has more consistent classification performance while maintaining high accuracy.

Fig. 16Box plot of different methods (SEU)

Fig. 17Accuracy analysis

a) S-CNN

b) MIV-LEA-BP

c) SVM

d) MIV-HO-BP

e) MIV-GA-BP

f) MIV-PSO-BP

g) MCB-Net

Fig. 16 is a box plot of each method under the SEU dataset. Fig. 16 shows that the overall distribution of MCB-Net is relatively concentrated, and the median position is relatively high. However, it must be pointed out that MCB-Net has only outliers compared with other algorithms, indicating that there is still room for improvement in stability under extreme conditions.

Fig. 17 plots the confusion matrix for the SEU dataset. From the confusion matrix, MCB-Net performs outstandingly in classification performance, and the prediction accuracy of each category is at a high level. Compared with other models, MCB-Net's overall accuracy and single category accuracy are better than most comparison models. For example, the 100 % accuracy for category 4 and 98.61 % for category 5 are far higher than the performance of some models in this category. At the same time, the accuracy of other categories such as categories 1, 2, and 3 also remains at a high level, and the misclassification rate is extremely low (for example, only 2.78 % of category 1 is misclassified as category 2, and only 2.78 % of category 3 is misclassified as category 4). This shows that MCB-Net has stronger discrimination between categories, better overall classification stability and accuracy, and can complete multi-category classification tasks more accurately than other models.

4.4.2. Error analysis

Table 15 and Table 16 show the comparison of error indicators under the SEU dataset. The MCB-Net algorithm demonstrated excellent performance. As shown in Table 15, MCB-Net also outperformed most of the comparison algorithms, with its average MAE only about 0.01 lower than the best-performing MIV-PSO-BP (0.037723) and S-CNN (0.036540) models. As can be seen from Table 16, the average value of MCB-Net is significantly lower than that of other comparison methods. Compared with MIV-LEA-BP, its MSE is reduced by 38.6 %, which fully demonstrates its core advantage in accuracy. At the same time, the maximum and minimum MSE values of MCB-Net are maintained at a low level, indicating that the algorithm has strong robustness and can effectively avoid extreme errors. Overall, MCB-Net has an absolute advantage in the MSE indicator, which more severely penalizes large errors, confirming its comprehensive superiority in improving the overall prediction accuracy and stability of the model.

Fig. 18 provides an intuitive display of MAE and MSE. The MCB-Net algorithm outperforms all other comparison models in terms of MSE. Its maximum, minimum, and average MSE values are all at the lowest, significantly lower than those of models like MIV-LEA-BP and SVM. In MAE, the average performance of MCB-Net is close to that of the optimal model, and the maximum value of MAE is significantly lower than that of BPNN optimized by other algorithms, showing a more stable error upper limit control ability.

Fig. 18MAE and MSE comparison of four algorithms (SEU)

a) MAE

b) MSE

4.4.3. Volatility analysis

Fig. 19 shows the SD of the convergence curve under the SEU dataset. As can be seen from the figure, the convergence error of MCB-Net shows a lower fluctuation amplitude and a more stable downward trend overall. In the first five cycles, the error decreased rapidly and remained at a low level, while the LEA model had obvious peak fluctuations. In the 5th to 15th cycles, although there are some fluctuations, the error fluctuation of MCB-Net is significantly smaller than that of PSO, S-CNN and other algorithms. Later in the cycle, the error continues to decrease steadily. This low volatility and high stability stems from MIV's effective screening of input information and CPO’s precise control of the optimization process. The synergistic effect of the two enables the algorithm to avoid local optimal traps during training, while improving convergence efficiency and robustness. Compared with other algorithms, MCB-Net not only achieves a faster convergence speed within 20 cycles, but more importantly, its error fluctuation is effectively suppressed, reflecting the algorithm's stronger anti-interference ability and generalization performance in complex optimization tasks.

Fig. 19Volatility across algorithms

4.5.1. Convergence performance test

To comprehensively evaluate the convergence performance of the CPO algorithm, this study selected commonly used multimodal benchmark test functions for experiments to verify its capabilities in global search and local development. All test functions solved in this paper have a dimension of 10, and the population size and number of epochs are 30 and 100, respectively. Multimodal functions contain multiple local extreme points and are often used to test the optimizer’s ability to explore and escape from local optimality.

Fig. 20 shows the convergence curves of each algorithm. Overall, the advantages of CPO are particularly significant. Taking F19 as an example, LEA and GA show an overall slow decline during the iteration process, always maintaining a high fitness level, and the final result is significantly inferior to CPO; although PSO and HO can gradually reduce the fitness value, there is a gap in both convergence speed and final accuracy, while CPO can decline rapidly in the initial stage and eventually reach convergence. In terms of convergence speed, CPO’s rate of decline on all functions is significantly greater than that of the other compared algorithms. In F23, although GA and HO have certain optimization trends, their rate of decline is slower than that of CPO, and their final results are much higher than CPO. CPO can converge quickly and reach the optimal value at an early stage. The results of F28 further highlight the differences. Except for CPO, the fitness values of the other optimization algorithms always remain at a high level. From the perspective of convergence accuracy, the final fitness value of CPO always remains at the lowest level and is relatively stable. Specifically, on F31, LEA maintains a high level and hardly converges. GA, PSO, and HO converge significantly slower than CPO.

In summary, CPO has both strong exploration efficiency and local convergence capabilities in the tested multimodal functions, and can obtain better solutions within a limited number of epochs, demonstrating its stability and advantages in complex optimization problems.

Fig. 20Multimodal benchmark function test

a) F19

b) F23

c) F28

d) F31

4.5.2. Ablation experiment

In this subsection, systematic ablation experiments are conducted to verify the synergistic effect of the proposed algorithm in this paper. Due to the comparison of other optimizations in subsection 4.1.3, there are only two groups of experiments: Experiment 1 using only BPNN for diagnosis and Experiment 2 using MIV-BPNN. Each group of experiments is run independently on the CWRU dataset with the same Network structural parameters to eliminate the effect of randomness.

Fig. 21 shows the confusion matrix of Experiment 1 and Experiment 2. As can be seen from the figure, the classification performance of a single BP model fluctuates to a certain extent. For example, the recognition rate for categories 6 and 7 is only 83.5 %, and the generalization ability is limited when processing certain complex patterns. The MIV-BP model improves the accuracy to 95.8 %, mainly due to the MIV algorithm's selection of characteristic indicators, which eliminates the influence of other interfering indicators on the results. The MCB-Net model achieved an even higher accuracy rate of 98.2 %. First, the MIV algorithm precisely selected the features that contributed most to classification, achieving 100 % accuracy for categories 2 and 8 while minimizing interference from other information. Second, the CPO algorithm further optimized network parameters, avoiding local optimal solutions through global search. This significantly increased the model’s recognition rate for complex categories (such as categories 6 and 7) from 83.5 % to 94.59 %, while also achieving enhanced recognition stability across all categories. In comparison, MCB-Net not only inherits the feature selection capability of MIV, but also enhances the robustness and convergence efficiency of the model through the dynamic parameter adjustment mechanism of CPO, and ultimately achieves near-optimal performance in classification tasks.

In addition, combining subsection 4.1.2 with other optimization algorithms, best-performing optimization algorithm, MIV-HO-BP, achieved an accuracy rate of 97.35 %, MIV-LEA-BP is 94.25 %, MIV-PSO-BP is 95.3 %, and BPNN performs the best after optimization with MIV feature selection and post-CPO algorithm. This result not only reflects the necessity of feature selection to reduce redundancy and improve model efficiency but also the advantages of the CPO algorithm.

Fig. 21Experiment 1 and Experiment 2 confusion matrices

a) BP

b) MIV-BP

c) MCB-Net