2.1 Principle of VMD

VMD is an excellent adaptive time-frequency signal decomposition method, which estimates the components of each signal by solving a frequency-domain variational optimization problem. VMD assumes that all the components are narrow-band signals concentrated around their respective center frequencies, so VMD builds up a constrained optimization problem based on the conditions that the components are narrow-banded, so as to estimate the center frequencies of the signal components and reconstruct the corresponding components. VMD is underpinned by a sound mathematical theory and is essentially an adaptive optimum Wiener filter [30].

The feature decomposition of the signal is accomplished by constructing the following variational problem. And the constraint is that the sum of all modals is equal to the original signal f(t).

(1)

where u_k(t) is the intrinsic mode function (IMF), which is essentially an FM signal, and ω_k(t) is the center frequency of u_k(t).

In order to solve the above variational problem, a quadratic penalty factor α and Lagrange operator λ(t) are introduced into the Lagrange multiplier method, which transforms the constrained variational problem into an unconstrained variational problem, and an augmented Lagrange expression is obtained.

(2)

Then the cross-multiplier method is utilized to constantly update the iterations u_k(t), ω_k(t), λ(t), until convergence to the requisite precision, finishing the signal decomposition and feature extraction.

2.2 Method of optimization

In the signal decomposition of the above VMD methods, the number of channels of the IMFs K and the quadratic penalty factor are the key elements of the decomposition objectives and effects. Usually, they are set based on a priori knowledge or expert experience, but such methods lack adaptivity and are susceptible to human subjective factors, so it is proposed to use the emerging intelligent optimization algorithm, dung beetle optimization (DBO) [31], to optimize the VMD parameters. In the process of parameter optimization, an appropriate fitness function should be chosen as the optimization aim of the process to assure the correctness and efficacy of the signal decomposition findings. The general single-indicator fitness function has a certain applicability, but the optimization results are only for a single type of feature indicator, and the results have not been mathematically verified in multiple indicator dimensions, which lacks sufficient adaptivity and robustness in the face of a variety of feature types of signals. In order to expand the applicability range of the fitness function, strengthen the application breadth of this optimization process and the information entropy value of the decomposition results, a composite optimization indicator function L is proposed to replace the single fitness function based on the perspective of multi-dimensional data fusion, and the DBO-VMD method is further optimized and validated. In the parameter optimization process of DBO-VMD, a specific fitness function L is constructed as an evaluation criterion, in which Tanimoto coefficient ensures signal consistency before and after the decomposition of vibration signals, permutation entropy extracts unusual changes in vibration signals caused by faults, and kurtosis detects changes in the intensity of the impact of the signals, which is one of the characteristics of fault signals. Therefore, the fitness function may successfully extract the features, such that the extracted features have more and richer information, and minimize the absence of information linked to fault characteristics.

Tanimoto coefficient is utilized to indicate the similarity of the data before and after the decomposition, if it is the closer to 1, it indicates that the greater the degree of similarity. Here it is introduced into the function can characterize the degree of similarity before and after the decomposition of vibration signals, to ensure that the retention of the key information, the formula is shown in Equation (3).

(3)

where T_c is the Tanimoto coefficient, x_i(t), y_i(t) are the time series data before and after the input and output respectively.

The value of permutation entropy of each component can effectively map the degree of change and complexity of the time series, and has a strong sense of the minor changes in the mechanical system response signal [32]. At the same time, the bigger the value of PE, the more chaotic the time series data distribution is. So it is one of the main factors for fault diagnosis.

(4)

where H_PE is the value of PE, m denotes the number of embedding dimensions used to reconstruct the time series, and P_i denotes the probability of occurrence of each component sequence alignment in the reconstructed vector. Thus from the preceding description we may deduce that the bigger the value of PE, the more prominent and chaotic the raw vibration signal fluctuation is, which is most likely owing to the fault.

kurtosis value is a measure of the peak state of the distribution of random variables dimensionless parameter, which may present the concave and flat degree of the top of sample function, therefore it can be used to determine that the shock signal strength. The bigger the value of kurtosis, the more irregular and powerful the bearing vibration shock is, which suggests that the bearing may be wrecked.

(5)

where x(t) is the IMF signal, x is the mean value of the IMF signal, and σ is the standard deviation of the IMF signal.

Based on the parameters previously discussed, the L function can be created:

(6)

where a and b are used as the weighting factors of the permutation entropy and the kurtosis, and a + b = 1. Based on the characteristics of kurtosis and permutation entropy in the fault vibration signal, for the faulty bearing vibration signal, the permutation entropy is more sensitive to the degree of integrity of the overall signal, and the kurtosis is for the characterization of faulty impact signals, so this method is to take a = 0.6, b = 0.4 as the final indexes.

From the above description we can learn that the higher value of L represents the more valid information it contains and the closer it is to the initial signal, which means the bearing which is the source of the time series is more likely to be wrecked. So we can derive the fitness function for use in the optimization of K and α in VMD:

(7)

According to Equation (7) we know that for the fitness function of VMD parameter optimization, it is a minimization problem. Therefore the minimization of the fitness function will be employed as the optimization target of the DBO.

Combined with the above composite indicator fitness function method, the specific steps of applying DBO to the adaptive parameter optimization of VMD method are shown below:

1. (1) Presetting the starting parameters of the DBO algorithm. The overall number of dung beetle populations is set to be 30, and the numbers of each behavioral population are 6 for rolling, 6 for breeding, 7 for foraging, and 11 for stealing, respectively.
2. (2) The VMD parameters that need to be optimized in this research are two parameters: the number of IMF channels K, and the quadratic penalty factor α, therefore the two-dimensional vector X = [X₁, X₂] is set as the population unit, and X₁, X₂ stand for K, α accordingly. Based on the a priori knowledge and experience, the value range of X₁ is [3,8], while the value range of X₂ is [300,5000].
3. (3) H_PE and K_u in Equation (7) are normalized and the X that minimizes the fitness function is computed as the starting population location.
4. (4) Through the four behavioral populations of diverse ways for repeated iterations to update the population position, and pick reducing Fitness as the ideal selection criterion for the population.
5. (5) If the current number of iterations t is less than the predefined maximum number of iterations T, return to (4) to continue the execution, and vice versa, output X as K, α.

2.3 Construction of feature vector

The CMPE algorithm improves the MPE algorithm's insufficient degree of coarse-graining through the composite coarse-graining method, and obtains multiple aligned entropy values for different time series, and uses the average value as the entropy value of the time series, so that the effectiveness and robustness of the feature extraction can still be maintained under the large-scale factor, and the feature vectors used for inputting into the deep learning network are constructed by combining this method and the above mentioned DBO-VMD method. The particular stages of the procedure are mentioned below:

(1). For the original data sequence X = {x(i), i = 1, 2, 3, …, N}, let it through the process of coarse-graining, the scale factor is preset to be τ, and the process is shown in Equation (8).

(8)

where y_k,j^(τ) denotes the jth sampling point of the kth coarse-grained sequence under the scale factor τ, and [ ] denotes a rounding operation that ensures that the sampling point is integer.

(2). For each coarse-grained sequence y_k,j^(τ), compute the alignment entropy value H_PE within the sequence, and then average the alignment entropy value under the scale factor as the output entropy value, as shown in Equation (9).

(9)

where m is the embedding dimension used to rebuild the time series and λ denotes the time lengthening in the reconstruction procedure. H_PE is the value of PE, which has been defined in Equation (4)

1. (3) Use the K and α acquired from the optimization approach as the preset parameters of the VMD method to decompose the original signal, and finish the first feature extraction, then we may get many IMFs having independent features whose total number is K. Next use the optimization indicator function to screen and extract the decomposed IMFs. Calculate the results of fitness function L_i of each IMF and the mean value‾L of all, and the filtering rule is L_i ≥ θ_L, then the IMF that is considered as the ith component which is legitimate, and vice versa is invalid. For the threshold coefficient, θ_L is originally selected as 0.75, which may assure the efficacy of feature extraction.
2. (4) For different samples, the number of valid IMFs p_i varies, and in order to keep the input dimensions consistent, a “leveling” strategy is adopted. Assuming that the number of preset component IMFs is p, if p_i< p, then the value of θ will be progressively lowered, and vice versa, the value of θ will be gradually raised until the number of output IMFs equals p.
3. (5) The original signal of a single sample is set to extract q sequences for feature extraction, thus the dimension of the feature vector of a single sample is p·q, and this feature vector is utilized as the input vector for future input to the SDAE network for training.

Table 1 shows the pseudocode of the method.

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Table 1. Feature vector extraction method based on CMPE and optimized VMD.

https://doi.org/10.1371/journal.pone.0337832.t001

2.4 IWCCSCA

The SCA generates the initial population position by means of a randomized scale division over the range of intervals, and iteratively updates the position by Equation (10):

(10)

Where x_i,j^(t), x_i,j^(t+1) are the position of individual of the jth dimension before and after the iterative update, respectively, and x_g,j is the component of the current global optimal solution in the jth dimension. r₂, r₃, and r₄ are all uniformly distributed random variables. and r₁ is the amplitude conversion factor as shown in Equation (11) shown. The principle of SCA is briefly shown in Fig 1.

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Fig 1. Principal of SCA.

https://doi.org/10.1371/journal.pone.0337832.g001

(11)

where T_max is the maximum number of iterations, t is the current iteration number, and a is a constant, which is selected as a = 2 here.

Although the above classic SCA approach has some optimization ability, it has poor global search ability and sluggish convergence time when addressing difficult multi-dimensional optimization issues. Therefore, it can be improved according to the amplitude adjustment factor update, inertia weight update, and Cauchy chaotic variability principle, which significantly improves the performance of the original SCA algorithm by adapting the optimization adjustment of multiple parameters in the iterative updating formula of the SCA method. The performance of the original SCA algorithm is significantly improved in terms of global optimization accuracy and convergence speed.

First, r₁ in the original technique is utilized to balance the weight between global search and local optimization. However, the initial r₁ value is updated depending on the number of iterations without considering the current population location and the overall optimization search method. If r₁ is too big, the global search ability is powerful, but it will result in inadequate search, which is easy to cause the local optimization; If r₁ is too small, the iterative step size will become smaller appropriately, which leads to the inability to converge rapidly. Therefore, an adaptive updating approach of amplitude adjustment factor based on exponential function will be employed, as illustrated in Equation (12):

(12)

Where e is a natural constant and k is a conditioning factor. In Equation (12), compared to the original approach, this r₁’s value declines slowly at the beginning, increasing the number of searches and improving the searching ability, also it speeds the reduction at a later stage, making the algorithm converge rapidly.

Secondly, it is the position update based on inertia weights. Referring to the update method of particle swarm optimization algorithm, the inertia weights are increased in the early stage to strengthen the global search ability, and the proportion of inertia weights is reduced in the later stage to improve the local optimization ability and accelerate the convergence. Therefore, the better inertia weight updating technique is indicated in Equation (13):

(13)

where w^(t) is the current update inertia weight; w_max is the initial inertia weight, i.e., the maximum value of inertia weight; and w_min is the ending inertia weight, i.e., the minimum value of inertia weight. After introducing the inertia weight w^(t) variable, the position update method is Equation (14):

(14)

Finally, it is to add mutation perturbation to avoid the population from sliding into local optimization. This technique employs a hybrid Cauchy chaotic variation method to disturb the optimum solution x_g by adding Cauchy variation individuals to the ideal solution x_g and combining it with the usage of chaotic systems to boost the population diversity and broaden the search space.

Then the Cauchy operator is added into the optimal solution to bring it out of local optimality, as illustrated in Equation (15):

(15)

where x_g is the current optimal solution of the original method, which can also be considered as an elite individual, Cauchy is the Cauchy operator that conforms to the Cauchy distribution, and x_new is the new elite individual after the resultant Cauchy variation is updated.

The chaotic variational system utilizes Logistic chaotic values, which are defined as illustrated in Equation (16):

(16)

where c is the chaos coefficient, generally selected as c = 4.

To recap, the individual Cauchy chaotic variational perturbation is defined as follows.

(17)

where r₅ is the Cauchy chaos weight coefficient, which is used to select the mutation method and takes the value of [0,1] as a random number.

2.5 Model of diagnosis

Stacked denoising auto-encoder is based on the creation of neural network Auto-Encoder (AE), and based on the AE network, Vincent et al. suggested [33] Denoising Auto-Encoder (DAE), which replicates the condition of the data being interfered with by noise by adopting a random zero-setting of the input data using the approach of Drop-out-like method. and stack many DAEs then we may acquire the SDAE with better generalizability and robustness.

In SDAE network, the principle of sparse coding is commonly utilized to establish a sparse parameter ρ so that the average activation value of the hidden layer node j is near to ρ, which fulfills the objective of capturing the critical information of the input while deleting part of the redundant information. Based on this notion, the loss function of this network model is:

(18)

where x⁽ⁱ⁾, y⁽ⁱ⁾ are the input and the output of the network, respectively. The first term denotes the reconstruction loss from input data to output data; and the second term is the regularization loss, which is used to prevent the network model from overfitting. and KL scattering serves as a penalty term to constrain the activation values around ρ. β denotes the sparse factor weight, in this paper, we take β = 3, and the specific calculation formula is shown below:

(19)

In SDAE networks, the design and values of plural hyperparameters directly affect the accuracy, noise immunity, generalizability and other properties of the final model. Currently, most of the SDAE hyperparameters are determined by empirical enumeration of multiple combinations to obtain a better set of hyperparameters, which need to be specially tested and selected for specific fault diagnosis problems. For fault classification tasks in diverse domains, the suggested hyperparameters show limited generalization performance [34]. In order to boost the generalization capacity of the network and strengthen the resilience of the model in the face of complex disturbances, a set of network hyper-parameters for SDAE are determined via adaptive optimization using the IWCCSCA approach. By tuning the hyperparameters of the SDAE network with IWCCSCA, the model’s generalization ability may be strengthened, while simultaneously obtaining higher resilience and interference resistance.

For deep learning network models, the more hidden layers, the greater accuracy of its output, however, but, when the number of hidden layers exceeds 4, the model generalization capacity will exhibit a considerable decline [35]. Therefore, the number of hidden layers of the SDAE network in this paper is chosen to be 3. Therefore, this paper needs to optimize 7 network hyperparameters of SDAE: 3 hidden layer nodes numbers, 3 hidden layer sparsity coefficients, and 1 input data zero-ratio. Therefore the stages for developing the SDAE network model optimized based on the IWCCSCA approach are described below:

1. (1) Setting up the target population, from the above elaboration we know that the network has seven hyperparameters to be optimized, so assume that each single population is a seven-dimensional vector Xi = [x_i1, x_i2, …, x_i7], where x_i1, x_i2, x_i3 are the number of nodes in the three hidden layers, x_i4, x_i5, x_i6 are the sparsity coefficients corresponding to the three hidden layers, and x_i7 is the proportion of the zeroes in the input data.
2. (2) Set the population size N = 20, the number of iterations is selected as T_max = 100, select a suitable target classification error rate R_error, initialize the population according to the range of values of each corresponding component of the population, and calculate the corresponding R_error for each individual.
3. (3) Updating of stock holdings according to the method in section 2.4.
4. (4) Calculate the updated population individual corresponding to R_error, and locate the population individual corresponding to the least R_error.
5. (5) Check whether the iteration reaches the convergence condition, if the minimum value of R_error is less than the set threshold or the number of iterations t is equal to the set maximum number of iterations T_max, then it is considered that the optimization objective is reached, and exit the loop, and vice versa, repeat the step (3).
6. (6) The population individuals corresponding to the smallest R_error achieved in step (5) are utilized as the optimization results of the SDAE hyperparameters, and the results are fed to the network hyperparameters to complete the network model.

The following Fig 2 depicts the model.

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Fig 2. Diagnosis network model flowchart.

https://doi.org/10.1371/journal.pone.0337832.g002

2.6 Introduction of data

The faulty bearing vibration signal dataset utilized for the experiment is a publicly available dataset from Case Western Reserve University (CWRU). It is an open data set, which is a benchwork dataset and frequently used in fault diagnosis and it can be acquired via the website: https://csegroups.case.edu/bearingdatacenter. The bearing type of the test rig is deep groove ball bearing, and the motor bearing model is SKF-6205, which is processed by EDM technology to mimic the failures of four different sizes of 0.007“, 0.014”, 0.021”, and 0.028”, respectively. The sampling frequency of the experimental data was classified into 12kHz and 48kHz, and the data categories include Normal, Inner Race fault, Rolling ball fault, and Outer Race fault. The final data collection collected the acceleration time series data of the bearings of this experimental platform under load conditions of 0 hp, 1 hp, 2 hp, and 3 hp.

Based on the real experimental demands, a total of 10 rolling bearing data types with load circumstance of 0 hp will be chosen, and their comprehensive information is provided in Table 2.

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Table 2. Bearing fault types and parameters.

https://doi.org/10.1371/journal.pone.0337832.t002

For the selected 10 groups of bearing vibration signals, the signal waveforms corresponding to different fault types are significantly different, and the features are complex, so the extraction of features is especially critical to complete the differentiation and diagnosis. For each group of bearing signals, 100 samples are picked, and the sampling points of each sample are set to 1600. So 1000 samples in total will be selected as experimental verification data. The ratio of the training set to the test set is 3:1.

3.1 Signal decomposition

The VMD parameter optimization is performed according to the description in Section 2.2, and the maximum number of iterations of the DBO algorithm T_max = 100 is set, and the optimization is performed for 10 groups of signal data respectively, and their optimal parameters obtained by the DBO method are shown in Table 3.

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Table 3. DBO-VMD optimization method parameters.

https://doi.org/10.1371/journal.pone.0337832.t003

Select the F1 data one of the samples as an example, the results of the time-frequency analysis after VMD decomposition are shown in Fig 3 and Fig 4, we can see that the modal aliasing phenomenon and the endpoint effect has a considerable degree of improvement, and decomposition of the modal distinction, indicating that the effect of decomposition is more satisfactory.

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Fig 3. Time domain diagram after decomposition.

https://doi.org/10.1371/journal.pone.0337832.g003

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Fig 4. Frequency domain diagram after decomposition.

https://doi.org/10.1371/journal.pone.0337832.g004

In the DBO-VMD decomposition, this research introduces the composite optimization indicator function L instead of the usually used single indicator function as the fitness function. Here use F3 group signals as an example, and the efficacy of the reconstructed signals is compared with that of the single indicator-based fitness function with minimal permutation entropy, kurtosis, and minimum envelope entropy, respectively. The signal-to-noise overlap ratio is used to measure the signal decomposition effect, and the “distorted” component of the decomposition process is treated as noise, therefore the following formula is obtained:

(20)

Where Y₁ is the reconstructed signal and Y₂ is the original signal. Through this equation we may know that the bigger the R value, the less the noise in the reconstructed signal and the more effective information it contains. The SNR metrics obtained by different fitness functions are shown in Table 4, from which we can see that compared with a single metric, the composite optimization metric function as fitness function has a stronger ability to extract effective information, and the SNR recombination degree is improved by 3.39, 6.18, and 8.37 dB compared with the minimum alignment entropy, kurtosis, and minimum envelope entropy, respectively, indicating that the use of composite metrics optimization metric function as the fitness function can effectively improve the feature extraction effectiveness and noise resistance of VMD method.

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Table 4. Signal-noise overlap ration comparison.

https://doi.org/10.1371/journal.pone.0337832.t004

3.2 Fault diagnosis and classification

As can be observed from Step (4) in Section 2.3, for a single set of vibration signal data, the floating threshold approach is employed to complete the normalization of the IMF sample dimensions. From the computational analysis in Table 2, it can be observed that p = 4 is adopted as the equivalent number of IMFs. The CMPE feature vector computation is carried out using the approach explained in Section 2.3, appropriate to the experimental circumstance with the aim. In this research, the embedding dimension size m = 6, temporal extension λ = 1, and scaling factor τ = 16 are chosen. After the calculation, the CMPE feature vector is supplied into the SDAE network model as an input layer.

The parameter settings of IWCCSCA-SDAE correspond to section 2.5, and the SDAE network hyperparameters may be acquired by parameter optimization search, and the particular results are provided in Table 4. For the SDAE network model training parameters, they are set as shown in Table 5. After the setup is completed, the training and testing verification is carried out, the verification process is carried out 50 times, the average of each result is taken for analysis, and the accuracy standard deviation is calculated to analyze the model stability. Table 6 presents the training parameters of the SDAE.

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Table 5. SDAE network hyperparameters.

https://doi.org/10.1371/journal.pone.0337832.t005

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Table 6. SDAE network training parameters.

https://doi.org/10.1371/journal.pone.0337832.t006

The method used in this paper involves data preprocessing such as signal extraction and feature vector construction, which may increase the computational complexity. Therefore, in order to verify the computational efficiency of our method, we compared it with the following feature extraction-diagnostic classification combination models in terms of model iteration time: VMD-SDAE, EMD-SDAE, and SVM-CNN. The comparative results are presented in the Fig 5.

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Fig 5. Time cost comparison.

https://doi.org/10.1371/journal.pone.0337832.g005

From the figure, we can observe that owing to the rather extensive data pretreatment method, the first iteration of the technique proposed in this paper is quite sluggish. However, since the data processing results have more obvious characteristics, the iteration efficiency can be improved in the subsequent iteration process, thereby alleviating the problem of excessive time spent on data processing in the early stages to a certain extent.

In order to demonstrate the benefit of this paper's technique in terms of convergence speed, diagnostic accuracy, resilience, etc., this paper will include other model as a comparison, and the selection of each algorithm as well as the parameter settings are presented below:

1. (1) SCA-SDAE: The parameter values are consistent with those in this paper.
2. (2) PSO-SDAE: In the PSO algorithm, the initialization population size N = 10, maximum number of iterations T_max = 100, learning factor c1 = c2 = 2, inertia weights w = 0.4, and parameter settings of the SDAE method are consistent with those of this research.
3. (3) GA-SDAE: In GA algorithm, the initialized population size N = 50, evolutionary periods T = 200, crossover probability K = 0.8, mutation probability τ = 0.05, and parameter settings of SDAE algorithm are consistent with this research.
4. (4) SVM: The radial basis kernel function is employed for classification, the penalty factor C = 500, kernel parameters σ = 0.85, and lastly the K-fold cross-validation technique, in the method K = 10.
5. (5) SVM-CNN: The SVM parameters are configured as stated above. The CNN parameters are as follows: 2 convolution layers, 2 pooling layers, convolution kernel size 5 × 5, pooling layer kernel size 2 × 2, stride 1.

The convergence speed is of significant relevance for the efficiency of the model operation, and the model convergence speed is represented by loss rate curve, and the loss rate convergence graphs of this paper's approach and each comparative method are displayed in Fig 6. From Fig 6, we can see that the IWCCSCA-SDAE method has a faster convergence speed at the beginning, fewer iterations are needed to complete the convergence, and the loss rate is more stable after convergence, which indicates that it has advantages in convergence speed, global optimization search, and local search, and can make the model have a stronger equilibrium search ability and higher running efficiency.

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Fig 6. Loss ratio of models.

https://doi.org/10.1371/journal.pone.0337832.g006

For the fault diagnosis method based on the combination of DBO-VMD and IWCCSCA-SDAE proposed in this paper, the final average diagnostic accuracy reaches 99.51% and the mean square error reaches MSE = 6.6396 ∙ 10⁻⁶, which can be preliminarily seen to have a high diagnostic accuracy after 50 simulation experiments. For the same dataset feature vector input, the diagnosis of 250 samples, a total of 10 types of faults in the test set data, its diagnostic results are shown in Fig 7.

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Fig 7. Model diagnosis results.

https://doi.org/10.1371/journal.pone.0337832.g007

In Fig 7, the red dots show the data with diagnostic mistakes, thus we may derive the following conclusions from Fig 7: the IWCCSCA-SDAE method diagnoses 3 wrong samples with a diagnostic accuracy of 0.988; the SCA-SDAE method diagnoses 14 samples with diagnostic errors with a diagnostic accuracy of 0.944; the PSO-SDAE method diagnoses 20 samples with diagnostic errors with a diagnostic accuracy of 0.920; GA-SDAE method has 17 sample diagnostic errors, with a diagnostic accuracy of 0.932; SVM-CNN method has 22 sample diagnostic errors, with a diagnostic accuracy of 0.912; SVM method has 36 sample diagnostic errors, with a diagnostic accuracy of 0.856. From the above conclusions, we can see that the improved IWCCSCA algorithm for the optimization of hyperparameters of the SDAE network can effectively improve the accuracy of the model fault diagnosis, which has certain advantages compared to the previous methods.

3.3 Noise immunity

In order to test the method proposed in this paper in the noise resistance, robustness and other aspects of the enhancement effect, in the original vibration signal to add Gaussian white noise and low-frequency periodic sinusoidal signals, whose frequency is f = 40 Hz, to simulate the natural environment of the environmental noise and the working environment of the pedestal vibration and other interference signals. A total of five sets of data comprising noise signals are created, and the signal-to-noise ratios are 2dB, 4dB, 6dB, 8dB and 10dB, respectively. The diagnostic accuracy of each model can be obtained by taking the average of diagnostic accuracy of 50 simulation experiments for each method. The comparison graph of each accuracy rate in various noise scenarios is provided in Fig 8.

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Fig 8. Diagnosis accuracy of models after adding noise.

https://doi.org/10.1371/journal.pone.0337832.g008

Through Fig 8, we can see that with the increase of noise intensity, the diagnostic accuracy of each method shows different degrees of diagnostic accuracy decline, but the decline trend of the IWCCSCA-SDAE method is relatively slower, and the average diagnostic accuracy also maintains a higher level compared to other methods, which indicates that its noise resistance and robustness have been improved compared to other methods.

In order to quantify the noise immunity of the method, the noise decrease rate α_N is proposed, which represents the decrease in the diagnostic accuracy of the model compared to the diagnostic accuracy of the un-noised model under specific noise conditions, and the expression is shown in Equation (21).

(21)

Where Accuracy₀ represents the diagnostic accuracy of the model for the original signal and Accuracy_N represents the diagnostic accuracy of the model after signal noise addition.

The results of each method and its six groups of anti-noise attenuation rate are shown in Table 7, from which we can easily see that the anti-noise attenuation rate of the IWCCSCA-SDAE method is relatively low, indicating that it is superior in terms of anti-noise, robustness, etc., and also indicating that the addition of the ring noise element in the self-coder network can effectively improve the model anti-noise, robustness, and generalization ability, and improve the diagnostic accuracy of the model in the complex environment. It also indicates that the inclusion of ring noise components to the self-encoder network may substantially increase the model's noise resistance, resilience and generalization ability, and improve the model's diagnostic accuracy in complicated situations.

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Table 7. Noise decrease rate comparison.

https://doi.org/10.1371/journal.pone.0337832.t007