A bearing fault diagnosis method for complex system based on improved extended belief rule base

Abstract

In the field of vibration monitoring for rotating machinery, high-precision bearing fault diagnosis faces challenges due to imbalanced measurement data and noise interference from complex environments. To address these issues, this paper proposes a bearing fault diagnosis method based on an ensemble undersampling extended belief rule base (EBRB-EU), aiming to improve the reliability and accuracy of equipment condition assessment. Specifically, to overcome the limitation of using Euclidean distance in EBRB, which fails to effectively capture the differences in the probability distribution of vibration signals, the paper introduces the Kullback-Leibler (KL) divergence to replace the Euclidean distance, providing a more accurate measure of the match between samples and rule antecedent attributes. An ensemble undersampling strategy is then employed to calibrate the training set, reducing the impact of majority class samples on model training. Subsequently, the data is divided into multiple subsets, and a set of parallel sub-EBRB models is constructed. The differential evolution algorithm is used to optimize the parameters of each sub-model, minimizing evaluation bias. Finally, an ensemble voting mechanism is applied to integrate the diagnostic results, enhancing the accuracy of fault state identification. Experimental results demonstrate that the KL divergence-based EBRB-EU model outperforms the Euclidean distance-based EBRB method on multiple datasets, effectively alleviates the data imbalance issue, and exhibits strong resistance to interference.

Introduction

During the operation of large rotating machinery, such as turbines, compressors, and generators, bearings, as critical components, directly affect the safety and operational efficiency of the entire system. These types of equipment typically operate under high-speed, continuous conditions for extended periods. When a bearing fails, it can easily lead to shutdowns or even catastrophic accidents. Therefore, developing reliable and accurate bearing fault diagnosis models is of paramount importance for effective condition monitoring and early warning systems.

Bearing fault diagnosis technology has evolved from threshold-based methods relying on expert knowledge to time-domain, frequency-domain, and time-frequency feature analysis based on signal processing, and currently to data-driven intelligent diagnostic methods¹. With the development of machine learning techniques, data-based diagnostic methods have gradually become mainstream, reducing reliance on specialized knowledge to some extent. However, these methods still face significant challenges in practical applications, particularly the issue of dependence on large-scale, high-quality labeled data. Among these, the problem of class imbalance in data distribution severely limits model performance, leading to biased diagnostic results and affecting the reliability and generalization ability of the model in real industrial scenarios. In summary, current fault diagnosis methods can generally be classified into three categories: knowledge-driven methods, data-driven methods, and hybrid methods that combine both knowledge and data. Each category has its own characteristics and limitations in addressing the issue of data imbalance.

Knowledge-driven methods heavily rely on the prior knowledge of domain experts, using this knowledge as the core foundation for system modeling and decision reasoning. These methods typically offer good interpretability, providing clear and transparent evaluation processes and results, making them particularly suitable for scenarios where the mechanisms are well-defined and expert experience is well-established. such as: Zhou et al. proposed an implicit knowledge-guided explainable machine learning framework, which quantifies knowledge inconsistencies through Monte Carlo simulations and optimizes model parameters². Li et al. proposed a fault diagnosis method for building energy systems based on a hierarchical object-oriented Bayesian network (HOOBN)³. Dou et al. proposed a method based on Empirical Mode Decomposition (EMD)⁴. Ge et al. proposed a knowledge graph-based framework for abnormal electrocardiogram (ECG) diagnosis, named ECG-KG⁵. Wang et al. proposed a supervised framework based on a knowledge-refined hybrid graph network, called the KSGraphLSTM model⁶. Guo et al. proposed a knowledge-driven spatiotemporal graph attention neural network (K-STGAT). This model transforms process mechanism knowledge into a graph structure to build a spatiotemporal graph. It introduces a symbolic attention mechanism to extract spatiotemporal features of variables, utilizes mutual information to select quality-related variables, and constructs a quality regression model, thereby enabling quality-related fault detection⁷. However, this approach also has notable limitations, as its performance is largely constrained by the completeness and accuracy of expert knowledge. Knowledge acquisition and representation often become bottlenecks in system construction. When faced with novel faults, complex systems, or outdated knowledge, the model’s adaptability and diagnostic effectiveness significantly decline. Furthermore, while knowledge-driven methods do not directly rely on data volume, knowledge acquisition itself is often based on balanced sample experiences, and insufficient recognition of rare faults can lead to diagnostic biases.

Next are data-driven methods, which learn directly from data. In terms of handling uncertainty, deep learning models can automatically extract features and exhibit a certain level of robustness to noise. such as: Wang et al. proposed a Relative Angle Matrix (RAM) method that converts one-dimensional time series into two-dimensional images⁸. Zheng et al. proposed an unsupervised transfer learning method (UTL-SOCNN-FBNM) based on structural optimization convolutional neural networks (SOCNN) and fast batch nuclear norm maximization (FBNM)⁹. Cui et al. proposed a self-attention-based contrastive learning method (SACL) for bearing fault diagnosis¹⁰. Cui et al. proposed an adaptive generalized domain data fusion-driven kernel sparse representation classification method¹¹. Xiang et al. proposed a novel variant of the one-dimensional convolutional neural network (CNN) called ISONet, which improves the conventional CNN by introducing over-parameterization in the input space dimension¹². Delanyo et al. proposed a data augmentation method based on analytical wavelets (DAAW)¹³. Prashant et al. proposed a bearing fault diagnosis model based on a multi-size wide kernel convolutional neural network (MWK-CNN)¹⁴. Wang et al. proposed various efficient bearing fault diagnosis methods and models, It includes mechanical few-shot fault diagnosis models based on multi-scale perception multi-level feature fusion image Quadrant entropy (MPMFFIQE) and Harris Eagle Optimized Support Vector Machine (HHOSVM)^15–17. Deng et al. proposed various efficient fault diagnosis methods, including fault diagnosis under variable operating conditions, as well as enhancing classification robustness by constructing a dual-model collaborative generalized learning system (SC-KSBLS) to maintain high accuracy even under noise^18–21. Zhang et al. proposed three distinct methods for remaining useful life (RUL) prediction across their studies: from a cross-domain perspective, they introduced two approaches named DAGCN-SFDA-DFL and TTAGPN; additionally, from a semi-supervised learning angle, they developed an RUL prediction method called SAM-SVAE^22–24. Although data-driven methods have demonstrated strong potential in condition monitoring and fault diagnosis, their performance is still severely limited by the reliance on large-scale, high-quality labeled data. Labeled data forms the foundation for establishing the mapping relationship between input features and target outputs, and its quantity and quality directly determine the accuracy and generalization ability of the model. However, in practical industrial measurement scenarios, especially when targeting specific equipment or rare fault modes, obtaining a sufficient and reliable number of labeled samples is often difficult and costly. This data scarcity can hinder the model’s ability to fully learn subtle fault characteristics and complex state boundaries in vibration signals, leading to a decline in generalization and a significant deterioration in diagnostic accuracy. Furthermore, the labeling process itself may introduce uncertainties due to human error or instrument limitations, further affecting the model’s reliability.

The final approach is the knowledge-and-data hybrid-driven method. This approach, on one hand, incorporates expert knowledge, physical mechanisms, and other prior information to constrain the model’s hypothesis space, enhancing its interpretability and generalization ability. On the other hand, it leverages data-driven techniques to automatically learn underlying patterns and complex relationships from historical data, improving the model’s accuracy and adaptability. This hybrid mechanism enables the model to not only maintain a clear reasoning path and transparent decision-making process but also retain high robustness in scenarios of data scarcity or noise interference, making it particularly suitable for condition monitoring and fault diagnosis in complex industrial systems. such as: Yang et al. implemented online detection of milling machine tool wear by applying Page’s cumulative sum (CUSUM) to the residuals²⁵. Shen et al. proposed a physics-informed deep learning method that incorporates physical knowledge of bearing faults into the model training process through a customized data- and physics-based loss function²⁶. Li et al. proposed an interpretable waveform segmentation model based on nested U-Net (NUWSM)²⁷. Liao et al. proposed a method called ClassBD, which combines blind deconvolution with a deep learning classifier²⁸. Matthew et al. combined deep convolutional autoencoders (DCAE) with a physics-informed loss term that incorporates domain knowledge²⁹. Seungyun et al. proposed a domain alignment method based on rotational difference and peak difference for bearing fault diagnosis under ultra-low-speed conditions³⁰. Qin et al. proposed a bearing fault data generation method based on digital twin technology, addressing the issue of sample imbalance and improving the accuracy of bearing fault diagnosis across different operating conditions³¹.

In conclusion, the work of previous researchers has made significant contributions to the field; however, there are still some challenges in bearing fault diagnosis. Current methods mostly focus on optimizing model structures or enhancing data, and have not systematically explored the modeling of vibration signal uncertainty and reliability assessment from the perspective of measurement science. Vibration signals are typically collected by accelerometers in industrial noise environments, and are significantly influenced by uncertainty and inherent class imbalance, with healthy state samples far outnumbering fault samples. This imbalance, combined with sensor noise and environmental interference, weakens the reliability of traditional diagnostic algorithms, often leading to estimation bias and increased diagnostic uncertainty. The BRB as a typical knowledge and data hybrid-driven modeling approach, is built upon foundations such as fuzzy theory, D-S evidence reasoning³², and decision theory³³. It effectively handles incomplete or imprecise information and is suitable for modeling complex decision-making problems, especially in scenarios involving complex datasets characterized by fuzziness, imprecision, or incompleteness. Additionally, the BRB offers good interpretability, addressing issues related to uncertainty, and has demonstrated high accuracy across various application domains, including fault diagnosis³⁴, health condition assessment³⁵, and medical decision-making³⁶. Although the BRB has demonstrated excellent performance across various domains, there are still some challenges in its practical application. In bearing fault diagnosis, complex systems typically involve multiple input indicators³⁷. When constructing the BRB using the Cartesian product form, the number of rules increases rapidly, leading to an exponential growth in the combination of rules. This not only increases the model’s complexity but may also significantly reduce its accuracy. To overcome this issue, the EBRB³⁸ was introduced.

The Extended Belief Rule Base (EBRB) is an advanced evolutionary form of the Belief Rule Base (BRB). It retains the core advantages of the BRB model, such as interpretability and uncertainty handling, while significantly enhancing the representation capability of imprecise and incomplete knowledge by introducing an input attribute extension mechanism based on belief distributions. This method transforms continuous or discrete sensor measurements into semantic information with a belief distribution structure, thereby more accurately characterizing the inherent uncertainty in measurement data and providing a new approach for uncertainty quantification in the construction of state indicators for vibration signals. Furthermore, the EBRB model has the following notable advantages: by explicitly modeling the cognitive and stochastic uncertainties in the rules, the EBRB can systematically address issues such as noise interference, sensor errors, and feature ambiguity present in vibration signals, significantly improving the robustness of the state recognition system and the reliability of the output results.This characteristic enables the EBRB to demonstrate strong adaptability and accuracy when handling complex and uncertain knowledge environments³⁹. Today, the EBRB has been widely applied across various domains. such as: Hou et al. applied the EBRB to the diagnosis of thyroid diseases⁴⁰. Yang et al. applied it to the classification of big data samples⁴¹. Fu et al. developed an EBRB cascade classifier that effectively handles imbalanced data by ensemble an adaptive data resampling strategy with a cascading model to dynamically adjust rule weights⁴².

In summary, although the EBRB performs excellently in various tasks, its modeling approach still has a critical limitation. When calculating the degree of match between a sample and a reference value, Euclidean distance is typically used as a similarity measure. However, Euclidean distance lacks a probabilistic foundation and cannot effectively measure the information loss between two distributions. Therefore, its discriminative ability is limited when dealing with measurement data, such as vibration signals, which possess complex uncertainties. In contrast, Kullback-Leibler (KL) divergence, rooted in information theory, quantifies the information loss that occurs when one probability distribution approximates another, making it naturally suited for comparing the similarity between probability distributions. Using KL divergence as a criterion for matching calculation allows for a more reasonable reflection of the distribution differences between the sample and the rule, thus improving the accuracy and reliability of EBRB in noise environments and under small sample conditions. Finally, based on the above discussion, the main contributions of this paper are as follows:

(1) A fault diagnosis method based on the integrated extended belief rule base (EBRB-EU) is proposed, aiming to solve the problem of model recognition deviation caused by the imbalance of data samples in practical engineering.

(2) A novel matching degree calculation method based on Kullback-Leibler (KL) divergence is proposed to replace the Euclidean distance used in EBRB, which can more accurately reflect the information loss between the sample and the reference value, and is particularly suitable for uncertainty modeling in vibration signals.

Problem description

Before establishing the EBRB-EU model to address the Bearing Fault diagnosis problem, the following issues must be considered:

Question 1: How to determine the input of the bearing fault diagnosis model. In practical engineering environments, the operating conditions of bearings are often subject to environmental noise, which can severely affect the diagnostic results of the model. How can this issue be addressed? To enhance the accuracy of the model, effective feature extraction for model input indicators should be performed prior to modeling. The extraction process can be expressed as:

1

In the equation, represents the initial signal sample dataset, denotes the feature extraction method, and represents the dataset after feature extraction.

Question 2: How to balance an imbalanced dataset, When the data samples are imbalanced, if the number of healthy samples exceeds the number of faulty samples, the model may become biased during the training process, classifying the state as that of the majority class. For example, when there are too many healthy bearing samples, the model tends to classify the bearing state as healthy. How can this imbalance be addressed? The relationship between healthy and faulty samples can be described as:

2

3

4

In this equation, represents the total initial sample size, y denotes the samples of healthy gears, represents the samples of faulty gears, indicates the faulty state, C refers to the number of healthy samples after balancing through ensemble methods, signifies the fluctuation in the number of extracted data, and represents the final total sample size.

Question 3: How to perform effective reasoning. When constructing the EBRB, it is essential to consider the impact of expert knowledge on the model’s reasoning process. The reasoning process can be described as follows:

5

In this equation, represents the final diagnostic result, denotes the diagnostic process of the model, and E signifies the expert knowledge.

Question 4: How to effectively optimize model initial parameters. Since the initial parameters of the EBRB are influenced by expert knowledge, which has certain limitations, this paper utilizes a Differential Evolution Optimization Algorithm to further optimize the parameters of the model. The optimization process can be described as follows:

6

In this equation, represents the optimized set of parameters, denotes the optimization function, and signifies the initial parameters before optimization.

The construction of a bearing fault diagnosis model based on the EBRB-EU

In this section, the entire process of bearing fault diagnosis using the EBRB-EU modeling approach is described. Section  3.1 details the implementation of the model, Sect.  3.2 discusses the use of an ensemble method to balance the data, Sect.  3.3 covers the construction of the EBRB model, and Sect.  3.4 explains the optimization of the model’s parameters using a Differential Evolution Optimization Algorithm to improve the accuracy of recognizing the various states of the bearings. Section  3.5 introduces the specific process of the voting algorithm. Finally, to enhance the transparency and reproducibility of the proposed method, Sect.  3.6 provides a detailed, step-by-step algorithmic description of the complete diagnostic pipeline, integrating all the aforementioned components from data preprocessing to final decision-making.

The bearing fault diagnosis process

The bearing fault diagnosis process can be carried out in the following four steps: first, the input indicators of the model are obtained; second, the model is constructed; third, inference is performed using an evidence reasoning algorithm; and fourth, an optimization algorithm is applied to optimize the model parameters. The modeling process of this model is shown in Fig. 1.

Fig. 1.

The modeling process of the bearing fault diagnosis method based on the EBRB-EU.

Step 1: Acquisition of model feature data.

The raw vibration signals collected from the bearings need to be processed, as these signals are difficult to analyze and compare directly. Therefore, it is generally necessary to process the vibration signals through feature extraction and analysis, extracting key characteristic parameters for effective fault diagnosis. After conducting time-domain, frequency-domain, and time-frequency domain analyses, numerous features are generated; however, not all features are effective in assessing the bearing condition. Consequently, a random forest method is used to rank the extracted features based on their importance, from which key features are selected. Finally, these features are employed to perform multi-dimensional analysis of the collected vibration signals to extract representative statistical indicators. Ultimately, the top-ranked features are used as input attributes for the model to improve its prediction accuracy and stability.

Step 2: Construction of the EBRB-EU model.

First, the data is balanced using an ensemble learning method, where the total dataset is randomly divided into multiple subsets. EBRB is an improved form of BRB, designed to address the combinatorial explosion problem that may arise in BRB models. Next, for each subset, a sub-EBRB model is employed for diagnostic analysis, generating multiple preliminary diagnostic results. Finally, a “voting” strategy is used to combine the preliminary results, leading to the final diagnostic conclusion. Through this process, the EBRB-EU model can effectively improve diagnostic accuracy.

Step 3: Evidence reasoning.

The evidence reasoning algorithm is employed for evidence inference. During the reasoning process, an approach that incorporates attribute reliability is utilized, enabling the model to effectively enhance its anti-interference capability and obtain accurate diagnostic results.

Step 4: Parameter optimization.

The EBRB-EU model utilizes the Differential Evolution (DE) algorithm to optimize the model’s initial parameters, aiming to enhance the model’s performance and diagnostic accuracy. By performing global optimization of the parameters using the DE algorithm, it effectively avoids the pitfalls of local optima, thereby achieving more precise and reliable diagnostic results. This optimization process not only accelerates the model’s convergence speed but also improves its adaptability and stability in complex data environments.

Ensemble undersampling process

Ensemble undersampling techniques have been widely applied to address class imbalance issues. In fault diagnosis, when the sample size of one class is significantly larger than that of others, the model typically tends to predict the majority class, thereby neglecting the recognition of the minority class. This bias can severely affect the model’s classification performance. To solve this issue, the ensemble undersampling method reduces the number of majority class samples and generates multiple different training subsets through repeated undersampling. These different training sets are then used as input data to train the ensemble model. This approach allows the model to learn more sample variability, thereby capturing different decision boundaries and enhancing the model’s ability to recognize the minority class. The training data partitioning is determined by fluctuations in model recognition accuracy. When the training data size increases to a level that stabilizes the model’s recognition accuracy, the dataset partition number at that point is adopted. The specific algorithm implementation process is shown in Fig. 2, and the overall process is as follows:

Fig. 2.

The implementation process of the ensemble undersampling algorithm.

Step 1: Bearing data definition.

The processed data samples are defined as follows: the dataset contains four health conditions of the bearings, specifically healthy condition, inner race fault, outer race fault, and ball fault. For the purpose of classification and labeling, each of these four conditions is assigned a label. The Nomal is assigned label 1, the ball fault is assigned label 2, the Inner race fault is assigned label 3, and the Outer race fault is assigned label 4.

Step2: Data partitioning.

After defining the dataset, it is divided into a training set and a test set. During the partitioning process, a random sampling method is employed, ensuring that the distribution of each label in both the training and test sets is consistent with that of the original dataset. A stratified sampling strategy is used to maintain the proportion of samples from each category in the training and test sets, ensuring consistency with the overall dataset.

Step 3: Define the undersampling target quantity.

First, the number of samples for each minority class is calculated, and the “average minority class sample size” is determined based on these quantities, serving as the target value for undersampling. Specifically, during each undersampling process, the number of minority class samples to be sampled from each majority class is determined to balance the sample sizes across all categories.

Step 4: Output the final dataset.

The minority class samples are combined with the undersampled majority class samples, and the resulting dataset is randomly shuffled. Subsequently, the undersampled data generated in each round is saved, and multiple training sets are created from these datasets. Each training set contains a different set of training samples, which helps enhance the model’s training performance and generalization capability.

Construction of the EBRB model

The representation form of EBRB

The EBRB model is an extension of the BRB model. The BRB model is an expert system constructed from IF-THEN rules, The extended belief rules can be expressed as follows:

7

where denotes the belief distribution of the premise attribute. denotes the belief degree of in the rule when the information of therule on the premise attribute is complete; otherwise, , denotes the belief distribution for the result. indicates the belief degree of each result. indicates the attribute weight , indicates the weight of the rule.

Set the reference point

For each input sample attribute, the value of the reference point for each input sample needs to be constructed using expert knowledge. Subsequently, the belief of the premise attributes is generated based on Eq. (8):

8

where represents the value of the reference point corresponding to premise attribute for . The reference value is provided by experts based on their knowledge. It is used to represent the regions with a higher data density within the input attributes, serving as the selection for the upper bound, significant points within the data range, and the lower bound.

Generate the extended attribute belief distribution and the resulting belief distribution

After calculating the belief corresponding to Eq. (8), the extended belief distribution of the attributes and the resulting belief distribution are generated, as shown in Eqs. (9) and (10).

9

10

Calculate the degree of match between the rule and the sample

The KL divergence between the membership vector of the test sample and the membership vector of the rule is then computed. This divergence is used to measure their match degree, where a larger distance indicates a lower degree of match.

11

12

where represents the match degree between the test sample and the rule on premise attribute . represents the symmetric KL divergence, is the true distribution, and is the predicted distribution. Here, To ensure numerical stability and avoid undefined situations when the probability is zero, a smoothing factor ( set to in this study) is applied to both distributions before calculating the divergence. where and are the smoothed probability vectors, This formulation guarantees that all probability values are positive and the divergence is well-defined and symmetric.

Here are two vector examples to further demonstrate the differences between Kullback-Leibler (KL) divergence and Euclidean distance. The vector examples are shown in Table 1.

Table 1.

Vector example.

Case1:uniform distribution vs. similar distribution	Case 2: asymmetric distribution
P1=[0.2,0.3,0.1,0.4]	P2 = [0.8, 0.1, 0.1]
Q1=[0.25,0.25,0.25,0.25]	Q3 = [0.1, 0.8, 0.1]
Q2=[0.1,0.4,0.3,0.2]	Q4 = [0.7, 0.15, 0.15]

1) Analysis of Case 1.

From the probability distribution in Fig. 3a, it can be observed that P vs. Q1: KL ≈ 0.0969, P vs. Q2: KL ≈ 0.0458, and on the right, P vs. Q1: Euclidean distance ≈ 0.1581, P vs. Q2: Euclidean distance ≈ 0.1414. This suggests that the KL divergence indicates P is more similar to Q2, whereas the Euclidean distance suggests P is more similar to Q1.

Fig. 3.

Comparative differences based on the vector case. (a) The comparative results under uniform steps and probability distribution. (b) The comparative results under asymmetric distribution.

2) Analysis of Case 2.

From Fig. 3b, it can be observed that P2 vs. Q3: KL ≈ 2.4079 (large), P2 vs. Q4: KL ≈ 0.0482 (small), P2 vs. Q3: Euclidean distance ≈ 0.9899, and P2 vs. Q4: Euclidean distance ≈ 0.1414. It can be concluded that the KL divergence is highly sensitive to the distribution order, whereas the Euclidean distance fails to effectively distinguish such informational discrepancies.

Rule activation weight calculation

The activation weight of each rule is calculated using the rule weight , attribute weight , and individual match degree . At this point, the activation weight of the input data for the rule can be expressed by the following formula:

13

In the equation, represents the individual match degree corresponding to the reference value of the premise attribute in the rule. when ,, at this point, the rule has not been activated.

Rule fusion and utility calculation

When a single rule is used as an evaluation metric, it is insufficient. In this case, rule fusion is required through the Evidence Reasoning (ER) algorithm to generate the final activated rule. The calculation method is as follows:

14

15

In the equation, represents the belief level of the diagnostic result under rule , N denotes the number of model output results, and signifies the combined belief. and , .

1) Formula calculation case.

The activation weight has a direct impact on the final result, which is calculated by using attribute weights. For the convenience of calculation, both attribute weights and rule weights are set to be equal: , This means that they have the same impact on the aggregation process. The two rules in EBRB can be expressed as follows:

  

Suppose that For Rule 1: It is calculated by formulas (9)-(13): Finally, it is calculated by formulas (14) and (15) ;

Parameter optimization

The initial fault diagnosis model is constrained by expert knowledge, and the initial parameters of the EBRB-EU model need to be further adjusted based on observed samples. In this study, a differential evolution algorithm is used to optimize the model parameters. To address rule redundancy and further improve the model’s recognition accuracy, Mean Squared Error (MSE) is employed as the loss function to optimize the weights of the activated rules and attribute weights. The mean squared error between the assessed health status and the true status demonstrates the accuracy of the EBRB-EU model.

16

Here, represents the actual health status, and t denotes the predicted output of the EBRB-EU model.

In contemporary optimization algorithms, classical methods such as Whale Optimization Algorithm (WOA), Particle Swarm Optimization (PSO), and Genetic Algorithm (GA) have been widely applied in various fields, such as (urban traffic planning)⁴³. However, this study selects the Differential Evolution (DE) algorithm as the parameter optimization method for the EBRB-EU model, primarily due to its distinctive advantages in handling high-dimensional, nonlinear, and complex optimization problems.

Differential Evolution (DE) is a population-based global heuristic search algorithm. Its core idea is to generate new individuals and replace the original ones through differential mutation, crossover, and selection operations among the population, thereby efficiently exploring the global optimal solution. Compared to algorithms such as PSO and ACO, DE exhibits several distinctive characteristics: on one hand, it demonstrates a strong balance between global search and local exploration; on the other hand, it has fast convergence speed, high optimization efficiency, and a limited number of controlled parameters. These attributes make it particularly suitable for models like EBRB, which require continuous parameter optimization. The optimization flowchart is shown in Fig. 4.

Fig. 4.

The optimization process of the differential evolution (DE) algorithm.

The overall process of the Differential Evolution optimization algorithm includes initializing the population, followed by mutation, crossover, and selection operations. The specific optimization process is outlined as follows:

1. Initialize the population.

Each individual in the population is initialized according to the equality and inequality constraint conditions. At this stage, individuals are independent of each other and include all parameters to be optimized. Additionally, the population size, the number of evolutionary iterations, and other DE parameters are set.

2. Mutation operation.

Individuals are randomly selected from the population, and mutation is performed using a Differential approach, generating intermediate mutant individuals. The mutation process can be described by the following formula::

17

In the equation, represents the mutant individual, refers to three randomly selected distinct individuals, and F is the scaling factor, which controls the magnitude of the mutation. Typically, the scaling factor is within the range of [0,1].

3. Cross operation.

Based on the crossover probability, a crossover operation is performed between the intermediate mutant individual and the current individual, resulting in a new individual. The crossover process can be described by the following formula:

18

where represents the individual after crossover, refers to the component of the mutant individual, and denotes the component of the current individual. is the crossover probability, which controls the likelihood of mutation and .

4. Select operation.

By comparing the adaptability of the current individual with the intermediate cross individuals, that is, the accuracy of classification, and then further selecting the better individual as the next generation of individuals, the selection process can be described by a formula:

19

In the equation, represents the current individual, refers to the individual after crossover, and denotes the individual that replaces in the new generation. and represent the fitness values of the current individual and the individual after crossover, respectively. If the individual after crossover exhibits a better fitness value, when, it replaces the original individual; otherwise, the original individual is retained.

Voting algorithm

To improve the accuracy of the model’s predictions, this study adopts the voting method as the final ensemble algorithm. Compared to other ensemble methods, this approach offers strong intuitiveness and ease of understanding, while also demonstrating high reliability in practical applications. The overall process of the voting method can be divided into three main steps, as illustrated in Fig. 5. The specific process is as follows:

Fig. 5.

Voting algorithm.

Step1: Summary of Prediction results.

First, the voting algorithm reads and organizes the prediction result files from each sub-model. Each file contains the preliminary prediction outputs for all categories corresponding to the respective model, ensuring that the prediction labels from each sub-model are correctly recorded. The primary objective of this step is to provide the foundational data for the subsequent voting decision.

Step 2: Voting integrated decision-making.

Subsequently, the voting algorithm performs the ensemble decision-making through a majority voting mechanism. For each sample, the algorithm counts the prediction labels from each sub-model and determines the final prediction label for the sample based on the most frequent category. In the event of a tie, where multiple categories have the same number of predictions, the voting algorithm automatically applies predefined rules (such as selecting the category with the higher frequency) to determine the final category. This process ensures the fairness and consistency of the ensemble decision.

Step 3: Result output.

After the voting decision is made, the algorithm constructs a confusion matrix based on the final prediction results and the true labels. The confusion matrix, as an essential performance evaluation tool, provides a clear visualization of the model’s accuracy and its misclassification patterns. Ultimately, the output includes a series of computed metrics to assist decision-makers in evaluating the model’s overall performance, primarily including Accuracy, Mean Precision, Mean Recall, and Mean F1-Score. These metrics comprehensively reflect the model’s performance across various dimensions, offering decision-makers a thorough basis for diagnostic assessment.

Algorithmic description of the EBRB-EU method

In this section, we provide a detailed pseudo-code description of the proposed EBRB-EU method to enhance the transparency and reproducibility of the algorithmic procedure. The pseudo-code systematically outlines the complete workflow, including data preprocessing, ensemble undersampling, sub-model construction and optimization, ensemble voting. The complete algorithmic workflow of the proposed EBRB-EU method is illustrated in Table 2, providing a visual overview of the entire diagnostic pipeline from data preprocessing to final decision-making.

Table 2.

Algorithm: the EBRB-EU method for bearing fault diagnosis.

Case study

In this section, to demonstrate the effectiveness of the proposed method, real-world datasets from bearing and gear cases are selected for experimentation. Multiple metrics are incorporated to assess the validity of the proposed approach, including Accuracy, F1-Score, Precision, and Recall. Additionally, disturbance analysis experiments are included to prove that the proposed method still provides reliable diagnostic results in noisy environments.

Section 4.2 presents the gear fault diagnosis experiments, including dataset description in Sect.  4.2.1, comparison experiments in Sect.  4.2.2, and experimental summary in Sect.  4.2.3.

Case1: bearing fault diagnosis

The bearing fault diagnosis experiments in Sect.  4.1 are systematically organized as follows: Sect.  4.1.1 describes the dataset; Sect.  4.1.2 details the construction of the fault diagnosis model based on EBRB-EU; Sect.  4.1.3 presents the comparison experiments, including comparisons with similar BRB models, data-driven models, and classical imbalance-handling methods; Sect.  4.1.4 provides an interpretability analysis of the model’s rule activation and decision mechanism; Sect.  4.1.5 conducts a four-fold cross-validation experiment to assess generalization capability; Sect. 4.1.6 presents the ablation experiments, systematically evaluating the contributions of three key components, Sect.  4.1.7 performs disturbance experiments to verify the model’s robustness in noisy environments. Finally, Sect.  4.1.8 provides a comprehensive summary of the experimental results, synthesizing the key findings from the bearing fault diagnosis study.

Dataset description

The proposed method is first evaluated using the publicly available bearing dataset from Case Western Reserve University, which can be accessed at https://engineering.case.edu/bearingdatacenter/download-data-file. The experimental setup comprises a three-phase asynchronous motor, a torque encoder, and a dynamometer, with vibration signals acquired via a 16-channel data recorder. In this study, we focus on the drive-end SKF6205 bearing under four health conditions: normal, ball fault, inner race fault, and outer race fault.

To ensure a leakage-safe evaluation protocol that rigorously assesses the model’s generalization capability, we adopt a cross-condition validation strategy based on different sampling frequencies. The dataset partitioning is explicitly designed to prevent any information leakage between training and testing sets. Specifically, vibration signals collected at a sampling frequency of 12 kHz under the 1730 RPM (3 HP load) operating condition are used exclusively for model training, while signals collected at 48 kHz under the same 1730 RPM (3 HP load) operating condition are used exclusively for model testing. It is important to emphasize that these two subsets originate from distinct acquisition sessions with no temporal or signal-level overlap, thereby ensuring complete separation at the raw-record level. No data from the 48 kHz condition is used during training, and no data from the 12 kHz condition is used during testing. This configuration yields a training set comprising only 12 kHz signals and a test set comprising only 48 kHz signals, both derived from the same operating condition but from different sampling frequencies, which realistically simulates scenarios where a diagnostic model may encounter data acquired under different sensor configurations or sampling rates.

In addition to the clear frequency-based split, we further ensure leakage safety by performing all data preprocessing steps—including windowing—after the dataset has been partitioned. Specifically, the raw vibration signals from the training set (12 kHz) and test set (48 kHz) are processed independently. Each raw signal is segmented into discrete samples using a sliding window approach. To ensure that each individual sample adequately captures the statistical characteristics of bearing vibration, the window length is set to 2048 data points. The sliding step is also set to 2048 points, meaning that consecutive windows are non-overlapping. This choice avoids overly dense sampling, which could introduce redundant information and potentially lead to overfitting. Crucially, because the sliding window segmentation is applied separately to the training and test sets after the dataset split, there is no possibility that any segment derived from a raw record in the training set could appear in the test set, nor vice versa. This guarantees that the evaluation is conducted on completely unseen data segments, preserving the integrity of the generalization assessment.Following the segmentation process, a comprehensive set of statistical features is extracted from each window to serve as input attributes for the fault diagnosis model. These features are systematically derived from three complementary analytical domains—time domain, frequency domain, and time-frequency domain—to provide a holistic characterization of the vibration signals. In each domain, a consistent set of statistical features is computed, including mean, standard deviation, skewness, kurtosis, maximum, minimum, peak-to-peak value, root mean square (RMS), crest factor, shape factor, impulse factor, margin factor and energy. Together, these features capture both the temporal dynamics and spectral characteristics of the vibration signals, which are essential for accurate and robust fault diagnosis. By training the model exclusively on 12 kHz data and testing it on 48 kHz data from the same operating condition, we establish a leakage-safe evaluation protocol that more realistically reflects real-world scenarios—such as sensor upgrades or changes in data acquisition settings—where the diagnostic model must generalize to data distributions not encountered during training.

The original dataset used in this study is presented in Table 3. In practical engineering settings, bearings operate under normal conditions most of the time, leading to a natural predominance of healthy samples. To simulate this realistic scenario, we performed random sampling on the original data, yielding an imbalanced dataset (Table 4) where the number of healthy samples significantly exceeds that of each fault category. This manipulated dataset better reflects the class imbalance challenges that diagnostic models face in real-world applications.

Table 3.

Distribution of the original dataset.

Four fault types	Sample length	Speed (r/min)	Sample quantity (12 kHz)	Sample quantity (48 kHz)
Ball fault	2048	1730	236	712
Inner race fault	2048		236	711
Normal	2048		237	237
Outer race fault	2048		413	1660

Table 4.

Dataset distribution under simulated real-world operating conditions.

Four fault types	Sample length	Speed (r/min)	Sample quantity (12 kHz)	Sample quantity (48 kHz)
Ball fault	2048	1730	42	18
Inner race fault	2048		42	18
Normal	2048		210	90
Outer race fault	2048		42	18

The dataset composition after random sampling is presented in Table 4. It can be observed that the number of healthy samples significantly exceeds that of each fault category, accurately reflecting the class imbalance characteristic commonly encountered in real-world industrial environments. Specifically, under the 12 kHz sampling frequency used for training, the dataset contains 42 ball fault samples, 42 inner race fault samples, 42 outer race fault samples, and 210 healthy samples. This yields a healthy-to-fault ratio of 5:1 for each individual fault type (210 healthy samples versus 42 samples per fault category). Similarly, under the 48 kHz sampling frequency used for testing, the dataset comprises 18 ball fault samples, 18 inner race fault samples, 18 outer race fault samples, and 90 healthy samples, also resulting in a healthy-to-fault ratio of 5:1 for each fault type (90 healthy samples versus 18 samples per fault category). This consistent imbalance across both training and test sets ensures that the model is trained and evaluated under conditions that faithfully simulate real-world scenarios.

Construction of fault diagnosis model based on EBRB-EU model

This section elaborates on the construction process of the bearing fault diagnosis model based on EBRB-EU, which primarily includes data preprocessing, training set partitioning and sub-model construction, as well as model inference and ensemble integration.

(1). Data preprocessing and input feature selection.

First, time-domain, frequency-domain, and time-frequency domain analyses are performed on the raw vibration signals to extract multiple statistical features. To ensure the effectiveness of the features and avoid data leakage, the Random Forest feature importance ranking algorithm is applied solely to the training set to evaluate the extracted features. A higher feature importance indicates a greater contribution to model construction. The feature extraction method for bearing vibration signals is presented in Table 5a, b).

Table 5.

Model input feature parameters for construction.

(a) Feature	Equation (time-domain and frequency-domain)
Mean
Std
Skewness
Kurtosis
Max
Min
Peak to peak
RMS
Crest factor
Shape factor
Impulse factor
Margin factor
Energy

(b) Feature	Equation (time-frequency domain)
Mean
Standard deviation
Skewness
Kurtosis
Max
Min
PeakToPeak
RootMeanSquare
CrestFactor
ShapeFactor
ImpulseFactor
MarginFactor
Energy

As illustrated in Fig. 6, the top seven ranked features are: Time-domain Mean (TMean), Time-domain Standard Deviation (TStd), Time-domain Maximum (TMax), Time-domain Minimum (TMin), Time-domain Root Mean Square (TRMS), Time-domain Skewness (TSkewness), and Time-domain Kurtosis (TKurtosis). These features are selected as the input attributes for the EBRB-EU model. Through this step, the raw vibration signals are transformed into representative feature vectors. denoted as .providing the foundation for subsequent fault diagnosis.

Fig. 6.

Feature importance ranking based on the random forest feature importance analysis algorithm.

Here, represents the sample point values of the bearing vibration signal in the time domain, and N denotes the total number of sample points. The feature values U1 to U13 in the time domain are calculated using the corresponding formulas. represents the amplitude values of the bearing vibration signal in the frequency domain obtained by Fourier transform, and M denotes the total number of frequency points. The feature values U14 to U26 in the frequency domain are calculated using the corresponding formulas. represents the time-frequency amplitude values of the bearing vibration signal obtained by time-frequency analysis. where t and f denote the time frame and frequency bin indices, respectively. T and F denote the total number of time frames and frequency bins. The feature values U27 to U39 in the time-frequency domain are calculated using the corresponding formulas.

(2) Ensemble undersampling and sub-model construction.

To address the data imbalance issue prevalent in real-world engineering scenarios, where healthy samples significantly outnumber fault samples, this paper employs an ensemble undersampling strategy to process the training set. Initially, the sample counts for each category are calculated, and the average number of minority class samples is used as the target value for undersampling. In each round of undersampling, a number of samples close to the target value are randomly drawn from the majority class (healthy state) and combined with all minority class (fault state) samples to form a new, balanced subset. To ensure the effectiveness and stability of the model, this process is repeated to generate 10 distinct balanced training subsets, each serving as the input for a parallel sub-EBRB model. This ensemble of 10 undersampled subsets allows the model to learn from diverse data distributions, effectively mitigating the bias introduced by class imbalance and enhancing the generalization capability of the final diagnostic system.

(3) EBRB Inference Based on KL Divergence and Ensemble Diagnosis.

Based on the constructed sub-training sets, the inference process for each sub-EBRB model proceeds as follows. First, reference points for each input attribute are established based on expert knowledge (see Table 6), and Eq. (8) is applied to convert the sample attribute values into belief distributions relative to these reference points. To more accurately capture the distributional discrepancy between the sample and the rule antecedents, this paper introduces the Kullback-Leibler (KL) divergence as a replacement for the traditional Euclidean distance in matching degree calculation. Using the calculated matching degree, along with the rule weights and attribute weights, the activation weight for each rule is determined via Eq. (13). The Evidential Reasoning (ER) algorithm (Eqs. 14–15) is then employed to fuse the activated rules, yielding preliminary diagnostic results for the bearing status from each sub-model. To address the limitation that initial parameters are constrained by expert knowledge, the Differential Evolution (DE) algorithm (Fig. 4) is introduced to globally optimize the parameters of each sub-model, using Mean Squared Error (MSE) as the loss function (Eq. 16), thereby further enhancing diagnostic accuracy. Finally, a voting mechanism (Fig. 5) is adopted to make an ensemble decision based on the preliminary results generated by the nine sub-models, selecting the most frequent category as the final diagnostic output. This enables accurate identification of the bearing health status. The reference points and corresponding values for the result attributes are provided in Table 7.

Table 6.

Reference points and reference values for every attributes.

Reference point	EL	ML	L	M	H	MH	EH
Reference value	− 0.0616	0.0031	0.0037	0.0043	0.0134	0.0167	0.0815
Reference point	EL	ML	L	M	H	MH	EH
Reference value	0.0591	0.0653	0.0739	0.1136	0.1136	0.6474	2.2875
Reference point	EL	ML	L	M	H	MH	EH
Reference value	0.1752	0.2143	0.2644	0.4527	3.0022	3.5014	9.7493
Reference point	EL	ML	L	M	H	MH	EH
Reference value	− 9.3009	− 3.2249	− 2.8433	− 1.8589	− 0.2476	− 0.2095	− 0.1602
Reference point	EL	ML	L	M	H	MH	EH
Reference value	0.0611	0.0668	0.0668	0.1144	0.5133	0.6474	2.2876
Reference point	EL	ML	L	M	H	MH	EH
Reference value	− 0.5238	− 0.2022	− 0.1191	− 0.0537	0.0891	0.1663	0.5353
Reference point	EL	ML	L	M	H	MH	EH
Reference value	− 0.4448	− 0.1544	− 0.0352	0.0768	6.3554	16.735	41.3903

Table 7.

Reference points and reference values for output results.

Reference point	Normal	Ball	Inner race fault	Outer race fault
Reference value	1	2	3	4

Among these, EL denotes extremely low, ML denotes moderately low, L denotes low, M denotes medium, H denotes high, MH denotes moderately high, and EH denotes extremely high.

After obtaining the diagnostic results from the 10 parallel sub-EBRB-EU models, a voting algorithm (process illustrated in Fig. 5) is used for ensemble integration. This algorithm counts the predicted labels for each sample from all 10 sub-models and determines the final diagnostic result as the category with the highest frequency. The complete bearing fault diagnosis process based on EBRB-EU is depicted in Fig. 7.

Fig. 7.

The overall process of bearing fault diagnosis based on EBRB-EU.

Comparison experiment

To comprehensively evaluate the performance of the proposed EBRB-EU method, a series of experiments were conducted on the bearing and gear fault datasets. A multi-metric evaluation strategy was adopted to assess the model’s diagnostic capability from different perspectives. Specifically, four widely-used classification metrics were employed: Accuracy, which measures the overall correctness of the model; Precision, which reflects the proportion of correctly identified positive samples among all samples classified as positive; Recall, which indicates the model’s ability to capture all actual positive samples; and the F1 score, which provides a harmonic mean of Precision and Recall, offering a balanced assessment when dealing with imbalanced data distributions.

To guarantee a fair and controlled comparison, each baseline model was carefully configured with reasonable hyperparameters selected based on grid search and recommendations from the literature. The detailed hyperparameter settings for all comparative models are summarized in Table 8.

Table 8.

Comparison of model hyperparameters.

Model	Hyperparameters
KNN	n_neighbors = 5, weights="uniform”, metric="minkowski”, p = 2;
BPNN	hidden_layer_sizes=(100,), activation="relu”, solver="adam”, alpha = 0.0001, learning_rate_init = 0.001, max_iter = 300;
RF	n_estimators = 300, max_depth=None, min_samples_split = 2, min_samples_leaf = 1;
SVM	C = 1.0, kernel="rbf”, gamma="scale”, class_weight=None,
DT	criterion="gini”, max_depth=None, min_samples_split = 5, min_samples_leaf = 2;
ELM	n_hidden = 500, activation="sigmoid”, alpha=1e-2;
CWSVM	C = 1.0, kernel="rbf”, gamma="scale”, class_weight="balanced”,
BRF	n_estimators = 300, max_depth=None, min_samples_split = 2, min_samples_leaf = 1, class_weight="balanced_subsample”;
CSL	C = 1.0, kernel="rbf”, gamma="scale”, class_weight=None;

1. Comparison with the similar BRB models.

First, following the ensemble undersampling strategy described in Sect.  3.2, the imbalanced training set was randomly partitioned into ten balanced subsets, each containing a comparable number of samples across all bearing health conditions. Based on these subsets, ten parallel sub-EBRB-EU models were constructed and independently trained. For each sub-model, the reference points and reference values for input attributes and output results were set according to Tables 6 and 7, respectively. During the training phase, the Differential Evolution (DE) optimization algorithm was applied to optimize the parameters of each sub-model, with the maximum number of iterations set to 200 to ensure convergence. To ensure the transparency of the optimization process, the key parameter settings for the Differential Evolution (DE) algorithm are summarized in Table 9. After training, each sub-model generated preliminary diagnostic results on the test set. These ten sets of preliminary predictions were then integrated using a majority voting mechanism to produce the final diagnostic decisionThe preliminary diagnostic results of the ten individual sub-models are presented in Fig. 8a–j, and the final integrated result after voting is shown in Fig. 8k.

Table 9.

Parameter settings for the DE optimization algorithm.

Parameter name	Initial population size	Lower limit of mutation operation	Upper limit of mutation operation	crossover probability
Parameter value	5	0.2	0.8	0.2

Fig. 8.

The bearing fault diagnosis results based on EBRB-EU.

To further validate the superiority of the proposed EBRB-EU method, it was compared with several representative belief rule base (BRB) variants, including Interval BRB (IBRB), which introduces interval uncertainty into rule parameters; Hierarchical BRB (HBRB), which organizes rules in a layered structure to reduce complexity; and Approximate BRB (ABRB), which simplifies rule generation for large-scale problems. All models were evaluated under the same experimental conditions using multiple performance metrics, including accuracy, F1-score precision, and recall, to ensure a comprehensive and fair comparison.

As shown in Fig. 9a-c and Table 10, each of the similar BRB models (IBRB, HBRB, and ABRB) exhibits significant classification errors, with overall accuracies ranging from 81.2% to 84.0% and average F1-scores between 0.718 and 0.744. These models struggle particularly with minority fault classes—for instance, the ABRB model achieves a recall of only 0.555 for Ball Fault (BF), and the HBRB model achieves a precision of just 0.631 for Inner Race Fault (IRF). Furthermore, Fig. 8 reveals that individual sub-EBRB-EU models also suffer from bias in bearing fault diagnosis due to class imbalance. Each of the 10 sub-models exhibits varying degrees of identification errors, with some sub-models misclassifying minority fault samples as the majority Normal state. This is expected, as each sub-model is trained on a different balanced subset and may capture different aspects of the fault patterns.

Fig. 9.

Diagnosis results of similar BRB models. (a) IBRB, (b) HBRB, (c) ABRB.

Table 10.

Experimental results based on the comparison of similar models.

Models	Accuracy	Average F1-Score	Ball fault		IRF		Normal		ORF
			Precision	Recall	Precision	Recall	Precision	Recall	Precision	Recall
EBRB-EU	0.93	0.892	0.875	0.777	0.809	0.944	0.977	0.977	1	0.833
IBRB	0.826	0.725	0.588	0.555	0.8	0.666	0.944	0.944	0.705	0.666
HBRB	0.84	0.744	0.714	0.833	0.631	0.631	0.933	0.933	0.714	0.555
ABRB	0.812	0.718	0.769	0.555	0.5	0.833	0.888	0.952	0.705	0.666

It is worth noting that during the training phase, each sub-model’s parameters were optimized using the Differential Evolution (DE) algorithm. As illustrated in Fig. 10, the optimization process gradually converges around 150 iterations, indicating that the model parameters have been effectively optimized and the sub-models have reached a stable state. This ensures that the subsequent ensemble learning is built upon well-trained base models with reliable parameter configurations.

Fig. 10.

Iterative process of the DE optimization algorithm.

However, when the voting algorithm is applied for final decision-making, the recognition accuracy of the ensemble EBRB-EU model reaches 93.0%, with an average F1-score of 0.892. More importantly, the ensemble model achieves significantly improved performance on minority fault classes: for Ball Fault, precision reaches 0.875 and recall reaches 0.777; for IRF, precision reaches 0.809 and recall reaches 0.944 ; and for ORF, precision reaches 0.833 and recall reaches a perfect 1.0. This demonstrates that the ensemble voting mechanism effectively aggregates the diverse perspectives of the 10 sub-models, compensating for individual sub-model biases and producing a more robust and accurate diagnostic outcome.

2. Comparison with the data-driven model.

To further demonstrate the effectiveness of the proposed method, a comparative analysis was conducted between the EBRB-EU and several classical machine learning models, including Backpropagation Neural Networks (BPNN), k-Nearest Neighbors (KNN), Support Vector Machines (SVM), Decision Trees (DT), Extreme Learning Machines (ELM), and Random Forests (RF). All models were comprehensively evaluated using multiple performance metrics, including accuracy, precision, recall, and F1-score, to ensure a multi-faceted assessment of diagnostic capability.

As shown in Fig. 11a-f and Table 11, data-driven models exhibit varying performance in bearing fault diagnosis. Decision Tree (DT) achieves the highest accuracy among them at 88.1%, followed by Random Forest (RF) at 86.8% and Support Vector Machine (SVM) at 84.0%. However, most data-driven models struggle with minority fault classes—for instance, KNN achieves a precision of only 0.222 for Ball Fault and 0.277 for Outer Race Fault (ORF).

Fig. 11.

Diagnosis results based on the data-driven learning model. (a) KNN. (b) BPNN. (c) RF. (d) SVM. (e) DT. (f) ELM.

Table 11.

Experimental results compared with the data-driven model.

Models	Accuracy	Average F1-Score	Ball Fault		IRF		Normal		ORF
			Precision	Recall	Precision	Recall	Precision	Recall	Precision	Recall
EBRB-EU	0.93	0.892	0.875	0.777	0.944	0.944	0.956	0.977	1	0.833
KNN	0.743	0.528	0.666	0.222	0.5	0.5	0.787	0.787	0.5	0.277
BPNN	0.805	0.682	0.571	0.444	0.388	0.388	0.81	1	0.916	0.611
RF	0.868	0.735	0.875	0.777	0.5	0.5	0.872	0.988	0.500	0.277
SVM	0.84	0.737	0.562	0.5	0.833	0.833	0.855	0.988	0.888	0.444
DT	0.881	0.761	0.625	0.833	0.666	0.666	1	1	0.666	0.555
ELM	0.784	0.622	0.466	0.388	0.333	0.333	0.81	1	0.909	0.555

In contrast, the proposed EBRB-EU model achieves the highest overall accuracy of 93.0% and an average F1-score of 0.892, substantially outperforming all data-driven comparators. More importantly, EBRB-EU demonstrates balanced performance across all fault classes: for Ball Fault (precision: 0.875, recall: 0.777), IRF (precision: 0.944, recall: 0.944), and ORF (precision: 1, recall: 0.833). This balanced performance highlights the effectiveness of the proposed ensemble undersampling strategy in mitigating class imbalance.

Furthermore, data-driven models are highly dependent on training sample quantity, contributing to their performance degradation on minority classes. In contrast, EBRB-EU integrates expert knowledge through its belief rule structure, reducing its reliance on large sample sizes. Achieving 93.0% accuracy despite limited fault samples demonstrates that the proposed method is particularly well-suited for small-sample scenarios in fault diagnosis.

3. Comparison with classical imbalance-handling methods.

To further illustrate the effectiveness of the proposed method in addressing imbalanced datasets, we will conduct a comprehensive comparative analysis between our proposed approach and three established methods specifically designed for handling class imbalance: Class-Weighted Support Vector Machine (CWSVM), Balanced Random Forest (BRF), and Cost-Sensitive Learning (CSL).These methods have gained recognition in the machine learning community for their ability to mitigate the adverse effects of class imbalance, The models will be comprehensively evaluated using multiple performance metrics to assess their capability in fault diagnosis under imbalanced data conditions.

As shown in Fig. 12; Table 12, all three imbalance-handling methods—CWSVM, BRF, and CSL—achieve reasonable accuracies (84.7%–86.8%) under small-sample and imbalanced conditions, confirming the effectiveness of imbalance mitigation strategies. However, they exhibit notable instability on minority classes: CWSVM achieves a recall of only 0.5 for Ball Fault, and BRF achieves just 0.555 for Outer Race Fault (ORF), indicating persistent susceptibility to misclassifying rare fault types.

Fig. 12.

Diagnosis results based on the classical imbalance-handling model. (a) CWSVM, (b) BRF, (c) CSL.

Table 12.

Experimental results compared with classical imbalance-handling model.

Models	Accuracy	Average F1-score	Ball fault		IRF		Normal		ORF
			Precision	Recall	Precision	Recall	Precision	Recall	Precision	Recall
EBRB-EU	0.93	0.892	0.875	0.777	0.809	0.944	0.956	0.977	1	0.833
CWSVM	0.854	0.853	0.5	0.642	0.777	1	0.988	0.855	0.611	0.916
BRF	0.861	0.732	0.636	0.777	0.785	0.611	1	1	0.555	0.555
CSL	0.847	0.862	0.5	0.692	0.722	1	0.988	0.839	0.611	0.916

In contrast, the proposed EBRB-EU model achieves the highest accuracy of 93.0% and an average F1-score of 0.892, with balanced performance across all fault classes—notably achieving a Precision of 0.875 for Ball Fault and a perfect 1.0 for ORF. This superiority stems from fundamental differences in how these approaches handle class imbalance.

Interpretability analysis: rule activation and decision mechanism of the EBRB-EU model

To further elucidate that the EBRB-EU model proposed in this paper is not a “black box” and to validate its strong interpretability, this subsection provides a qualitative analysis of the model’s internal inference mechanism through a representative example.

A core advantage of the EBRB model lies in the transparency of its reasoning process. After calculating the matching degree based on KL divergence, each sub-EBRB model computes the activation weight for every rule based on the input sample. Table 13 presents the most highly activated rule for a typical “Inner Race Fault” (IRF) test sample in the bearing fault diagnosis experiment.

Table 13.

Activated rules and their weights for an inner race fault sample.

Rule ID	Rule antecedents (selected key features)	Activation weight	Rule consequent (belief distribution)
	Wenergy=(MH = 0.6188,LH = 0.3812) TMax=(H = 0.2953,MH = 0.7047) TStd=(L = 0.988,M = 0.012) Wenergy= (ML = 0.7037,L = 0.2963) TMax=(M = 0.9676,H = 0.0324) TStd=(EL = 0.139,ML = 0.861) Wenergy=(M = 0.2526,H = 0.7474)	0.01001	(Ball:0), (IRF:1), (Normal:0), (ORF:0)

From Table 13, it can be observed that this highly activated rule assigns the highest confidence to “Inner Race Fault” (IRF) in its consequent belief distribution, with a belief value of 1.0 for IRF. This demonstrates that when the antecedent attributes match the specific combination of reference points shown in the table, the model strongly supports an IRF diagnosis. After fusing this rule with other activated rules using the Evidential Reasoning (ER) algorithm, the final diagnostic result is determined based on the collective evidence. This example intuitively demonstrates how the model learns interpretable patterns from data, with a clear and traceable reasoning path at the individual rule level.

It is worth noting that the activation weight in Table 13 is relatively small (0.01001). This occurs because the transformation of training samples into rules generates a large number of rules, resulting in relatively small weight values assigned to each individual rule. However, this does not affect the fault diagnosis outcome. The activation weight represents the relative contribution of a rule to the final decision, and the ER algorithm aggregates all activated rules based on their weights to produce a collective belief distribution. As long as the relative ranking of rule contributions is preserved, the magnitude of individual weights does not impact the final diagnostic result. Thus, the overall inference process of the model remains unaffected.

It is important to note that while the EBRB-EU employs an ensemble of multiple sub-models with a voting mechanism, the interpretability of the overall system is not compromised. Each sub-model maintains its own transparent rule structure, and the voting process allows users to trace which sub-models supported a conclusion and examine the specific rules activated within them. This multi-perspective validation strengthens confidence in the final diagnostic outcome.

Four-fold cross-validation experiment

To further assess the generalization capability and robustness of the proposed method across varying operating conditions, we adopt a Leave-One-Group-Out Cross-Validation (LOGOCV) strategy based on motor speed. Unlike conventional random K-fold cross-validation, which may inadvertently mix samples from different operating conditions in both training and test sets, LOGOCV provides a more rigorous and realistic evaluation by ensuring that the model is tested on operating conditions entirely unseen during training.

Specifically, the dataset is stratified into four mutually exclusive groups according to motor speed, each corresponding to a distinct load condition: 1730 RPM (3 HP load), 1750 RPM (2 HP load), 1772 RPM (1 HP load), and 1797 RPM (0 HP load). In each iteration of the cross-validation process, one complete speed group is reserved as the test set, while the remaining three groups are combined to form the training set. This procedure is repeated four times, with each speed group serving as the test set exactly once.

It enables a systematic assessment of the model’s cross-speed generalization performance, which is critical for practical deployment in variable-speed industrial environments.

An examination of the four-fold cross-validation results in Table 14 demonstrates the stability and reliability of the proposed EBRB-EU model across different operating conditions. The model achieves a mean accuracy of 91.9% with a standard deviation of only 1.5%, while the average F1-score reaches 0.880 ± 0.017, indicating consistent and robust diagnostic performance with minimal fluctuation across all four folds.

Table 14.

Four-fold cross-validation experimental data.

Accuracy	Average F1-Score	Ball fault		IRF		Normal		ORF
		Precision	Recall	Precision	Recall	Precision	Recall	Precision	Recall
0.919±0.015	0.880±0.017	0.916±0.071	0.847 ± 0.036	0.833±0.045	0.908 ± 0.105	0.960±0.023	0.958 ± 0.010	0.805±0.056	0.838 ± 0.112

Per-class metrics further corroborate the model’s balanced capability in handling both majority and minority classes. For the Normal class, the model maintains consistently high precision (0.960 ± 0.023) and recall (0.958 ± 0.010), demonstrating reliable identification of healthy bearing conditions. The minority fault classes exhibit slightly higher variability, as expected given their limited sample sizes. Ball Fault achieves a mean precision of 0.916 ± 0.071 and recall of 0.847 ± 0.036, while Inner Race Fault shows precision of 0.833 ± 0.045 and recall of 0.908 ± 0.105. Outer Race Fault presents precision of 0.805 ± 0.056 and recall of 0.838 ± 0.112.Crucially, the narrow fluctuation range across all four folds—with accuracy varying by less than 3.5% and most metrics exhibiting standard deviations below 0.10—reinforces the model’s credibility and robustness.

Ablation experiment

To systematically isolate and validate the individual contributions of the three key components in the proposed EBRB-EU method—namely, the KL divergence-based matching degree calculation, the ensemble undersampling strategy, and the differential evolution (DE) optimization algorithm—we conducted a series of ablation experiments under the same data split and experimental protocol. Four variant models were designed for comparative analysis: EBRB-EU (proposed full model) incorporates all three components, including KL divergence for matching degree calculation, ensemble undersampling strategy, and DE optimization; EBRB-EU0 employs KL divergence and ensemble undersampling but excludes the DE optimization algorithm, retaining the initial parameters derived from expert knowledge; EBRB-EU1 employs ensemble undersampling and DE optimization but replaces KL divergence with traditional Euclidean distance for matching degree calculation, serving as a baseline to evaluate the effectiveness of KL divergence; and EBRB-EU2 employs KL divergence and DE optimization but excludes the ensemble undersampling strategy, using a single balanced subset for training instead of multiple parallel sub-models, thereby isolating the impact of the ensemble mechanism.

Under identical experimental conditions, all four variants were trained and evaluated on the same dataset to ensure a fair and controlled comparison, with the detailed experimental results presented in Table 15.

Table 15.

Comparison results of ablation experiments.

Models	Accuracy	Average F1-Score	Ball fault		IRF		Normal		ORF
			Precision	Recall	Precision	Recall	Precision	Recall	Precision	Recall
EBRB-EU	0.93	0.892	0.875	0.777	0.809	0.944	0.956	0.977	1	0.833
EBRB-EU0	0.819	0.705	0.5	0.818	0.667	0.666	0.966	0.861	0.555	0.714
EBRB-EU1	0.854	0.764	0.722	1	0.666	0.705	0.966	0.915	0.611	0.578
EBRB-EU2	0.833	0.797	0.666	0.888	0.727	0.888	0.96	0.811	0.681	0.833

From the ablation study results presented in Table 15, the following observations can be made:

Impact of KL divergence: Comparing EBRB-EU1 (Euclidean distance), which achieves an accuracy of 85.4% and an average F1-score of 0.764, with EBRB-EU (KL divergence), which achieves 93.0% accuracy and an average F1-score of 0.892, reveals that the introduction of KL divergence leads to a substantial improvement in diagnostic performance. Notably, EBRB-EU demonstrates superior performance on minority fault classes, achieving a recall of 0.777 for Ball Fault and a perfect recall of 1.0 for ORF, compared to EBRB-EU1’s recall of 1.0 for Ball Fault but only 0.578 for ORF. This confirms that KL divergence, rooted in information theory, is more effective than Euclidean distance in capturing the distributional differences between vibration signal samples and rule antecedents, making it particularly suitable for uncertainty modeling in bearing fault diagnosis.

Impact of ensemble undersampling: Comparing EBRB-EU2 (single subset), which achieves 83.3% accuracy and an average F1-score of 0.797, with EBRB-EU (ensemble of 10 subsets) demonstrates that the ensemble strategy effectively mitigates the bias introduced by class imbalance. While the single-subset model shows reasonable performance, the ensemble model achieves more balanced and robust diagnostic results across all categories—particularly for IRF (precision: 0.809, recall: 0.944) and ORF (Precision: 1.0)—validating the effectiveness of the ensemble undersampling approach in aggregating diverse perspectives from multiple balanced subsets.

Impact of DE optimization: Comparing EBRB-EU0 (without optimization), which achieves only 81.9% accuracy and an average F1-score of 0.705, with EBRB-EU (with optimization) indicates that the DE optimization algorithm plays a crucial role in enhancing model performance. The optimized model shows consistent improvements across nearly all metrics, with particularly notable gains in precision for Ball Fault (from 0.5 to 0.875) and recall for ORF (from 0.714 to 8.33). By globally optimizing the initial parameters constrained by expert knowledge, the optimization process significantly improves the model’s recognition accuracy and stability.

In summary, the ablation experiments demonstrate that each of the three proposed components contributes positively to the overall performance of the EBRB-EU model. The combination of KL divergence-based matching, ensemble undersampling, and DE optimization yields the best diagnostic results (93.0% accuracy), confirming the effectiveness and necessity of each component in addressing the challenges of imbalanced data and noisy environments in bearing fault diagnosis.

Disturbance experiment

In this section, a disturbance injection analysis experiment is conducted. The experiment is designed to simulate common noise interference scenarios in real industrial measurement environments, aiming to evaluate the applicability of the proposed method under adverse measurement conditions. In summary, white noise with different signal-to-noise ratios will be injected to quantitatively analyze the diagnostic reliability of EBRB-EU when processing contaminated signals, thereby clarifying its potential for practical engineering applications. Based on the random forest importance ranking algorithm described in Sect.  4.1.2, the two most influential attributes—TRMS and Tmax—are selected for noise injection. The disturbance intensity introduced in our experiments can be expressed as parameter , The aim is to test the effectiveness of the model in four different noise intensity environments. To more intuitively observe the performance fluctuations of various models under noisy conditions, a multi-round comparative analysis was conducted using accuracy as a key evaluation metric. Specifically, the comparative study focused on analyzing fluctuations in the accuracy rates across different models. The experiment was repeated over 50 rounds, and the average values were taken to evaluate the anti-interference capability of each model.

As can be observed from Fig. 13; Table 16, all models exhibit relatively minor accuracy fluctuations when the noise level increases from 0.001 to 0.005, indicating that low-intensity disturbance has limited impact on model performance. However, as the disturbance intensity continues to increase from 0.005 to 0.02, most models experience substantial performance degradation. For instance, the accuracy of IBRB drops from 81.0% to 73.9%, ABRB declines from 80.0% to 71.9%, and KNN falls from 72.5% to 64.9%—all representing decreases of approximately 7–8% points. This indicates that these models are significantly affected by noise interference and struggle to consistently identify equipment states or provide reliable results for decision-makers under higher noise levels.

Fig. 13.

The accuracy dynamics of different models under noisy conditions.

Table 16.

Detailed comparison results of each model in a noisy environment.

Models	Average accuracy disturbance: (0.001)	Average accuracy disturbance: (0.005)	Average accuracy disturbance: (0.01)	Average accuracy disturbance: (0.02)
EBRB-EU	0.92	0.92	0.905	0.894
IBRB	0.809	0.81	0.779	0.739
HBRB	0.829	0.826	0.8	0.764
ABRB	0.8	0.8	0.76	0.719
KNN	0.73	0.725	0.689	0.649
BPNN	0.789	0.78	0.748	0.69
RF	0.86	0.845	0.809	0.757
SVM	0.823	0.819	0.779	0.734
DT	0.877	0.867	0.829	0.789
ELM	0.75	0.75	0.709	0.66
CWSVM	0.84	0.824	0.795	0.759
BRF	0.85	0.839	0.82	0.77
CSL	0.829	0.82	0.778	0.71

In contrast, although the EBRB-EU model also experiences some accuracy fluctuation as noise intensifies, its overall variation remains relatively limited across all four noise levels. Specifically, EBRB-EU maintains 92.0% accuracy at both 0.001 and 0.005 disturbance levels, then declines modestly to 90.5% at 0.01 and 89.4% at 0.02—a total decrease of only 2.6% points from the lowest to highest noise level. This represents the smallest performance degradation among all compared models. Even under increasing disturbance intensities, the model continues to deliver stable and dependable diagnostic results to support decision-making.

This stability can be attributed to the EBRB-EU method proposed in this paper, which employs evidence reasoning and belief distribution modeling to effectively suppress identification deviations caused by noise, thereby enhancing the accuracy of state assessment. It is evident that, even in noisy environments with varying disturbance magnitudes, the model maintains high precision and reliable evaluation capability. These results confirm that EBRB-EU possesses strong and consistent noise resistance, with performance degradation substantially lower than that of other models under high-noise conditions.

Experimental summary

1. The ensemble undersampling strategy effectively mitigates the bias caused by imbalanced data. Through multiple comparative experiments, the proposed EBRB-EU model consistently outperforms data-driven models and classical imbalance-handling methods, particularly in identifying minority fault classes, confirming its capability to address sample imbalance in real-world scenarios.

2. The four-fold cross-validation experiments under different motor speeds yield consistent accuracy ranging from 90.2% to 93.7%, with an average of 91.9%. The narrow performance fluctuation across all four folds demonstrates the model’s strong generalization capability and stability across varying operating conditions.

3. The Differential Evolution (DE) optimization algorithm demonstrates strong compatibility with the EBRB model, substantially improving recognition accuracy compared to the non-optimized EBRB-EU0. This validates the effectiveness of parameter optimization in enhancing model performance.

4. The ablation experiments reveal that replacing traditional Euclidean distance with KL divergence for membership calculation leads to a significant improvement in diagnostic accuracy. Furthermore, the ensemble strategy combining multiple balanced subsets produces more robust and balanced results than any single sub-model, confirming the contribution of both key components.

5. The disturbance experiments validate the robustness of the proposed method under noisy conditions. As noise intensity increases from 0.001 to 0.02, EBRB-EU maintains stable performance with only a 2.6% point decrease in accuracy—the smallest degradation among all compared models—demonstrating its reliability in real industrial environments with significant noise interference.

Case 2: gear fault diagnosis

In this section, the open-source gear dataset is used for further evaluation of the EBRB-EU method. Section  4.2.1 provides a description of the dataset and the experimental setup; Sect.  4.2.2 presents comprehensive comparison experiments with multiple models, including similar BRB variants, data-driven models, and classical imbalance-handling methods; and Sect.  4.2.3 summarizes the experimental findings and discusses the overall performance of the proposed method in gear fault diagnosis tasks.

Dataset description

As key components responsible for transmitting torque and motion, gear performance directly influences the overall efficiency, stability, and lifespan of the machinery. Therefore, effective monitoring of gear conditions and fault diagnosis is crucial for maintaining the health and reliability of rotating machinery. Recognizing the potential impact of gear-related failures on operational efficiency and safety, researchers have proposed a variety of methods aimed at accurately diagnosing gear faults. Among the commonly explored techniques are: Dong et al. proposed an interpretable multiscale lifting wavelet contrast network for planetary gearbox fault diagnosis with small samples⁴⁴. Zheng et al. proposed a novel signal decomposition method called empirical reconstruction Gaussian decomposition (ERGD) for gear fault diagnosis⁴⁵. Zhang et al. proposed a multi-modal data cross-domain fusion network for gearbox fault diagnosis under variable operating conditions⁴⁶.

The dataset utilized in this study is the publicly available gearbox fault diagnosis dataset from Southeast University, China. This dataset is widely recognized in the field of rotating machinery fault diagnosis and contains two subdatasets, including bearing data and gear data, both acquired on a Drivetrain Dynamics Simulator (DDS). In our experiments, we specifically employed the gear subdataset to validate the effectiveness of the proposed method for gear fault diagnosis tasks.

Two different working conditions are configured in this dataset, with rotating speed-load settings of 20 Hz-0 Nm and 30 Hz-2 Nm. Within each data file, there are eight channels of signals representing: 1-motor vibration, 2,3,4-vibration of the planetary gearbox in the x, y, and z directions, 5-motor torque, and 6,7,8-vibration of the parallel gearbox in the x, y, and z directions. According to previous studies, the signals from channels 2, 3, and 4 (planetary gearbox vibrations) are considered the most effective for fault diagnosis.

For our experiments, we selected channel 1 (motor vibration) as the data source, as it provides comprehensive vibration information of the entire drivetrain system. To evaluate the generalization capability of the proposed method under varying operating conditions, we adopted a cross-domain diagnostic strategy: the dataset collected under 20 Hz-0 Nm condition was used for model training, while the dataset collected under 30 Hz-2 Nm condition was used for testing. This experimental setup allows us to validate the robustness and transferability of the EBRB-EU method when applied to gear fault diagnosis tasks under different working conditions—a common challenge in real-world industrial applications.

Comparison experiment

In this section, a comprehensive evaluation of the proposed EBRB-EU method is conducted using multiple performance metrics, including accuracy, precision, recall, and F1-score for each fault class, as well as the overall accuracy and macro-averaged F1-score to provide a holistic assessment of model performance. The experimental data samples used in the experiments are presented in Table 17.

Table 17.

Unbalanced gear dataset for the experiment.

The number of training set samples	Chipped	Miss	Root	Surface	Health
	108	108	108	108	357
The number of test set samples	Chipped	Miss	Root	Surface	Health
	45	45	45	45	154

As shown in Tables 18 and 19, the proposed EBRB-EU method demonstrates superior performance in gear fault diagnosis across all evaluation metrics. The model achieves the highest overall accuracy of 92.8%, substantially outperforming all comparable approaches—including similar BRB variants (IBRB: 82.0%, HBRB: 84.1%, ABRB: 82.6%), classical data-driven models (the next best being BPNN at 86.2%), and specialized imbalance-handling methods (CWSVM: 84.0%, BRF: 81.7%, CSL: 83.2%). This significant margin of at least 6.6% points over the next best performer underscores the effectiveness of the proposed method in handling complex gear fault diagnosis tasks.Beyond overall accuracy, the model achieves a macro-averaged F1-score of 0.929, with an average precision of 0.910 and average recall of 0.956. The high recall value is particularly noteworthy, indicating that the model excels at identifying actual fault instance.

Table 18.

Based on the detailed experimental data for gear fault diagnosis of each models.

Models	Accuracy	Average F1-score	Average precision	Average recall
EBRB-EU	0.928	0.929	0.910	0.956
IBRB	0.820	0.815	0.790	0.856
HBRB	0.841	0.829	0.812	0.858
ABRB	0.826	0.807	0.793	0.836
KNN	0.754	0.734	0.716	0.777
BPNN	0.862	0.832	0.833	0.833
RF	0.811	0.784	0.791	0.795
SVM	0.838	0.788	0.786	0.794
DT	0.856	0.847	0.826	0.877
ELM	0.802	0.774	0.767	0.788
CWSVM	0.84	0.829	0.845	0.821
BRF	0.817	0.812	0.794	0.857
CSL	0.832	0.801	0.802	0.804

Table 19.

Detailed data for gear fault diagnosis based on each model.

Models	Chipped		Miss		Root		Surface		Health
	Precision	Recall	Precision	Recall	Precision	Recall	Precision	Recall	Precision	Recall
EBRB-EU	0.878	0.953	0.848	0.966	0.989	0.861	0.86	0.946	0.895	0.973
IBRB	0.678	0.888	0.833	0.888	0.958	0.746	0.846	0.977	0.636	0.777
HBRB	0.811	0.955	0.769	0.888	0.953	0.805	0.678	0.888	0.85	0.755
ABRB	0.603	0.777	0.661	0.866	0.976	0.805	0.829	0.755	0.898	0.977
KNN	0.597	0.888	0.566	0.666	0.964	0.707	0.684	0.866	0.772	0.755
BPNN	0.925	0.822	0.778	0.777	0.928	0.922	0.877	0.955	0.659	0.688
RF	0.549	0.866	0.818	0.8	0.935	0.844	0.864	0.711	0.790	0.755
SVM	0.682	0.622	0.744	0.777	0.953	0.928	0.814	0.777	0.735	0.866
DT	0.755	0.755	0.918	1	0.954	0.817	0.771	0.977	0.73	0.844
ELM	0.886	0.866	0.69	0.644	0.907	0.831	0.631	0.8	0.72	0.8
CWSVM	0.889	0.888	1	0.75	0.85	0.922	0.711	0.711	0.777	0.833
BRF	0.654	0.8	0.925	0.822	0.974	0.733	0.745	0.977	0.671	0.955
CSL	0.673	0.733	0.692	0.8	0.913	0.889	0.868	0.733	0.866	0.866

These results not only highlight the model’s ability to maintain both high precision and high recall simultaneously, but also further confirm the overall effectiveness, reliability, and practical applicability of the proposed EBRB-EU method in real-world gear fault diagnosis scenarios.

Experimental summary

In the gear fault diagnosis task, although the EBRB-EU model still exhibits some classification errors in certain gear condition recognition tasks, it demonstrates optimal performance in handling imbalanced samples compared to other models. Specifically, EBRB-EU maintains consistently high and balanced performance across all five gear health conditions—with recall values exceeding 0.95 for four out of five classes and precision values above 0.848 for all five classes. In contrast, other comparative models (such as IBRB, HBRB, KNN, RF, etc.) generally exhibit significant performance degradation on minority fault classes, manifesting as low precision or recall. This indicates that the EBRB-EU model, through its ensemble undersampling strategy and KL divergence matching mechanism, effectively mitigates recognition errors caused by data imbalance, demonstrating stronger robustness and reliability in real-world gear fault diagnosis scenarios.

Conclusion and discussion

Proposes an ensemble undersampling extended belief rule base (EBRB-EU) method for bearing fault diagnosis to address class imbalance and noise interference in real-world industrial environments, validated through bearing and gear fault diagnosis case studies. The experimental results demonstrate that the ensemble undersampling strategy effectively mitigates model bias toward the majority class, achieving balanced performance across all fault categories with significant improvements in minority class recognition. Disturbance experiments confirm strong noise robustness, with EBRB-EU exhibiting only a 2.6% point accuracy decrease under increasing noise—the smallest degradation among all compared models. Four-fold cross-validation under different motor speeds yields consistent accuracy from 90.2% to 93.7% (average 91.9%), validating the model’s generalization capability. Ablation experiments show that replacing Euclidean distance with KL divergence for membership calculation significantly enhances diagnostic accuracy by better capturing distributional differences in vibration signals, while Differential Evolution optimization substantially improves recognition accuracy compared to the non-optimized version.A limitation of the current approach lies in determining the optimal number of data partitions, which currently requires manual observation. Future research will focus on developing an adaptive method to autonomously determine the partition number and extending the framework to other rotating machinery components.

Author contributions

**XingChi Yan** : Writing – original draft, Software, Methodology, Formal analysis, Data curation.**Ning Li** : Writing – original draft, Methodology, Investigation. **ShuYin Nan** : Conceptualization, Investigation. **Yan Yu** : Writing – review & editing, Supervision, Funding acquisition, Conceptualization. Resources. **Run Dai** : Investigation.

Funding

This work was supported in part by Key Laboratory of Equipment Data Security and Guarantee Technology, Ministry of Education under Grant No.GDZB2024050100, in part by the Natural Science Foundation of Heilongjiang Province under Grant No. PL2024G009, in part by the Basic Research Support Program for Outstanding Young Teachers in Provincial Undergraduate Universities of Heilongjiang Province under Grant No. YQJH2024116, in part by Shandong Provincial Natural Science Foundation under Grant No. ZR2023QF010, in part by National Science Foundation of China Grant No. 72471067.

Data availability

Data will be made available on request.

Declarations

Competing interests

The authors declare no competing interests.