Nuisance Attribute Projection (NAP) was originally employed to suppress interferences in tasks such as face recognition and image recognition. In this study, interferences induced by variable operating conditions are regarded as redundant attributes. NAP is introduced to eliminate such interference in the feature space, mitigate speed-related effects, and enhance feature representation for improved bearing fault diagnosis.

The schematic of the NAP principle is depicted in Figure 1. The vector S denotes redundant attributes from varying operating conditions, while the target vector C contains the essential information related to bearing fault modes and severity.

Figure 1. Nuisance attribute projection.

Consider an N -dim feature space, n samples from various operating conditions can be represented as a N × n sample matrix F = [ f 1 , f 2 , ⋯ , f n ] , and the feature matrix F ′ is calculated from F and the projection matrix P as follows:

where P is an N × N matrix.

where I is an N × N unit matrix, i is the i th NAP direction, w i is an orthonormal vector, and d is the number of major directions extracted from the original feature space with the d ≤ N .

The parameter d is selected to balance computational complexity and projection performance. A larger d improves the projection effect but increases computational cost, which is evaluated using the sum of distances between projected features. NAP aims to minimize the projection error projection effect via the projection matrix P, where a smaller projection effect indicates better performance. PE is computed as follows.

where W i j is the weight matrix representing the relation between every feature sample.

W i j is positive when x i and x j come from different conditions or zero when these pairs come from the same condition. In this paper, different operating conditions refer to different rotational speeds or channels, which require removing interfering attributes in the feature vectors. W i j in NAP algorithm is shown in Equation (4):

where the w c is operation condition.

The main feature vectors of the following feature problems can be obtained by transforming the result P in Equation (3):

where the matrix Z ( W ) = diag ( W ⋅ L ) − W , diag ( ⋅ ) is a factor that forms a diagonal matrix by vectors, and L is the length column vector of every vector.

The matrixP can be calculated by solving for w i in Equation (5) to remove nuisance attributes from the characteristic space.

In this study, the samples for constructing the working condition matrix W are the multi-speed vibration feature samples of healthy bearings in the training set. The parameter d is determined by traversing integers from 1 to N, calculating the projection error (PE) for each integer, and selecting the value that minimizes the PE. The solution of the projection matrix P is performed using only the training set data throughout the process, with no test set data involved, thus avoiding data leakage.

Invertible Gaussian Reparameterization (IGR, Invertible Gaussian Reparameterization) [27] is a distribution technique that maps Gaussian noise onto a simplex, providing a continuous relaxation of discrete distributions. The IGR defined on the simplex S ( K − 1 ) is determined by parameters ( μ , σ ) , where μ ∈ R K − 1 and σ ∈ ( 0 , ∞ ) K − 1 , σ > 0 . A Gaussian noise ε ~ N ( 0 , I K − 1 ) is used for the affine transformation:

Then, perform an invertible smooth mapping (where τ > 0 is a temperature hyperparameter):

Obtain the density function through variable transformation:

where J g ( y , τ ) is the Jacobian matrix of g ( y , τ ) . The Jacobian term cancels out in the KL divergence ratio, simplifying to the Gaussian KL divergence:

Due to the lack of parameter interpretability, solve for the reference parameters by optimization (where p 0 is the target distribution):

The NDACN method achieves accurate fault diagnosis of variable-speed rolling bearings through five progressive steps, which are closely connected in accordance with the flowchart logic as shown in Figure 2. The specific implementation is as follows:

Step1:NAPProjectionProcessing. Twelve time-domain features, six frequency-domain features, eight node energies from three-layer wavelet packet decomposition and one envelope spectrum entropy are extracted from the vibration signals of variable-speed bearings to construct a 27-dimensional original feature matrix F. A working condition matrix W is built using multi-speed healthy signals, which are all derived from the training set and correspond to fault-free feature samples under different rotational speeds.

The orthogonal projection directions are obtained by solving the characteristic equation to construct the projection matrix p = I − ∑ i d w i w i T . The stripping of rotational speed interference is achieved via K = P × F , converting the non-stationary variable-speed features into approximately stationary features under constant speed.

Step2:DataPreprocessing. Z-score standardization is adopted to normalize the de-conditioned feature matrix, eliminating the influence of different dimensions. The standardized data are divided into a training set and a validation set A i , which are used for model training, hyperparameter tuning, and diagnostic performance evaluation respectively to ensure the rationality of data utilization.

Step3:DFTDynamicFeatureScreening. Neighborhood features [ L 0 , L 1 ] = Conv 1 ( A i ) are extracted via one-dimensional convolution, and Gaussian noise is injected to adapt the dimensions through parameter broadcasting. A temperature hyperparameter is introduced to construct a discrete distribution, and effective features are screened via Softmax soft filtering and argmax hard threshold decision. Invertible Gaussian reparameterization (IGR) is utilized to retain gradients, and the sparse degree is controlled via the loss filtering function L S F , outputting a high-discrimination feature matrix B i .

Step4:AttentionEnhancementMechanism. The feature sparsity metrics M i and Ω i are calculated based on high-order statistics, and a dynamic sparsity control loss is constructed to guide the model to focus on key fault features. Q, K, and V vectors are extracted via one-dimensional convolution, and the similarity is calculated to generate an attention weight matrix, which is weighted and aggregated to obtain enhanced features. A learnable scaling coefficient is introduced, and gated residual fusion G i = B i + r i ⊙ E i is adopted to strengthen the representation ability of fault features.

Step5:FaultDiagnosisviaConvolutionalNetwork. The attention-enhanced features are input into the convolutional network, where local features are captured via convolutional layers, the generalization ability is improved pooling layers, and the features are mapped to the fault category space via fully connected layers. The model is trained using the cross-entropy loss function and the Adam optimizer, and an adaptive learning rate strategy is set to avoid overfitting. Finally, four types of diagnostic results are output: healthy, inner race fault, outer race fault, and cage fault.

Figure 2. Framework of the NDACN method.

After NAP deconditioning, noise interference and redundant information may still remain in the feature matrix. Moreover, traditional hard threshold screening methods face challenges in sparsity control, which tend to cause the loss of key fault features or excessive residual noise. In addition, the direct application of hard threshold operations leads to gradient loss, which impairs the stability and convergence speed of subsequent model training. Dynamic filtering dynamically adjusts the feature retention rate through a gradient retention mechanism and a loss function.

In the establishment of threshold decision-making, the core of threshold processing is to effectively remove noise interference while preserving key fault information, which transforms feature selection into a binary mask operation:

where s o f t i denotes the convolutional coding value of the i-th feature, x i − 1 , j represents the j-th feature value of the (i − 1)-th layer, w j is the convolution kernel weight, and b is the bias term.

After obtaining the convolutional coding, the feature information μ (feature mean) and ξ (feature standard deviation) are extracted, and Gaussian noise ε ∼ N ( 0 , 1 ) is generated. Then:

After calculating k, transform λ = k (feature scaling value) and perform probability calculation:

Parameter scaling is performed via the temperature parameter T (temperature hyperparameter):

Finally, feature screening is realized through a hard threshold operation:

where Hard denotes the hard mask value, B i is the output high-discrimination feature matrix, and A i is the input standardized feature matrix.

During training, the corresponding hard-optimized coding results are backpropagated for gradients via the IGR technique to preserve the differentiability of feature screening. In this study, the hard mask (Hard) is applied to each feature dimension; specifically, argmax hard threshold decision-making is performed dimension by dimension on the 27-dimensional feature matrix after soft filtering, enabling adaptive screening of feature dimensions.

The features screened by the DFT module still suffer from problems such as a low proportion of key information and insufficient discrimination of fault features across different frequency bands. Traditional feature extraction methods treat all frequency band features equally, making it difficult to focus on the key frequency bands related to faults, thus leading to insufficient sensitivity of diagnostic models to weak faults. The parameterized attention convolution enhancement can strengthen the representation of key fault features and suppress the interference of irrelevant features. The core idea is to reallocate attention weights, enabling the model to automatically focus on frequency bands and feature regions that contribute more to fault diagnosis, thereby improving the discriminability of weak faults. Meanwhile, combined with a gated residual enhancement strategy, it realizes the adaptive fusion of original features and attention-enhanced features, avoiding the loss of key information. Its core components include: defining sparsity metrics based on high-order statistics: M i = ∑ l = 1 L B i ( l ) 4 , Ω i = ∑ l = 1 L B i ( l ) 2 , where B i is the input feature vector, and ℒ s k is the feature sparsity loss. The dynamic sparsity control loss is:

where ϒ ( e ) ∈ ( 0 , 1 ) is the proportional factor decaying with training epochs, N is the number of feature channels, and M i Ω i 2 is the kurtosis of the feature, reflecting sparsity. Then, one-dimensional convolution is used to extract attention elements (where convolution kernels act independently on each channel):

To ensure the consistency of feature information before and after, and to ensure that key information is attended to, the similarity matrix is calculated as:

and the features are aggregated by weighting: E i = D i V i . To achieve gated residual enhancement, a learnable scaling coefficient r i is introduced to realize adaptive feature fusion. The final implementation effect can be expressed as:

This structure suppresses noise while retaining key fault features, as shown in Figure 3.

Figure 3. Parameterized attention convolution enhancement.

The “imbalanced data scenario” mentioned in this study refers to the slight quantitative differences among various types of samples (non-extreme imbalance) caused by the varying difficulties in collecting fault samples under variable-speed operating conditions in actual industrial settings. In contrast to the samples collected in equal proportions in the laboratory, this scenario is more consistent with practical engineering applications. To comprehensively and objectively evaluate the fault diagnosis performance of the NDACN method, especially its ability to identify minority-class fault samples under the variable-speed and imbalanced data scenario, accuracy (Acc), precision, recall and F1-score are selected as the core evaluation metrics. All metrics are derived based on the four-class confusion matrix, covering both the overall diagnosis performance and the recognition performance of minority-class samples. Their specific definitions and calculation methods are as follows:

1) Confusion Matrix Definition

To evaluate the algorithm performance more effectively, the confusion matrix is commonly adopted. The evaluation method based on the four-class confusion matrix is shown in Table 1, and the calculation of related parameters is given in Equations (21) to (234).

2) Core Evaluation Metric Calculation

To comprehensively and objectively evaluate the model’s fault diagnosis performance and avoid the one-sidedness of a single metric, classification accuracy, precision, recall, and F1-score are selected as the core classification indicators. Meanwhile, the classification result confusion matrix and feature clustering visualization are adopted as qualitative evaluation indicators. The calculation formulas for each quantitative metric are as follows:

In the multi-class scenario, the F1-score is calculated as the weighted average of the F1-scores of each class, and precision and recall are also computed using the weighted average method, with the weights being the proportion of samples in each class.

Table 1. Four-class confusion matrix.

Note:TP_k denotes the number of true positive samples for the k-th class, and FN_kl denotes the number of false negative samples where the k-th class is misclassified as the l-th class.

Classification accuracy reflects the model’s overall recognition ability; precision and recall reflect the recognition accuracy and coverage of the model for various types of faults; F1-score is the harmonic mean of precision and recall, which is more suitable for performance evaluation on imbalanced Dataset. The confusion matrix intuitively displays the model’s recognition results for each type of fault. For feature clustering visualization, the t-SNE algorithm is used to reduce high-dimensional fault features to a two-dimensional space, and the model’s feature extraction ability is verified through the degree of aggregation and separation of feature clusters.

The energy recovery bearing dataset contains experimental data from simulated bearing fault tests. As illustrated in Figure 4, the test rig is primarily composed of a speed-regulating motor, test bearings with implanted faults, a B&K accelerometer, and a supporting data acquisition system. Acceleration data under eight health states of bearings were collected in the experiment, including normal state, inner race fault, outer race fault, and cage fault. The experimental platform for data acquisition consists of core components: a variable-speed AC motor, a coupling, support bearings, a housing for mounting test bearings, an accelerometer, and a data acquisition system. During operation, the test bearings are driven to rotate by the motor; if faults exist in the bearings, periodic shocks are generated. These shocks are captured as vibration signals by the accelerometer and digitized by the data acquisition system.

Figure 4. Data acquisition experimental platform.

The motor speed was regulated by a PLC (Programmable Logic Controller) to achieve increase, decrease, and increase-then-decrease variations, from which vibration datasets under three operating conditions were acquired. The sampling duration ranged from 10 seconds to 30 seconds, and data of healthy bearings, inner race faulty bearings, outer race faulty bearings, and cage faulty bearings were recorded respectively. The corresponding sampling frequency was 8192 Hz, and the bearing model was 6206. The faulty components of the tested rolling bearing and its disassembled parts are shown in Figure 5.

Figure 5. Rolling bearing fault components and configuration.

To verify the effectiveness of the NDACN method, three types of variable operating condition data were selected in the experiments, namely increasing speed, decreasing speed, and increasing-then-decreasing speed, which correspond to Experiment 1, Experiment 2, and Experiment 3, respectively. Based on the 27 dimensions of feature extraction and selection, and to prevent model overfitting, 80% of the original data was extracted as the training set and 20% as the test set after extracting significant features, with a total of 352 samples. Detailed information is presented in Table 2.

Table 2. Bearing feature dataset.

In this study, the original data was divided into a training set and a test set at a ratio of 80:20 in the feature screening stage, with an independent validation set further split from the training set at a ratio of 20% for hyperparameter tuning and early stopping determination. In the model performance verification stage, the training-test set division ratios of 4:6, 3:7 and 2:8 were adopted to further verify the generalization ability of the model under different sample sizes. All division schemes ensured that the validation set was independent of the training and test sets, and the validation set was only used for hyperparameter tuning without participating in model training. A total of 352 samples were generated from the original vibration data through non-overlapping framing with a frame length of 1024 sampling points and an overlap rate of 0%, including 88 samples for each state of healthy condition, inner race fault, outer race fault and cage fault. The data division strictly followed the principle of “source data independence”, where the framed windows from the same original collected data were assigned to either the training set or the test set exclusively to avoid data leakage. As shown in Figures 6-8 are the 10-second vibration data under the three operating conditions, corresponding to the time-domain waveforms of the healthy bearing, inner race faulty bearing, outer race faulty bearing and cage faulty bearing in sequence.

Figure 6. Time-domain diagram of experiment 1.

Figure 7. Time-domain diagram of experiment 2.

Figure 8. Time-domain diagram of experiment 3.

In the fault feature extraction, 27 features which are frequently used in bearing fault diagnosis are extracted in this section, including 12 time-domain features, 6 frequency-domain features, and 8 node energy and amplitude spectral entropy of wavelet packet three-layer decomposition. The detailed information of 18 time-domain and frequency-domain features and their computation methods are shown in Table 3.

The calculation of various features is based on the original vibration signals and spectrum data. Among them, time-domain features are directly extracted from the original signals through statistical analysis, frequency-domain features are calculated after converting the signals to the frequency domain via Fourier transform, wavelet packet features are obtained by acquiring the energy of each node through three-layer wavelet packet decomposition, and the envelope spectrum entropy is obtained by calculating the entropy of the envelope signal, comprehensively covering the time-domain, frequency-domain, and nonlinear characteristics of fault signals.

To verify the purification effect of the NAP (Noise-Attenuation Projection) method on vibration signals under variable speed conditions, three comparative experiments were performed under increasing speed, decreasing speed, and increasing-then-decreasing speed conditions. A 27-dimensional feature set was adopted, and the feature distributions before and after NAP projection are compared in Figure 9, aiming to eliminate speed interference and enhance fault features.

In all three experiments, consistent improvements were obtained after NAP projection. Before projection, features of healthy and faulty bearings showed obvious fluctuations and overlaps, which made different bearing conditions hard to distinguish. After projection, feature curves were smoothed, speed interference was suppressed, and the boundaries between health and fault classes became clearer.

Table 3. Time-domain and frequency-domain features.

Note: x ( n ) ( n = 1 , 2 , ⋯ , N ) is the original signal, N is the number of data points; s ( k ) ( k = 1 , 2 , ⋯ , M ) is the amplitude of the original signal spectrum, M is the number of spectral lines; f ( k ) is the amplitude of the k_th spectral line frequency.

Figure 9. Feature comparison before and after NAP projection.

Experimental results show that NAP exhibits strong adaptability to different operating conditions. In the three experiments, feature overlap was greatly reduced, weak fault features were effectively enhanced, and key fault characteristics were well preserved. NAP can suppress speed interference without reducing feature dimensions, thus ensuring the integrity of input features for fault diagnosis.

To intuitively verify the optimization effect of NAP projection on feature distribution, PCA dimensionality reduction combined with K-means clustering (number of clusters k = 3) is adopted to project the 27-dimensional features into a two-dimensional space for visual analysis. The clustering results under three different variable-speed conditions are shown in Figure 10, focusing on verifying the improvement effect of NAP on the intra-class aggregation and inter-class separation of features.

Figure 10. Visualization of feature projection matrix.

After projection, the four types of samples from independent clusters with clear boundaries and compact distributions, showing strong intra-class aggregation and little inter-class overlap. NAP projection effectively improves intra-class similarity and amplifies inter-class differences, providing highly discriminative input features for subsequent feature screening and attention enhancement modules.

The ReliefF algorithm is adopted to quantitatively evaluate the importance of the 27-dimensional features after NAP projection, with the results presented in Figure 11. Higher feature weights indicate greater contributions to fault identification, while lower weights suggest redundancy or interference. In Data-1, high-weight fault features are distinct, with the 18th-dimensional feature reaching 0.38. In Data-2, feature weights vary gently, but core features remain effective. In Data-3, the 23^rd-25th-dimensional features show significantly higher weights, capturing the key information of weak faults.

Figure 11. Feature dimension weight distribution.

The results of feature weight analysis verify the effectiveness of NAP in filtering features. It eliminates speed interference, significantly improves the contribution of core features, and maintains stability under different working conditions. Based on these results, subsequent steps can further select high-weight features to construct a simplified feature set, reducing the computational complexity of the model while retaining key fault information, and improving diagnostic efficiency and generalization ability.

To comprehensively verify the effectiveness and superiority of the NDACN method in variable-speed bearing fault diagnosis, this paper conducts comparative Data based on three gradient working conditions of Data 1, Data 2, and Data 3, with training set and test set ratios of 4:6, 3:7, and 2:8 respectively. F1-score, precision, recall, and accuracy are taken as the main evaluation indicators. The NDACN is compared with seven diagnostic methods: CNN, LSTM, CNN-LSTM, WDCNN, DANN, ResCNN, and NAP-CNN. The results are shown in Tables 4-6. Overall, under various working conditions and different data division conditions, all diagnostic indicators of the NDACN method are superior to those of the comparison models, and it also shows stronger stability, which effectively verifies the reliability and advantages of the proposed method in variable-speed bearing fault diagnosis.

Specifically, the network architecture of NDACN is configured as follows: 3 layers of 1D convolutional layers (with the number of channels being 32, 64, and 128 in sequence, kernel size of 3 for all layers, stride of 1, and padding of 1); 2 layers of max-pooling layers (pooling kernel of 2 and stride of 2); 2 layers of fully connected layers (with 256 and 4 neurons respectively); and a Dropout layer (dropout rate of 0.2). For the comparative models (CNN/LSTM/CNN-LSTM/WDCNN/DANN/ResCNN), the network structure parameters adopted are classic and publicly available in the same research field, consistent with those reported in the original literature, to ensure the fairness of comparison.

Table 4. Experimental results of different models under different ratios in data 1.

Table 5. Experimental results of different models under different ratios in data 2.

Table 6. Experimental results of different models under different ratios in data 3.

The training settings are as follows: batch size = 32; the optimizer is Adam with an initial learning rate of 0.001, and the cosine annealing learning rate scheduler is adopted ( T max = 50 , η min = 0.0001 ); the number of training epochs = 100; the early stopping strategy is implemented (patience = 10, i.e., training stops if the validation set loss does not decrease for 10 consecutive epochs); L₂ regularization is applied (weight decay = 0.0005); all models are implemented based on the PyTorch framework.

Under the 4:6 training-test set division ratio, the comprehensive diagnostic performance of NDACN can reach the relatively optimal level in all three data: In Data 1, the F1-score, precision, recall, and accuracy are 0.9896, 0.9895, 0.9898, and 0.9892, respectively. In Data 2, the values are 0.9854, 0.9853, 0.9855, and 0.9850, respectively. In Data 3, the values are 0.9938, 0.9937, 0.9940, and 0.9933, respectively. Compared with the suboptimal model ResCNN, all indicators of NDACN are improved by more than 9%. Compared with models such as DANN, WDCNN, and NAP-CNN, the advantages are more obvious, and the performance gap with basic models such as CNN, LSTM, and CNN-LSTM is further expanded.

As the training set ratio gradually decreases to 3:7 and 2:8, the diagnostic performance of all models declines to varying degrees, but the performance degradation of NDACN is always the smallest: In Data 1, its accuracy decreases from 0.9892 to 0.9663, with a decline of only 2.31%.In Data 3, under the 2:8 small-sample training ratio, it can still maintain a high accuracy of 0.9700. In contrast, among the comparison models, the performance degradation of CNN and CNN-LSTM exceeds 4.5%, that of LSTM, DANN, and WDCNN is above 3.5%, and the accuracy of NAP-CNN under the 2:8 ratio is only 0.8343, which is 13.57% lower than that of NDACN.

To intuitively analyze the diagnostic performance of the 8 models under the 4:6 training-test set ratio, the specific results are shown in Figure 12.

Figure 12. Model accuracy under 4:6 training-test ratio.

Comparison with other models shows that NDACN has better performance stability. NAP-CNN (incorporating NAP de-conditioning) generally outperforms CNN, LSTM, and CNN-LSTM without interference processing, confirming that variable-speed interference is critical to diagnostic accuracy. Although DANN alleviates condition differences better than WDCNN, it lacks targeted fault feature screening and enhancement. NDACN achieves a good balance between precision and recall, with an F1-score above 0.96 and a precision-recall difference of less than 0.5% across all three datasets.

Through the collaborative framework of NAP de-conditioning, DFT dynamic feature screening, attention enhancement, and convolutional network diagnosis, NDACN effectively addresses key issues (interference and low feature discrimination) in variable-speed bearing fault diagnosis, and its performance is superior to other comparison models under different variable-speed conditions and training set ratios.

To verify the enhancement and discrimination effect of the attention enhancement module in NDACN on fault features, the fault feature channel information processed by this module is reduced to a two-dimensional space using the t-SNE algorithm for visual analysis. Feature visualization is completed in Data 1, Data 2, and Data 3 respectively, and the results are shown in Figure 13.

Figure 13. Visualization of NDACN after attention enhancement under different data.

In the three datasets, healthy samples are clustered in distinct regions and far from fault clusters, while different fault samples exhibit regular distributions. This demonstrates that the attention enhancement module can effectively capture fault specificity, enhance inter-class differences via adaptive weight assignment, and reduce the influence of speed fluctuations and noise. Compared with NAP projection only, the addition of the attention mechanism significantly improves intra-class compactness and inter-class separation, with clearer clustering boundaries and more concentrated distributions. On the basis of NAP-based interference elimination, the module further focuses on key fault features and improves the quality of feature representation.

As shown in Figure 14, the confusion matrices of the three Data all show high classification accuracy, with the proportion of samples on the diagonal exceeding 84%. Among them, the classification effect of outer race fault (ORF) is relatively optimal: the recall rate is 99.1% in Data 1, 99.1% in Data 2, and 99.2% in Data 3, with only a few misclassified samples. In terms of precision, the precision of ORF is higher than 99.4% in the three Data, and the precision of cage fault (CF) exceeds

Figure 14. Confusion matrices of NDACN under different data.

93.5%, indicating that the model has high prediction reliability for these two types of faults, and the probability of misclassification into other categories is low. The precision of normal state (NB) and inner race fault (IRF) is relatively low, mainly because samples from other categories are easily misclassified into these two types. This phenomenon is basically consistent with the recall analysis results and conforms to the actual characteristics of fault diagnosis under variable-speed conditions.

From the confusion matrix results of the three Data, it can be seen that NDACN can achieve accurate identification of various faults under different working conditions. In particular, it exhibits nearly perfect classification ability for outer race faults with clear features, and can maintain high recognition accuracy for weak cage faults. Although there is a certain proportion of misclassification between the healthy state and inner race faults, the overall diagnostic performance is still superior to the comparison models.