Abstract
Fault diagnosis is essential for the safe and stable operation of industrial equipment and for improving maintenance efficiency. The belief rule base (BRB) is well suited to fault diagnosis under complex conditions because of its strengths in knowledge representation and uncertainty reasoning. However, traditional BRB still faces three limitations in multi-class diagnosis: rule-scale expansion, insufficient use of feature reliability during inference, and inadequate exploitation of output reliability from multiple submodels. To address these issues, this study proposes a reliability-enhanced one-versus-rest BRB method, termed RE-OvR-BRB. First, the OvR strategy decomposes the multi-class task into binary subtasks and constructs parallel BRB submodels to reduce rule-combination complexity. Second, a multidimensional reliability modeling scheme is proposed, which incorporates feature-level reliability into matching calculation, rule activation, and evidence fusion, and further integrates submodel output-level reliability into the final decision fusion. Third, a decision fusion mechanism considering submodel output reliability is introduced to improve the robustness of multi-class decisions. Experiments on the CWRU bearing dataset and the UConn gear dataset show that the proposed method outperforms competing methods in classification accuracy, decision confidence, and noise robustness, demonstrating its effectiveness under complex industrial conditions.
Keywords: Belief rule base, Fault diagnosis, Multi-class diagnosis, Reliability-aware modeling, Evidential reasoning
Subject terms: Engineering, Mathematics and computing
Introduction
Fault diagnosis is essential for ensuring the safe operation of industrial equipment, improving production efficiency, and reducing maintenance costs1,2. In complex industrial environments, uncertain factors such as measurement noise, fluctuations in operating conditions, and sensor drift commonly cause reliability levels to vary across samples2–4. Failure to effectively characterize these reliability differences in input features can result in inference distortions and unstable diagnostic outcomes2,3.
The belief rule base (BRB) is a rule-based inference model that integrates expert knowledge with observational data. It conveys knowledge through a rule-based structure and belief distributions, while handling uncertain information via an evidential reasoning mechanism5. By integrating knowledge representation with uncertainty reasoning, BRB proves highly effective for fault diagnosis in small-sample, complex, and uncertain operational environments. In recent years, BRB has been extensively used in fault diagnosis, health condition evaluation, and reliability analysis6–8, providing a robust approach for state identification and uncertainty management in complex systems.
However, several limitations persist in current BRB studies for multi-class fault diagnosis under complex operating conditions. First, as the number of fault categories and input attributes increases, the number of rules tends to grow rapidly, posing challenges for rule design, parameter tuning, and inference computation9,10. Second, most existing studies rely on static reliability descriptions, which makes it difficult to capture dynamic variation in input feature reliability across samples7,8,11.This limitation also hinders the effective incorporation of such variation into BRB inference. Additionally, in parallel multi-class decision scenarios, the output reliability of different submodels may vary, yet this information is often neglected in the final decision process. Consequently, the stability and reliability of diagnostic outcomes are further compromised. Therefore, current BRB methods still lack a unified modeling framework that can simultaneously address rule complexity, feature reliability utilization, and multi-submodel decision fusion.
To address these issues, the One-versus-Rest (OvR) strategy provides a systematic way to decompose multi-class tasks. This technique partitions the original multi-class problem into several binary subproblems, allowing each submodel to focus on distinguishing one target class from the others. Consequently, the complexity of rule integration in unified modeling is reduced9,10. Nonetheless, structural decomposition via OvR alone cannot effectively utilize input feature reliability across diverse samples. Additionally, it still poses challenges for the robust fusion of outputs from multiple submodels.
Based on the above analysis, a reliability-enhanced RE-OvR-BRB fault diagnosis method is proposed. Under the OvR strategy, multiple parallel BRB submodels are constructed to reduce the complexity of rule combination in multi-class modeling. Meanwhile, sample-varying feature reliability is incorporated into the BRB inference process. Additionally, the reliability level of each submodel output is further integrated at the decision layer. In this way, the stability and reliability of multi-class diagnosis under complex operating conditions are enhanced. The main contributions of this study are summarized as follows:
(1) A RE-OvR-BRB framework for multi-class fault diagnosis is proposed. Multiple parallel BRB submodels are constructed to achieve multi-class extension and reduce the modeling burden associated with rule-scale growth.
(2) A sample-varying feature reliability modeling method is designed and incorporated into the BRB inference process, enabling feature reliability differences to be reflected in matching calculation, rule activation, and evidence fusion.
(3) A decision fusion mechanism that accounts for submodel output reliability is constructed. By jointly leveraging the outputs of multiple submodels and their reliability levels, the stability and reliability of multi-class diagnostic results are improved. The effectiveness of the proposed method is further validated through experiments on bearing and gear fault diagnosis.
The remainder of this paper is organized as follows. Section 2 reviews studies on fault diagnosis, with an emphasis on data-driven, knowledge-driven, and hybrid-driven methods, as well as recent progress in BRB-related research. Section 3 presents the problem description and analyzes the main challenges of multi-class fault diagnosis under complex operating conditions, including rule-combination complexity, feature-reliability utilization, and multi-submodel decision fusion. Section 4 introduces the proposed RE-OvR-BRB model, covering model construction, inference mechanism, parameter optimization, and modeling procedure. Section 5 presents the case study and experimental analysis, validating the effectiveness of the proposed method on bearing and gear datasets. Section 6 concludes the paper and discusses future research directions.
Related work
Existing fault diagnosis methods can generally be classified into three categories: data-driven, knowledge-based, and hybrid approaches. Data-driven approaches automatically extract statistical patterns and deep feature representations from sample data. As a result, they possess strong capabilities in feature learning and nonlinear modeling. These methods have been widely applied in fault diagnosis, condition monitoring, and remaining useful life prediction2. For instance, Wang et al. employed integrated fault features and support vector machines for roller bearing fault diagnosis12. Mao et al. proposed a deep auto-encoder with fused discriminant information for bearing fault diagnosis13. Ding et al. developed a time-frequency Transformer for rolling bearing fault diagnosis14. Shi et al. employed bidirectional-convolutional LSTM networks for planetary gearbox fault diagnosis15. Meanwhile, Lin et al. investigated a generalized MAML framework for few-shot cross-domain bearing fault diagnosis driven by heterogeneous signals16, and Magadán et al. presented an explainable and interpretable method for bearing fault classification and diagnosis under limited data17. In general, although data-driven approaches offer strong feature-learning and nonlinear-modeling capabilities, they typically require high-quality samples. Their stability and interpretability may still be limited under small-sample, noisy, and complex operating conditions2,3.
In contrast, knowledge-based approaches perform fault diagnosis by leveraging expert knowledge, rule-based structures, probabilistic reasoning, and fuzzy inference. Consequently, they exhibit strong traceability and knowledge representation capabilities3. For instance, Nan et al. proposed a real-time fault diagnosis method using a knowledge-based expert system that integrates expert knowledge, real-time sensor data, and fuzzy inference18. Berredjem and Benidir developed a fuzzy expert system for bearing fault diagnosis, in which fuzzy rules were induced from numerical data and combined with an improved range-overlap strategy for feature selection19. Chu et al. established a gray-correlation-based Bayesian network model for fault source diagnosis in multistage manufacturing systems under small-sample conditions20. Rajabi et al. further employed a multi-output neuro-fuzzy classifier together with signal processing and permutation entropy for fault diagnosis in industrial rotating equipment21. These approaches can, to a certain extent, reflect the knowledge logic involved in the diagnostic process. However, they usually depend on relatively complete prior knowledge. Their adaptability may still be constrained under complex operating conditions or changing data distributions3.
To integrate the advantages of data-driven and knowledge-based approaches, hybrid methods have gradually garnered increasing attention for fault diagnosis under complex operating conditions22–26. For instance, Xu et al. proposed a hybrid knowledge-based and data-driven method for the aging-dependent reliability evaluation of high-voltage circuit breakers22. Li et al. developed a model-data-hybrid-driven diagnosis method for open-switch faults in power converters23. Zeng et al. enhanced the adaptability of hybrid approaches to new scenarios by introducing easy transfer learning into model-data-hybrid-driven fault detection for battery inverters24. Lian et al. constructed an interpretable large-scale BRB modeling method for complex industrial systems with expert knowledge and limited data25. Geng et al. further examined data- and knowledge-driven belief rule learning for hybrid classification26. Overall, hybrid methodologies offer a valuable means to balance diagnostic accuracy with knowledge representation amidst complex operating environments. Nevertheless, how to further achieve unified handling of uncertain information and diagnostic reasoning remains an open issue.
As a typical hybrid-driven model, a BRB represents expert knowledge through a rule base with belief distributions. It also handles uncertain information by using an evidential reasoning mechanism5,26,27. Based on these characteristics, BRBs have been applied to fault diagnosis, health assessment, and reliability analysis6–8. For instance, Li et al. proposed an adaptive and interpretable BRB model for fault diagnosis6. Shi et al. developed a fault diagnosis method based on a belief rule base with adaptive attribute weights, in which extracted data features were incorporated into BRB modeling to improve diagnostic performance7. Yin et al. and Zhang et al. demonstrated the potential of BRB for health condition assessment and interpretable state assessment8,11.
Recent studies have further extended BRB-based modeling from the perspectives of reliability characterization and unreliable knowledge representation. For example, Lian et al. proposed a transparent fault diagnosis method for complex equipment by incorporating expert reliability into a BRB framework with linguistic Z-numbers28. Lian et al. also developed a linguistic Z-number rule-based modeling framework considering knowledge reliability based on evidential reasoning rules29. In addition, a large-scale linguistic Z-number BRB methodology was introduced for multidimensional and unreliable knowledge representation and learning30. Moreover, BRB has also been applied to the performance evaluation of complex systems under sensor disturbance conditions31. These studies have enriched BRB research in reliability-aware modeling and unreliable knowledge processing. However, they mainly focus on expert reliability, knowledge reliability, unreliable knowledge representation, or disturbance-aware evaluation, whereas the present study emphasizes sample-varying feature reliability and multi-submodel output reliability in multi-class fault diagnosis.
These studies indicate that BRB has a solid foundation for applications in the diagnosis and assessment of complex systems. However, existing studies have focused more on specific applications and performance validation, while collaborative modeling for rule complexity control and reliability utilization in complex multi-class scenarios has received less attention.
In addition, existing studies have begun to extend BRB in two directions: classification complexity control and reliability modeling. For classification tasks, Chang et al. applied BRB to modeling classification problems9, and Gao et al. further investigated rule generation and reduction10. For reliability-related studies, Feng et al. proposed a safety assessment model based on BRB and the power set32, while Li et al. developed a BRB-based health prediction model with attribute reliability to handle unreliable monitoring information33.
Overall, existing studies have provided useful foundations for BRB-based fault diagnosis from the perspectives of classification modeling and reliability characterization. However, several limitations still remain in multi-class fault diagnosis under complex operating conditions. First, although some studies have explored classification-oriented BRB modeling and rule reduction, the problem of rule-scale growth in multi-class scenarios has not been sufficiently addressed. Second, existing reliability-related studies mainly focus on static knowledge or attribute reliability, while the sample-varying reliability of input features and its progressive utilization throughout the inference process have received less attention. Third, the reliability differences among multiple parallel submodels are rarely incorporated into the final decision stage. Therefore, a unified BRB framework that can simultaneously address rule-scale control, feature reliability utilization, and multi-submodel decision fusion is still lacking. These limitations motivate the problem formulation in the next section.
Problem description
Industrial fault diagnosis is typically formulated as a multi-class classification problem under complex operating conditions. The modeling process involves both class discrimination and the management of uncertain information. By combining rule-based representation with uncertainty reasoning, BRB provides an effective foundation for modeling such tasks. However, when applying existing BRB methods to multi-class diagnosis in complex operating conditions, several important issues remain to be addressed, as outlined below.
Problem 1:
In multi-class fault diagnosis, how can the OvR-BRB structure be used to alleviate rule combination complexity?
In multi-class fault diagnosis, the rule scale of the traditional BRB model often grows rapidly as the number of fault categories and input attributes increases. This growth further increases the burden of rule design, parameter optimization, and inference computation. This problem becomes more pronounced when multiple categories are modeled within a unified rule base.
To address this issue, the One-versus-Rest (OvR) strategy is considered in this study. The original multi-class task is decomposed into multiple binary subtasks, and a corresponding BRB submodel is constructed for each subtask. For a single BRB submodel, the number of rules can be expressed as
 |
1 |
where M denotes the number of input attributes, K denotes the number of reference values assigned to each attribute, and L denotes the number of rules.
Equation (1) shows that the rule scale increases rapidly with the number of input attributes and reference values. Therefore, in multi-class fault diagnosis, the first issue to address is how to construct a rational OvR-BRB structure that alleviates the modeling complexity caused by rule-scale growth while preserving class discrimination capability25,34.
Problem 2:
Given that feature reliability may vary across samples, how can it be effectively reflected during the BRB inference process?
Under noise disturbance and varying operating conditions, different input features within the same sample may be affected to varying degrees. Consequently, their reliability is often inconsistent and varies across samples. If such differences in feature reliability are not fully accounted for during inference, low-reliability features may still interfere with the diagnostic result, thereby undermining model stability and reliability.
To characterize sample-dependent feature reliability, the feature reliability vector of sample x is defined as
 |
2 |
where 
denotes the reliability of the m-th input feature in sample x, and M denotes the number of input attributes with m=1,2, … ,M.
Equation (2) indicates that feature reliability varies not only across samples but also among features within the same sample. However, the final output of the BRB is not determined directly by the input features; it is jointly influenced by matching correction, rule activation, and evidential reasoning (ER) fusion. Therefore, if feature reliability is not effectively integrated into these key stages, the adverse effects of low-reliability features on matching, activation, and fusion cannot be adequately suppressed.
Problem 3:
When multiple submodels produce outputs simultaneously, how can output reliability be incorporated to improve the final decision?
Following the one-versus-rest (OvR) decomposition, the original multi-class task is decomposed into multiple binary subtasks, each handled by a corresponding BRB submodel running in parallel. Since each submodel is trained to discriminate only one target class from the rest, its confidence in classifying a given sample may vary across submodels, leading to differences in the reliability of their outputs. If the final class is determined solely by comparing the raw outputs of these submodels, low-reliability outputs may still adversely affect the final decision, making it difficult to ensure the stability and reliability of multi-class diagnosis.
Let the output of the c-th submodel for sample x be 
, and let its output reliability be 
. Then, the final classification result can be expressed as:
 |
3 |
As shown in Eq. (3), the final classification result depends not only on the magnitude of each submodel output but also on the reliability associated with that output. Therefore, how to effectively incorporate submodel output reliability into the decision fusion process remains a key issue, closely related to the stability and reliability of multi-class decision-making.
In summary, multi-class fault diagnosis under complex conditions faces not only the challenge of rapidly increasing rule combination complexity, but also the insufficient utilization of feature reliability and the inadequate fusion of multiple submodel outputs. To address these issues, the next section develops a RE-OvR-BRB fault diagnosis model and presents its implementation, covering model construction, inference mechanism, decision fusion, and parameter optimization.
RE-OvR-BRB method
Based on the foregoing problem analysis and research objectives, this section presents the systematic modeling of the proposed RE-OvR-BRB method. The main contents include the overall framework, feature reliability modeling and its inference process, the decision fusion mechanism with submodel output reliability, and the parameter optimization procedure. These parts together provide a complete description of the implementation path of the proposed method.
As shown in Fig. 1, a multi-class fault diagnosis framework based on RE-OvR-BRB is proposed in this study. The framework first performs data normalization, reference grade generation, and class-relevant feature selection. The original multi-class problem is then decomposed into multiple one-versus-rest BRB sub-tasks. After that, each submodel outputs the class support, which is normalized as 
, together with the model reliability 
. In the decision layer, the normalized support is weighted by the reliability factor to obtain the integrated decision score 
. The final classification result is then determined accordingly.
Fig. 1.
Overall framework of the proposed RE-OvR-BRB model.
As shown in Fig. 2, each RE-OvR-BRB submodel incorporates dynamic reliability calculation and an evidential discounting mechanism into the conventional BRB inference process. In this way, reliability information is incorporated into the matching degree calculation, rule activation, and evidence fusion, thereby mitigating the adverse effects of low-reliability evidence. Moreover, the CMA-ES algorithm is employed to optimize parameters. The final outputs include the class support 
and the model reliability 
. The proposed framework integrates internal reliability modeling within each submodel and reliability-aware fusion across submodels into a unified architecture, enabling stable and interpretable multi-class fault diagnosis.
Fig. 2.
Inference and optimization flow of each RE-OvR-BRB submodel.
Overall framework and submodel construction
At the task modeling stage, the one-versus-rest (OvR) strategy is used to decompose the C-class fault diagnosis problem into C binary subtasks. Under this strategy, each subtask is formulated as a binary classification problem. The label of the c-th sub-task is defined as follows:
 |
4 |
Each subtask corresponds to one RE-OvR-BRB submodel. All submodels perform inference in parallel, each outputting a belief distribution along with a support degree indicating the likelihood that a sample belongs to the target class. The support degrees from all submodels are then fused with model-level reliability to produce the final classification result.
Let the training set be 
, where 
is the input feature vector and 
is the class label. For each class c, the task is transformed into a binary sample set 
. The initial construction of each submodel is then completed through feature selection, reference grade generation, and rule-based initialization.
Class-relevant feature selection
Different fault classes are usually associated with different discriminative features. In view of this, feature selection is conducted separately for each OvR sub-task rather than using the same input attributes for all submodels. For the current binary task, a small set of features with strong discriminative ability for the target class is selected from the original feature set and taken as the input attributes of the corresponding submodel. The input vector of the c-th submodel is then defined as follows:
 |
5 |
where 
denotes the k-th input feature of the c-th submodel, and K is the number of selected features. A random forest is used to evaluate the importance of each feature for the current binary task on the training samples. The top K features are then selected as the input attributes of the c-th submodel. Here, K=2.
With this treatment, each submodel can focus on the most discriminative features for separating the target class from the remaining classes. As a result, the rule scale is reduced, and the resulting rule base becomes more specific and interpretable.
Reference grade generation
The BRB model maps continuous input attributes into a semantic space. After the input attributes of each submodel are determined, reference grades need to be generated. These reference grades provide semantic anchors for continuous features and serve as the basis for attribute matching and rule antecedent construction. For the m-th input attribute of the c-th submodel, the reference grade set is defined as:
 |
6 |
where 
denotes the reference grade set of the current input attribute, and R is the number of reference grades. In this study, five reference grades are generated using a quantile-based method, with typical values corresponding to the 0.1, 0.3, 0.5, 0.7, and 0.9 quantiles.
Through this process, each continuous attribute is mapped into a discrete reference space with clear semantic meaning, which lays the foundation for matching degree calculation and rule antecedent construction.
Rule base structure and parameter representation
After the class-relevant input attributes and their reference grades are determined, the corresponding belief rule base is established for each OvR sub-task. This rule base is used to characterize the relationship between input attribute combinations and output grades in the current binary classification problem. The c-th RE-OvR-BRB submodel contains L belief rules. The k-th rule is defined in the following IF-THEN form:
 |
7 |
where 
denotes the antecedent reference grade of the m-th input attribute in the k-th rule. 
denotes the set of output grades. 
denotes the belief degree assigned to grade 
in the consequent of the k-th rule. 
and 
denote the rule weight and attribute weight, respectively. In this study, five output grades are used, and thus N=5.
Since the conclusion of a rule may not be completely distributed over all output grades, the sum of the belief degrees does not necessarily equal 1. It satisfies
 |
8 |
Equation (8) shows that the total belief degree of the rule consequent does not exceed 1. This property allows the rule base to represent incomplete information. On this basis, the consequent belief degrees, rule weights, and attribute weights together form the parameter set to be optimized for the c-th submodel, which is written as
 |
9 |
where 
denotes the parameter set of the c-th submodel. This representation provides a unified basis for the subsequent model training and parameter optimization.
Model inference
After model construction, the inference process describes how attribute-level reliability is incorporated into rule matching and final decision-making. For a test sample 
, each submodel first extracts the corresponding input vector 
and computes the matching degree between the input attributes and reference grades. The matching degrees are then corrected by attribute-level reliability, and the attribute weights are simultaneously reallocated to reflect credibility differences across attributes. Based on the revised matching information, rule activation and ER evidence fusion are performed to obtain the output belief distribution and the support degree of each submodel. Finally, the support degrees of all submodels are combined according to submodel reliability to produce the final classification result. For notational brevity, the submodel index c and sample index i are omitted in the following subsections where the context is clear.
Matching degree calculation
In the inference process, each continuous input attribute is first mapped to the predefined reference grade space. This step is implemented by calculating its matching degree with respect to each reference grade. When the value of an input attribute lies between two adjacent reference grades, the matching degree is determined as follows:
 |
10 |
where 
denotes the m-th input attribute of the current submodel, 
denotes the matching degree of 
with respect to the j-th reference grade, and 
and 
denote the two reference grades adjacent to 
.
As indicated by Eq. (10), an input attribute value falling between two adjacent reference grades yields non-zero matching degrees for those two grades only, and the two values sum to one. This design achieves a smooth mapping from the continuous attribute space to the discrete semantic space, and provides a computational foundation for subsequent rule activation.
Dynamic attribute reliability
Matching degree alone can characterize the proximity between input attributes and reference grades. However, it fails to reflect the differences in sample-specific input feature representativeness across different samples. Even when two samples are numerically well-matched to a reference grade, the typicality and reliability of their input attributes for the corresponding class may still differ. To address this limitation, attribute-based reliability is further introduced to quantify the degree to which the current input attributes support the classification of the sample into each class. Unlike a fixed constant, this reliability varies with the input sample, and is therefore referred to as dynamic attribute reliability. For class c and submodel m, the dynamic attribute reliability of the m-th input attribute is defined as:
 |
11 |
where 
and 
denote the mean and standard deviation of class c on attribute m, respectively, both estimated from the training samples of that class. ψ is a sensitivity factor that controls the rate at which reliability decays as the attribute distribution deviation increases, and \varepsilonis a small positive constant introduced to prevent numerical instability caused by near-zero denominators.
As indicated by Eq. (11), dynamic attribute reliability is computed on a per-sample basis, and its value is jointly determined by the deviation between the input attribute and the typical distribution of the current class. When 
approaches class c in the statistical distribution of attribute m, 
takes a higher value. Conversely, when the attribute deviates substantially from the class-typical distribution, the reliability decreases accordingly. In other words, for the same input attribute within a given submodel, different samples are assigned different reliability values. This is also the reason why this quantity is termed dynamic attribute reliability in this study.
Unlike fixed submodel weights, the reliability yielded by Eq. (11) can be adjusted in real time as the input sample changes, making it better suited for handling complex engineering fault diagnosis scenarios with dynamic fluctuations. This design enables the model not only to determine which reference grade an input falls into, but also to further assess the reliability of that input under the current class. For notational convenience, 
is subsequently abbreviated as 
.
On this basis, the obtained 
values are incorporated into the subsequent matching degree refinement, rule activation, and evidence fusion processes, so as to reduce the adverse influence of low-reliability inputs on the inference results.
Reliability-aware matching degree correction and weight reallocation
In the original BRB, neither matching degrees nor attribute weights explicitly account for sample-specific reliability differences. Consequently, low-reliability evidence may exert an excessive influence on the inference result if these quantities are used without modification. To alleviate this issue, reliability is introduced into the inference stage to revise the original matching degrees and adjust the attribute weights. First, the original matching degree is corrected by reliability to obtain the reliability-aware matching degree:
 |
12 |
where 
denotes the original matching degree of input attribute 
with respect to the j-th reference grade, 
denotes the dynamic attribute reliability of 
, and 
denotes the reliability-refined matching degree.
On this basis, to further reflect differences in reliability levels across attributes, reliability is combined with the original attribute weights and normalized to obtain the refined attribute weights:
 |
13 |
where 
denotes the original weight of the m-th input attribute, 
denotes the refined attribute weight, and M denotes the total number of input attributes in the current submodel.
As shown by Eq. (12), the contribution of low-reliability inputs to the reference grade matching results is reduced at the local matching level. Equation (13) further suppresses the influence of low-reliability attributes during rule activation at the attribute weight allocation level. Through these two refinements, attribute reliability is enabled to act simultaneously on both matching degree computation and weight distribution, and hence enhances the overall ability of the inference process to suppress the impact of low-reliability inputs.
Rule activation weight calculation
With the matching degrees and attribute weights refined, the activation degree of each rule under the current sample needs to be characterized. Since different rules correspond to different combinations of antecedent reference grades, their matching degrees with the current input sample also differ. To this end, the comprehensive matching degree of each rule is first computed, and the normalized rule activation weight is then obtained by incorporating the rule weight. The comprehensive matching degree of the k-th rule is defined as:
 |
14 |
where 
denotes the reference grade index corresponding to the k-th rule on attribute m, 
denotes the reliability-refined matching degree of that rule on attribute m, and 
denotes the refined attribute weight.
The activation weight of the k-th rule is obtained by incorporating the rule weight 
:
 |
15 |
where 
denotes the rule weight of the k-th rule, 
denotes the normalized rule activation weight, and L denotes the total number of rules in the current submodel.
Equation (14) is used to comprehensively characterize the matching degree between the current input sample and the rule antecedents. Equation (15) additionally incorporates rule weights on this basis, yielding the normalized activation weight of each rule under the current sample. As a result, the rule activation weight simultaneously reflects both the input-matching information and the rule importance, providing a reliable basis for ER evidence fusion.
ER-based evidence fusion
After the activation weights of all rules are obtained, the conclusion information from multiple rules needs to be fused to form a unified output from the submodel. Since incomplete or conflicting information may exist among different rules, relying on a single rule is insufficient to reliably reflect the overall judgment of the current sample. To handle such conflicts, each rule activation is treated as an independent source of evidence, and the ER evidence reasoning algorithm is adopted to perform the fusion. The belief degree of output grade 
is defined as:
 |
16 |
where 
denotes the belief degree of output grade 
after fusion, 
denotes the activation weight of the k-th rule, 
denotes the belief degree assigned by the k-th rule to output grade 
, L denotes the total number of rules in the current submodel, and N denotes the total number of output grades.
The normalization factor μ in Eq. (16) is defined as:
 |
17 |
where μ is the normalization factor, ensuring that the fused results satisfy the normalization requirement of a belief distribution.
Equation (16) and Eq. (17) together achieve the unified fusion of multi-rule activation conclusion information, enabling the submodel to produce a consistent belief distribution even under incomplete or conflicting evidence, and providing a foundation for support degree computation.
Support degree computation
After the belief distribution of the submodel output is obtained, a further transformation is needed to convert it into a scalar result that is more amenable to comparison, so as to support subsequent cross-submodel decision fusion. For this reason, the belief distribution output is mapped to a support degree in this study. The support degree is defined as:
 |
18 |
where 
denotes the support degree, 
denotes the utility value of output grade 
, and 
denotes the belief degree of output grade 
after fusion.
Equation (18) performs a utility-weighted aggregation over the belief distribution, converting the distributional inference result into a unified scalar representation, thereby providing a basis for subsequent cross-submodel comparison and decision fusion.
Submodel output reliability and decision fusion
After the dynamic attribute reliability of all input attributes is obtained, the overall inferential reliability needs to be further characterized. If linear weighted averaging were applied directly for aggregation, low-reliability attributes could disproportionately suppress the reliability of other attributes. To avoid this issue and better capture the influence of the least reliable attribute—which tends to dominate uncertainty at the classification boundary—the soft-min aggregation method is adopted here to consolidate the attribute-level dynamic reliability into submodel reliability. The reliability is defined as:
 |
19 |
where 
denotes the reliability of submodel c on sample 
, 
denotes the dynamic attribute reliability of the m-th input attribute, 
denotes the refined attribute weight obtained from Eq. (13), and β>0 is the aggregation sensitivity parameter.
As shown by Eq. (19), submodel reliability is not a simple average of individual attribute reliabilities, but rather places stronger emphasis on constraining low-reliability attributes. The parameter β controls this sensitivity: when β is small, the aggregation result approaches a weighted average; when β is large, the result becomes increasingly sensitive to the minimum reliability attribute.
With submodel reliability obtained, the original support degree is refined through reliability-based weighting to yield the final weighted support degree:
 |
20 |
where 
denotes the original support degree of submodel c for sample 
, 
denotes the weighted support degree, and γ is an adjustment factor controlling the degree to which submodel reliability corrects the original support degree. When γ is small, the correction effect is moderate, preserving the original support degree to a greater extent. When γ is large, the correction becomes stronger, rendering the result more sensitive to submodel reliability.
As expressed in Eq. (20), the final decision score is influenced not only by the original support degree but also by the overall reliability of the submodel. The final classification result is determined by selecting the class with the maximum weighted support degree:
 |
21 |
The hyperparameters β and γ are determined through internal data-driven search on the training set. Experimental results show that the model achieves favorable overall performance when β =10 and γ =0.6. These values are therefore adopted as the final settings.
Model optimization
After the model structure and inference mechanism are established, the rule consequent belief degrees, rule weights, and attribute weights still need to be determined. Since these parameters directly affect model output, empirical initialization alone cannot guarantee satisfactory performance. A data-driven approach is therefore adopted here to optimize the parameters of each submodel. For submodel c, the mean squared error between the weighted support degree and the corresponding binary supervision label is used as the objective function:
 |
22 |
where 
denotes the parameter set to be optimized for submodel c, 
denotes the weighted support degree of sample 
in submodel c, 
denotes the corresponding binary supervision label, and N denotes the number of training samples.
To ensure the rationality of BRB inference and the feasibility of the parameters, the following constraints must be satisfied during optimization:
 |
23 |
Equation (23) specifies the value ranges of the rule consequent belief degrees, rule weights, and attribute weights. The sum of consequent belief degrees does not exceed 1, ensuring that the rule base is capable of representing incomplete information in the form of a belief distribution.
Given the non-linearity, non-convexity, and parameter-coupling characteristics of the BRB parameter optimization problem, a sufficient number of iterations is required for the algorithm to converge to a stable solution. The CMA-ES algorithm is therefore employed here for optimization. This algorithm requires no gradient information and exhibits strong global search capability, making it well suited for the parameter requirements of the proposed model. Upon satisfaction of the termination criterion, the optimal parameters of submodel c are obtained.
To further illustrate the effect of parameter optimization, the before-and-after comparisons of key parameters, including attribute weights, rule weights, and representative belief distributions, are provided in Appendix C.
Model procedure
To facilitate implementation of the proposed RE-OvR-BRB multi-class fault diagnosis model, its training and inference procedures are summarized as follows.
Step 1. Data preprocessing and task preparation
The training set and test set are first loaded, and the original features are normalized to reduce the influence of scale differences on subsequent feature selection, reference grade generation, and model inference. Meanwhile, the sample feature vectors and class labels are specified to provide the basis for subsequent multi-class task decomposition and submodel construction.
Step 2. OvR task decomposition
For a fault diagnosis task with C classes, the original multi-class problem is decomposed into C mutually independent binary subtasks according to Eq. (4) by using the one-versus-rest (OvR) strategy. Each subtask only focuses on discrimination between the current class and the remaining classes, and a corresponding RE-OvR-BRB submodel is constructed.
Step 3. Class-relevant feature selection
For the k-th binary subtask, class-related feature evaluation is conducted on the training samples, and a small number of input attributes with strong discriminative ability are selected according to feature importance to form the input vector of the submodel, as shown in Eq. (5). This step enables each submodel to focus on the features that are most discriminative for the current class, while reducing the subsequent rule combination complexity.
Step 4. Reference grade generation
After the input attributes of each submodel are determined, the corresponding reference grade set is generated for each input attribute according to Eq. (6). Continuous features are thereby mapped into a discrete semantic space, which provides the basis for subsequent matching degree calculation, rule antecedent construction, and activation weight calculation.
Step 5. Initialization of the submodel rule base
For each submodel, a BRB rule base is constructed according to the combination relationships of the reference grades of the input attributes, and the rule structure is defined by Eq. (7). Then, the constraint form of the consequent belief degrees is given by Eq. (8), and the parameter set to be optimized is uniformly represented by Eq. (9), including rule weights, attribute weights, and consequent belief degrees. In this way, the initial modeling of each RE-OvR-BRB submodel is completed.
Step 6. Submodel parameter optimization
During the training stage, the objective function of each submodel is constructed according to Eq. (22), where the mean squared error between the weighted support and the corresponding binary supervisory label is taken as the optimization objective. At the same time, constraints are imposed on the consequent belief degrees, rule weights, and attribute weights according to Eq. (23). Then, CMA-ES is employed to iteratively optimize the parameters and obtain the optimal parameters of each submodel.
Step 7. Submodel inference for test samples
For a test sample, the input vector under the k-th submodel is first extracted, and the original matching degrees between the input attributes and the reference grades are calculated according to Eq. (10). Then, the dynamic attribute reliability of each input attribute is calculated according to Eq. (11). Subsequently, matching degree correction and attribute weight redistribution are performed according to Eqs. (12) and (13), respectively, so that the suppression ability for low-reliability inputs can be enhanced. On this basis, the combined matching degree and activation weight of each rule are calculated according to Eqs. (14) and (15), and ER-based evidence fusion is conducted by Eqs. (16) and (17) to obtain the belief distribution of the submodel output. Finally, the belief distribution is mapped into the support degree of the target class for the submodel according to Eq. (18).
Step 8. Submodel reliability calculation and decision fusion
After the dynamic reliability of all input attributes is obtained, the attribute-level reliability is aggregated into submodel-level reliability according to Eq. (19) to characterize the overall inference reliability of the current sample under that submodel. Then, the original support degree is weighted by reliability according to Eq. (20), and the weighted support degree of each submodel is obtained. Finally, the weighted support degrees of all submodels are compared according to Eq. (21), and the class corresponding to the maximum value is taken as the final diagnosis result.
Step 9. Output of the diagnosis result
The final class label of the test sample is output. Meanwhile, the support degree, reliability, and weighted support degree of each submodel are retained to support subsequent result analysis, interpretability display, and model performance evaluation.
To facilitate reproducibility, the pseudocode of the key implementation procedures of the proposed RE-OvR-BRB method is provided in Appendix B.
Case study
This section evaluates the effectiveness of the proposed method through case studies and comparative experiments. The experiments cover diagnostic performance, fusion strategy comparison, interpretability analysis, method comparison, noise robustness, and data perturbation visualization.
Dataset description and experimental setting
To verify the effectiveness of the proposed method, experiments were conducted on the Case Western Reserve University (CWRU) bearing dataset. The dataset was collected from the Bearing Data Center test platform, with faults introduced by electrical discharge machining. It includes four health conditions, namely normal condition, inner race fault, outer race fault, and ball fault. Each fault has a defect diameter of 0.014 in. The data used in this study were collected under a load condition of 0 hp, a sampling frequency of 12 kHz, and a shaft speed of 1797 rpm, with the outer race fault located at the 6 o’clock position.
To construct diagnosis samples, the drive-end acceleration signals were divided into equal-length segments with a window length of 1024 sampling points. No overlap was introduced between adjacent segments, and each segment was treated as one sample. For each segment, 14 handcrafted time-domain and frequency-domain features with physical meaning were extracted for subsequent modeling. Since the raw signal length of the normal condition was substantially longer than that of each fault condition, the number of segmented normal samples was much larger than that of each fault class. To alleviate the influence of class imbalance on classification, random down-sampling was applied to the normal samples so that the four classes contained approximately the same number of samples.
After preprocessing, the final dataset contained 472 samples in total, with about 118 samples in each class. Among them, 377 samples were used for training and 95 samples were used for testing, corresponding to an approximately 8:2 split. To mitigate the influence of feature-scale differences on model training, all input features were standardized using z-score normalization based on the statistics of the training set, and the same transformation was subsequently applied to the test set.
For fault diagnosis, this study adopts the RE-OvR-BRB submodel mechanism proposed in Sect. 4. A random forest is used for feature selection in each submodel, and the selected features are summarized in Table 1. The reference values for each submodel are generated from the training samples using the quantile-based method described in Sect. 4.1.2, and the corresponding results are presented in Table 2. Based on these settings, the subsequent experiments are conducted to evaluate the effectiveness of the proposed method from the perspectives of fusion strategy comparison, mechanism analysis, comparative experiments, robustness, and generalization performance.
Table 1.
Feature selection results for each RE-OvR-BRB submodel.
| Submodel | Selected features |
|---|---|
| BRB 0 (Normal vs. the rest) | Standard deviation, Peak-to-peak value |
| BRB 1 (Inner Race vs. the rest) | Mean, Shape factor |
| BRB 2 (Outer Race vs. the rest) | Mean, Root mean square (RMS) |
| BRB 3 (Ball vs. the rest) | Mean, Tkurtosis |
Table 2.
Reference values of selected features for each RE-OvR-BRB submodel.
| Submodel | Selected feature | Reference values |
|---|---|---|
| BRB 0 (Normal vs. the rest) | Standard deviation | [−1.0200, −0.6714, −0.4594, 0.7542, 1.4979] |
| Peak-to-peak value | [−0.8625, −0.7045, −0.6137, 0.5871, 1.5868] | |
| BRB 1 (Inner Race vs. the rest) | Mean | [−0.9951, −0.5566, −0.2626, 0.0937, 1.7307] |
| Shape factor | [−0.7976, −0.7256, −0.6267, 0.3496, 1.6482] | |
| BRB 2 (Outer Race vs. the rest) | Mean | [−0.9951, −0.5566, −0.2626, 0.0937, 1.7307] |
| Root mean square (RMS) | [−1.0123, −0.6796, −0.4580, 0.7326, 1.4776] | |
| BRB 3 (Ball vs. the rest) | Mean | [−0.9951, −0.5566, −0.2626, 0.0937, 1.7307] |
| Tkurtosis | [−0.6868, −0.6612, −0.6261, 0.1188, 1.5168] |
In this study, class labels 0, 1, 2, and 3 represent four health states of rolling bearings, namely, Normal, Inner Race Fault, Outer Race Fault, and Ball Fault. In the reference-grade generation process, VL (Very Low), L (Low), M (Medium), H (High), and VH (Very High) denote five semantic levels obtained by quantile-based partitioning of each feature over the training samples. These semantic levels characterize the relative magnitude of feature values from “very low” to “very high”, thereby improving the semantic interpretability of the rule antecedents.
Fusion strategy comparison
To examine the effect of different model-level reliability fusion strategies on the final classification results, comparative experiments were conducted using multiple fusion methods under the same submodel outputs. The results are presented in Table 3. In this table, accuracy represents the overall classification accuracy, Macro-F1 represents the macro-averaged F1 score across all classes and is used to evaluate the overall recognition performance in the multi-class task, and Low-margin ratio represents the proportion of low-confidence decision samples. A lower Low-margin ratio indicates more stable final decisions and clearer inter-class discrimination.
Table 3.
Comparison of different model-level reliability fusion strategies.
| Fusion method | Accuracy (%) | Macro-F1 | Low-margin ratio (%) |
|---|---|---|---|
| Soft-min + mild gate | 96.84 | 0.9683 | 7.37 |
| Weighted mean | 94.74 | 0.9475 | 14.74 |
| Weighted harmonic mean | 95.79 | 0.9581 | 11.58 |
| Soft-min | 95.79 | 0.9579 | 14.74 |
| Soft-min + shift gate | 93.68 | 0.9372 | 21.05 |
| Reliability contrast | 93.68 | 0.9370 | 15.79 |
Compared with Accuracy and Macro-F1, margin-related metrics are more suitable for reflecting decision stability under close class competition. In particular, they provide additional information on whether the predicted class can maintain a clear output gap from its strongest competitor, which cannot be fully revealed by correctness-based metrics alone. Therefore, Low-margin ratio is introduced here to further evaluate whether different fusion strategies can reduce low-confidence decisions and improve the stability of multi-class diagnosis.
The methods listed in Table 3 correspond to different strategies for aggregating submodel reliability and applying gating correction to the final decision scores. Among them, Soft-min + mild gate is the fusion strategy proposed and ultimately adopted in this study. As shown, the choice of fusion strategy has a notable impact on final performance. Compared with the other methods, Soft-min + mild gate achieves the best results across all three metrics, with accuracy, Macro-F1, and low-margin ratio reaching 96.84%, 0.9683, and 7.37%, respectively. These results indicate that the proposed fusion strategy not only improves overall classification performance but also reduces the proportion of low-confidence samples, thereby enhancing the stability and reliability of the final decision. Therefore, Soft-min + mild gate is adopted as the default model-level reliability fusion strategy in subsequent experiments.
Mechanistically, the Soft-min + mild gate strategy is designed to balance reliability sensitivity and support preservation in multi-submodel decision fusion. Specifically, the soft-min aggregation places greater emphasis on low-reliability submodels, thereby preventing overly optimistic decisions caused by unreliable outputs when class competition is close. On this basis, the mild gate further introduces a moderate reliability-aware correction to the original support scores, so that misleading low-reliability support can be suppressed without excessively weakening useful evidence from relatively reliable submodels. In this way, the proposed fusion strategy not only improves the discrimination of confusing samples near decision boundaries, but also helps maintain stable decisions for high-confidence samples. Therefore, it provides a more balanced and robust decision mechanism than direct averaging or overly strong gating schemes.
Figure 3 presents the grid-search results of parameters β and γ under the Soft-min + mild gate fusion strategy. The best performance is mainly observed in the region where γ =0.6 and 
. In this region, the model achieves high accuracy and Macro-F1 while maintaining a relatively large mean margin, indicating that the proposed fusion strategy improves classification performance while enhancing inter-class discriminability. Moreover, for γ between 0.6 and 0.8 and β between 10 and 14, the overall variation of all metrics remains relatively small, suggesting that the method exhibits a degree of parameter robustness rather than relying on a single isolated optimum.
Fig. 3.
Effect of soft-min and mild-gate parameters on model performance. (a) Accuracy; (b) Macro-F1; (c) Mean Margin; (d) Low-Margin Ratio.
In contrast, when γ =0.9, accuracy and Macro-F1 drop noticeably, mean margin continues to decline, and low-margin ratio increases sharply. This suggests that an excessively strong gating mechanism may weaken useful support information, thereby reducing the decision margin of boundary samples and compromising model stability. Considering classification performance, discriminative margin, and decision stability collectively, β =10 and γ =0.6 are selected as the final parameter settings.
Reliability mechanism analysis
To further illustrate the role of the proposed reliability mechanism in multi-class decision-making, this section examines its impact from four perspectives: overall classification performance, decision confidence distribution, internal inference behavior of typical versus boundary samples, and representative case analysis. Here, typical samples and boundary samples are selected as two representative diagnosis scenarios, because they respectively correspond to high-certainty and high-uncertainty decision states, and thus are suitable for revealing how the proposed reliability-aware mechanism affects inference behavior under different levels of classification difficulty. Together, these analyses provide a comprehensive view of how the reliability-aware mechanism improves both diagnostic accuracy and decision stability.
As shown in Fig. 4, the proposed method produces clearer category boundaries at the decision level, substantially reducing class overlap in the visualized decision region. As shown in Table 4, the diagnostic performance is significantly improved after the reliability-aware mechanism is introduced. OvR-BRB-Std refers to the baseline OvR-BRB without any reliability-aware mechanism, the proposed framework with reliability modeling components removed. Compared with the OvR-BRB-Std method, the proposed RE-OvR-BRB achieves higher accuracy (97.895% vs. 93.684%), Macro-F1 (0.9787 vs. 0.9359), and mean margin (0.3738 vs. 0.2185), while reducing the low-margin ratio from 54.737% to 6.3158%. These results indicate that the proposed method not only improves classification accuracy but also enhances inter-class discriminability, leading to more stable multi-class decisions.
Fig. 4.
Comparison of normalized confusion matrices for RE-OvR-BRB and OvR-BRB-Std. (a) RE-OvR-BRB; (b) OvR-BRB-Std.
Table 4.
Overall performance comparison.
| Method | Accuracy (%) | Macro-F1 | Mean margin | Low-margin ratio (%) |
|---|---|---|---|---|
| RE-OvR-BRB | 97.895 | 0.9787 | 0.3738 | 6.3158 |
| OvR-BRB-Std | 93.684 | 0.9359 | 0.2185 | 54.7370 |
The decision margin measures the difference between the target class output and the highest competing class output, with a larger margin indicating a more confident and stable decision that is less sensitive to disturbances. Conversely, a smaller margin suggests that the sample lies near the decision boundary, where class competition is stronger and the output is more susceptible to noise or input fluctuations.
As shown in Fig. 5, the margin score distribution of RE-OvR-BRB clearly shifts toward higher values, whereas that of OvR-BRB-Std is mainly concentrated in the low-margin region. Specifically, the median margin score of OvR-BRB-Std is only 0.095, while that of RE-OvR-BRB reaches 0.394, indicating that the proposed method yields larger decision margins across a greater proportion of samples. These results suggest that the reliability enhancement mechanism not only improves classification accuracy but also enlarges the output gap between the correct class and competing classes, thereby reducing uncertainty near the decision boundary and enhancing overall discriminative ability and decision reliability.
Fig. 5.
Comparison of discriminative margin distributions of the two methods.
Four indicators are compared to investigate the internal inference behavior of typical versus boundary samples: mean dynamic reliability, top-2 belief gap, activation entropy, and belief entropy. Mean dynamic reliability quantifies the average reliability level across all input features; top-2 belief gap characterizes the separation between the two highest output support values; activation entropy measures the dispersion of the rule activation distribution; and belief entropy captures the uncertainty of the output belief distribution. Higher values of the first two and lower values of the latter two are generally desirable.
As reported in Table 5 and shown in Fig. 6, typical samples exhibit substantially higher mean dynamic reliability and top-2 belief gap, along with lower activation entropy. The belief entropy results in Table 5 are consistent with this trend. Taken together, these results indicate that typical samples are associated with stronger input reliability and more concentrated inference behavior: low-reliability inputs are suppressed, rule activation becomes more focused, and the final decision is clearer. Boundary samples, by contrast, show lower reliability, more dispersed activation, and higher output uncertainty, making them more susceptible to noise and input fluctuations. These findings confirm that dynamic reliability not only reflects the intrinsic quality of input features but also actively shapes rule activation and belief distribution throughout the inference process, thereby influencing decision stability.
Table 5.
Comparison of internal statistical indicators between the boundary-sample group and the typical-sample group.
| Group | Mean dynamic reliability | Activation entropy | Top-2 belief gap | Belief entropy |
|---|---|---|---|---|
| Boundary group | 0.3199 | 0.8392 | 0.1849 | 1.2755 |
| Typical group | 0.7062 | 0.6445 | 0.2778 | 0.8053 |
Fig. 6.
Comparison of internal statistical characteristics of the boundary-sample group and the typical-sample group. (a) mean dynamic reliability (b) top-2 belief gap (c) activation entropy.
As shown in Fig. 7, the belief distributions of RE-OvR-BRB and OvR-BRB-Std differ markedly for this representative improved sample. For OvR-BRB-Std, the belief degrees are spread across multiple grades, including VL, H, and VH, with a relatively high belief degree assigned to VL. This indicates that strong uncertainty and evident grade competition still persist in its inference result for this sample. In contrast, the belief distribution of RE-OvR-BRB becomes much more concentrated, being mainly assigned to grade M, while the belief degrees of the other grades are substantially reduced. These results demonstrate that, after introducing the reliability enhancement mechanism, dispersed and conflicting low-reliability evidence can be effectively suppressed. Consequently, the inference result shifts from multi-grade competition to a more focused dominant judgment, which improves both the clarity of the output distribution and the stability of the final decision.
Fig. 7.
Comparison of final belief distributions for a representative improved sample under OvR-BRB-Std and RE-OvR-BRB.
As shown in Fig. 8, the InnerRace submodel gives the highest raw support for this sample. Therefore, InnerRace is taken as the initial winner before reliability correction. However, clear differences can be observed in the output reliability of different submodels. The Ball submodel has higher model reliability, whereas the reliability of the InnerRace submodel is relatively lower. After model-level reliability weighting is introduced, the raw support of each submodel is further adjusted. The contributions of highly reliable submodels are strengthened, while those of less reliable submodels are suppressed. As a result, the weighted support of Ball exceeds that of InnerRace, and Ball becomes the final winner. These results indicate that the proposed method does not rely solely on raw inference outputs for classification. Instead, the reliability of submodel outputs is further incorporated to revise the final decision. In this way, low-reliability yet misleading support can be suppressed when class competition is close, thereby improving both the reliability and correctness of the final classification result.
Fig. 8.
Reliability-weighted re-ranking of submodel supports for a representative corrected sample.
In summary, the proposed reliability-aware mechanism adaptively reweights class support at the submodel decision level, thereby improving the final class ranking. As a result, stable discrimination can be maintained for high-confidence samples, while misclassifications for some confusing samples can still be effectively corrected. Therefore, both classification performance and decision stability are enhanced simultaneously.
Comparative experiments
To validate the effectiveness of the proposed method, comparative experiments were conducted between RE-OvR-BRB and several baseline methods, including OvR-BRB-Std, EBRB (Extended Belief Rule Base), IBRB (Interval Belief Rule Base), SVM (Support Vector Machine), RF (Random Forest), MLP (Multi-Layer Perceptron), XGBoost (Extreme Gradient Boosting), KNN (K-Nearest Neighbors), LR (Logistic Regression), and DT (Decision Tree). Accuracy, Macro-Recall, and Macro-F1 are used as evaluation metrics. Accuracy measures overall classification correctness; Macro-Recall evaluates the average per-class recall and reflects the model’s ability to correctly identify each fault type; Macro-F1 provides a balanced measure of precision and recall across all classes. To reduce the effect of data partitioning on the results, the dataset was randomly split into training and test sets at a ratio of 8:2 in each run, and all compared methods were trained and evaluated under the same split. The mean and standard deviation over 20 independent runs are reported.The experimental results for all compared methods on these three metrics are summarized in Table 6.
Table 6.
Comparison of different methods on the test set over 20 independent runs.
| Method | Accuracy (%) | Macro-Recall | Macro-F1 |
|---|---|---|---|
| RE-OvR-BRB | 96.6842 ± 1.0851 | 0.9672 ± 0.0206 | 0.9669 ± 0.0209 |
| OvR-BRB-Std | 93.6834 ± 3.3663 | 0.9375 ± 0.0333 | 0.9359 ± 0.0353 |
| EBRB | 93.2105 ± 1.9053 | 0.9325 ± 0.0111 | 0.9322 ± 0.0109 |
| IBRB | 90.5263 ± 2.2228 | 0.9063 ± 0.0530 | 0.9029 ± 0.0646 |
| SVM | 91.5789 ± 1.0531 | 0.9167 ± 0.0400 | 0.9145 ± 0.0256 |
| RF | 92.8947 ± 1.9705 | 0.9297 ± 0.0176 | 0.9267 ± 0.0182 |
| MLP | 89.2632 ± 1.6422 | 0.7156 ± 0.1153 | 0.6573 ± 0.1460 |
| XGBoost | 91.1527 ± 1.3268 | 0.9123 ± 0.0215 | 0.9108 ± 0.0227 |
| KNN | 90.8634 ± 1.8752 | 0.9079 ± 0.0286 | 0.9061 ± 0.0293 |
| LR | 89.7651 ± 1.5439 | 0.8982 ± 0.0317 | 0.8957 ± 0.0324 |
| DT | 89.2478 ± 2.1673 | 0.8915 ± 0.0429 | 0.8893 ± 0.0436 |
From the repeated experimental results, it can also be observed that the proposed method achieves the best average performance while maintaining relatively small fluctuations, indicating good stability. Overall, the proposed method shows clear advantages in both classification performance and experimental stability.
Perturbation experiments
To evaluate the robustness of the proposed method under noisy perturbations, comparative experiments were conducted on RE-OvR-BRB, OvR-BRB-Std, EBRB, and IBRB across different noise levels. In these experiments, noise was added only to the test set, while the training set was kept unchanged. The noise levels were set to 0.03, 0.05, 0.08, 0.10, 0.12, 0.15, 0.18, and 0.20. Additive Gaussian noise was adopted, and its intensity was scaled by the standard deviation of each feature in the training set. Specifically, the perturbation on the j-th feature satisfies 
, where η denotes the noise level and 
denotes the standard deviation of the corresponding feature in the training set.
Figure 9 presents the robustness comparison among the BRB-based methods, including RE-OvR-BRB, OvR-BRB-Std, EBRB, and IBRB. As the noise level increases, the performance of all methods declines to varying extents. However, the proposed RE-OvR-BRB consistently maintains a relatively high performance level across the entire noise range and exhibits better stability than the other BRB-based methods. These results indicate that the introduced reliability-aware mechanism can effectively suppress the adverse effects of low-reliability evidence under noisy conditions, thereby improving diagnostic robustness. In addition, Table 7 further summarizes the robustness results for AUNC and Retention, showing that RE-OvR-BRB achieves the best values across all four metrics among the BRB-based methods.
Fig. 9.
Accuracy and macro-F1 of different methods under varying noise levels over 10 independent runs. (a) Robustness of accuracy(%) (b) Robustness of Macro-F1.
Table 7.
Comparison of AUNC and Retention robustness metrics for different methods.
| Method | AUNC-Acc | Ret.-Acc (%) | AUNC-MF1 | Ret.-MF1 (%) |
|---|---|---|---|---|
| RE-OvR-BRB | 0.9202 | 86.8602 | 0.9194 | 84.3340 |
| OvR-BRB-Std | 0.9135 | 80.3488 | 0.9049 | 80.2445 |
| EBRB | 0.8895 | 74.9412 | 0.8924 | 75.4821 |
| IBRB | 0.8851 | 79.0698 | 0.8816 | 78.1480 |
To provide a more comprehensive robustness comparison, the traditional machine-learning baselines introduced in Sect. 5.4, including SVM, RF, MLP, XGBoost, KNN, LR, and DT, were further incorporated into the perturbation analysis. The corresponding results are shown in Fig. 10. It can be observed that, although some machine-learning models maintain competitive performance under low-noise conditions, their performance degrades more rapidly as the noise level increases. By contrast, RE-OvR-BRB still preserves a clearer performance advantage over the entire noise range in both Accuracy and Macro-F1. This phenomenon suggests that the proposed method is not only effective in noise-free classification but also more robust as perturbation intensity increases.
Fig. 10.
Accuracy and Macro-F1 of RE-OvR-BRB and traditional machine-learning models under varying noise levels over repeated runs. (a) Robustness of accuracy(%) (b) Robustness of Macro-F1.
Among the traditional machine-learning baselines, RF and XGBoost show relatively stronger robustness in the low- and medium-noise regions, whereas KNN, LR, and DT are more sensitive to perturbations and exhibit more pronounced performance degradation. Overall, the results in Figs. 9 and 10 jointly demonstrate that the proposed RE-OvR-BRB achieves favorable robustness not only against other BRB-based methods, but also against representative traditional machine-learning models. Therefore, the perturbation experiments further verify that the proposed reliability-aware mechanism can effectively enhance both robustness and decision stability in noisy environments.
To evaluate the stability of different methods under noise perturbations, two robustness metrics—AUNC and Retention—were adopted. AUNC denotes the normalized area under the performance curve across the entire noise range, reflecting overall performance stability; Retention measures the proportion of performance preserved under high-noise conditions relative to the noise-free case, reflecting robustness under severe perturbations. As reported in Table 7, RE-OvR-BRB achieves the best results on all four metrics, with AUNC-Acc, AUNC-MF1, Ret.-Acc, and Ret.-MF1 reaching 0.9202, 0.9194, 86.8602%, and 84.3340%, respectively.
These results indicate that the proposed method maintains stronger overall performance throughout the noise variation process and better preserves diagnostic performance under severe noise conditions. In contrast, the other BRB-based methods show performance degradation to varying extents, further demonstrating that the proposed reliability enhancement mechanism effectively improves model robustness in complex noisy environments.In summary, the perturbation experiments verify that the proposed reliability-aware mechanism can suppress low-reliability evidence and enhance both robustness and decision stability through reliability-weighted fusion.
Generalization experiment
To further evaluate the generalization ability of RE-OvR-BRB on an external dataset, experiments were conducted on the gear fault dataset released by the University of Connecticut (UConn). This dataset focuses on gears, which are typical key components in rotating machinery, and differs clearly from the bearing dataset used in the main experiments in terms of fault object and signal distribution. Thus, it serves as a suitable testbed for examining the applicability of the proposed model in cross-component scenarios. All vibration signals were collected by a dSPACE system at a sampling frequency of 20 kHz.
The dataset contains 416 samples in total, with 104 samples for each of the four states. Class labels 0, 1, 2, and 3 correspond to the healthy state, missing-tooth fault, crack fault, and spalling fault, respectively. The dataset was divided into training and test sets at a ratio of 8:2 to evaluate the generalization performance on unseen samples.
To maintain consistency with the main experiments, the same data processing and modeling procedure was adopted. First, statistical features were extracted from the raw vibration signals as candidate input attributes. Then, under the OvR framework, random forest was used to perform feature selection for each binary subtask, with the top two most discriminative features retained for each submodel. Subsequently, five antecedent reference values were generated for each selected feature using the quantile-based method to construct the rule antecedents of each RE-OvR-BRB submodel. The feature selection results and reference values of the submodels are presented in Tables 8 and 9, respectively.
Table 8.
Feature selection results for each RE-OvR-BRB submodel.
| Submodel | Selected features |
|---|---|
| BRB 0 (Healthy vs. the rest) | Tstandard, Tkurtosis |
| BRB 1 (Missing-tooth fault vs. the rest) | Tstandard, Tkurtosis |
| BRB 2 (Crack fault vs. the rest) | Tstandard, Tskewness |
| BRB 3 (Spalling fault vs. the rest) | Tstandard, Tkurtosis |
Table 9.
Reference values of the selected features for each RE-OvR-BRB submodel.
| Submodel | Selected feature | Reference values |
|---|---|---|
| BRB 0 (Healthy vs. the rest) | Tstandard | [−0.9855, −0.7700, −0.3314, 0.2094, 1.6289] |
| Tkurtosis | [−1.1794, −0.4731, 0.0248, 0.5139, 1.1708] | |
| BRB 1 (Missing-tooth fault vs. the rest) | Tstandard | [−0.9855, −0.7700, −0.3314, 0.2094, 1.6289] |
| Tkurtosis | [−1.1794, −0.4731, 0.0248, 0.5139, 1.1708] | |
| BRB 2 (Crack fault vs. the rest) | Tstandard | [−0.9855, −0.7700, −0.3314, 0.2094, 1.6289] |
| Tskewness | [−1.2111, −0.4439, 0.0793, 0.4552, 1.2402] | |
| BRB 3 (Spalling fault vs. the rest) | Tstandard | [−0.9855, −0.7700, −0.3314, 0.2094, 1.6289] |
| Tkurtosis | [−1.1794, −0.4731, 0.0248, 0.5139, 1.1708] |
As shown in Table 8, three of four submodels share the same feature combination (Tstandard and Tkurtosis), which is consistent with the physical characteristics of gear faults—missing tooth, crack, and spalling all notably affect vibration amplitude dispersion and peak behavior. The exception is the Crack fault submodel, which selects Tskewness instead of Tkurtosis, confirming that the feature selection mechanism remains sensitive to inter-class differences. Moreover, since each submodel is trained on a distinct binary label partition, the optimized parameters and belief distributions are still class-specific despite the feature overlap.
On this basis, the proposed method was further trained and tested by following the same modeling and inference procedure as that used in the main experiments, and comparative experiments were conducted against OvR-BRB-Std, EBRB, IBRB, and the same conventional machine-learning baselines used in the main experiments, including RF, SVM, MLP, XGBoost, KNN, LR, and DT. The same grid search procedure for the soft-min and mild-gate parameters was also conducted on the gear dataset, yielding β = 14 and γ = 0.70 as the optimal settings, which differ slightly from those of the bearing dataset due to differences in signal characteristics and fault distributions. To comprehensively evaluate the performance of each method on unseen data, accuracy, Macro-Recall, and Macro-F1 were adopted as the evaluation metrics, and the corresponding results are reported in Table 11.
Table A1.
Main symbols used in the manuscript.
| Symbol | Description |
|---|---|
| C | Number of fault classes in the multi-class diagnosis task |
| x | Input feature vector of a sample |
| y | Class label of a sample |
| c | Index of the current OvR subtask/submodel |
| M | Number of selected input attributes in a submodel |
|
The m-th input attribute |
| j | Number of reference grades for an input attribute |
| L | Number of rules in a BRB submodel |
| N | Number of output grades |
|
Rule weight of the k-th rule |
|
Original weight of the m-th input attribute |
|
Reliability-refined weight of the m-th input attribute |
|
Belief degree assigned to the n-th output grade in the k-th rule |
|
Matching degree of attribute  to the j-th reference grade |
|
Dynamic reliability of the m-th input attribute |
|
Reliability-refined matching degree |
|
Activation weight of the k-th rule |
|
Fused belief degree of the n-th output grade |
|
Original support degree of submodel c |
|
Reliability-weighted support degree of submodel c |
|
Output reliability of submodel c |
| β | Aggregation sensitivity parameter in the soft-min strategy |
| γ | Adjustment factor in the mild-gate correction |
|
Parameter set to be optimized for submodel c |
Table 10.
Comparison of different methods on the gear fault dataset over 20 independent runs.
| Method | Accuracy (%) | Macro-Recall | Macro-F1 |
|---|---|---|---|
| RE-OvR-BRB | 97.6786 ± 1.0172 | 0.9768 ± 0.0206 | 0.9768 ± 0.0204 |
| OvR-BRB-Std | 91.0712 ± 3.1271 | 0.9107 ± 0.0417 | 0.9102 ± 0.0422 |
| EBRB | 92.3214 ± 1.3084 | 0.9232 ± 0.0131 | 0.9226 ± 0.0129 |
| IBRB | 82.0238 ± 3.4954 | 0.8202 ± 0.0350 | 0.8187 ± 0.0352 |
| RF | 90.8947 ± 1.1633 | 0.9097 ± 0.0577 | 0.9067 ± 0.0872 |
| SVM | 91.9048 ± 0.9771 | 0.9190 ± 0.0098 | 0.9189 ± 0.0100 |
| MLP | 88.5714 ± 1.1659 | 0.8467 ± 0.2117 | 0.7589 ± 0.2455 |
| XGBoost | 90.2418 ± 1.4256 | 0.9031 ± 0.0243 | 0.9017 ± 0.0251 |
| KNN | 89.6735 ± 1.9874 | 0.8958 ± 0.0306 | 0.8942 ± 0.0319 |
| LR | 87.9362 ± 1.6731 | 0.8786 ± 0.0352 | 0.8769 ± 0.0367 |
| DT | 85.1479 ± 2.3146 | 0.8503 ± 0.0478 | 0.8481 ± 0.0492 |
Conclusion
This study addresses three key challenges in multi-class fault diagnosis under complex operating conditions: rule combination complexity, feature reliability utilization, and multi-submodel decision fusion. To tackle these issues, the RE-OvR-BRB fault diagnosis model is proposed, with three main contributions. First, the OvR decomposition strategy is adopted to partition the multi-class task into multiple binary subtasks, effectively alleviating the rule-scale growth problem in unified multi-class modeling. Second, a sample-varying feature reliability modeling method is designed and integrated into the BRB inference process, enabling reliability differences to be reflected in matching degree correction, rule activation, and evidence fusion. Third, a submodel output reliability-aware decision fusion mechanism is constructed, which jointly leverages the support degrees and reliability levels of all submodels to enhance the stability of multi-class decisions. Experimental results on both the CWRU bearing dataset and the UConn gear dataset demonstrate that the proposed method achieves competitive performance in classification accuracy, class discrimination, decision stability, and noise robustness, confirming its effectiveness under complex industrial conditions.
Nevertheless, the current study has certain limitations. First, how dynamic reliability is progressively transmitted through matching correction, rule activation, evidence fusion, and decision fusion has not been systematically explained. Second, the feature reliability estimation relies on training-set statistics, which may degrade under significant operating condition shifts. Third, as the number of fault classes increases, the number of parallel submodels grows linearly, potentially raising training cost and decision ambiguity. Future work will explore rule activation visualization, reliability transmission analysis, adaptive reliability recalibration, and more scalable decomposition strategies, with the aim of improving the transparency, robustness, and applicability of the proposed model.
Appendix A. List of main symbols and abbreviations
The following tables summarize the main symbols and abbreviations used in the manuscript. These tables are arranged in a three-line style for direct insertion into the revised manuscript.
Table A2.
Main abbreviations used in the manuscript.
| Abbreviation | Full name |
|---|---|
| BRB | Belief Rule Base |
| OvR | One-versus-Rest |
| RE-OvR-BRB | Reliability-Enhanced One-versus-Rest Belief Rule Base |
| OvR-BRB-Std | Baseline OvR-BRB without reliability-aware mechanisms |
| ER | Evidential Reasoning |
| RF | Random Forest |
| SVM | Support Vector Machine |
| MLP | Multi-Layer Perceptron |
| LR | Logistic Regression |
| DT | Decision Tree |
| KNN | K-Nearest Neighbors |
| XGBoost | Extreme Gradient Boosting |
| EBRB | Extended Belief Rule Base |
| IBRB | Interval Belief Rule Base |
| Macro-F1 | Macro-averaged F1 score |
| Mean Margin | Mean decision margin |
| Low-Margin Ratio | Ratio of low-confidence decision samples |
| AUNC | Area Under the Noise Curve |
| Retention | Performance retention under high-noise conditions |
| CWRU | Case Western Reserve University dataset |
| UConn | University of Connecticut gear fault dataset |
| RMS | Root Mean Square |
| VL / L / M / H / VH | Very Low / Low / Medium / High / Very High |
Appendix B. pseudocode of the proposed RE-OvR-BRB method
This appendix provides the pseudocode of the proposed RE-OvR-BRB method, including submodel training, reliability-aware inference, and model-level decision fusion. The purpose is to clarify the implementation process and improve reproducibility.
Algorithm B1.
Training procedure of RE-OvR-BRB.
Algorithm B2.
Reliability-aware inference and decision fusion.
Appendix C. Parameter comparisons before and after optimization
This appendix presents the key parameter comparisons before and after optimization for the proposed RE-OvR-BRB. Representative tables illustrate the changes in attribute weights, rule weights, and belief distributions.
Table C1.
Before-and-after comparison of attribute weights for all submodels.
| Class | No. | Feature | Initial 
|
Optimized 
|
Change 
|
|---|---|---|---|---|---|
| 0 | 1 | Standard deviation | 1.0 | 1.2622 | 0.2622 |
| 0 | 2 | Peak to peak value | 1.0 | 0.8926 | -0.1074 |
| 1 | 1 | Mean | 1.0 | 2.1882 | 1.1882 |
| 1 | 2 | Shape factor | 1.0 | 0.9715 | -0.0285 |
| 2 | 1 | Mean | 1.0 | 1.2923 | 0.2923 |
| 2 | 2 | Root mean square (RMS) | 1.0 | 1.5932 | 0.5932 |
| 3 | 1 | Mean | 1.0 | 0.7675 | -0.2325 |
| 3 | 2 | Tkurtosis | 1.0 | 1.8958 | 0.8958 |
Table C2.
Before-and-after comparison of rule weights and consequent belief distributions in the representative submodel.
| No. | Antecedent | Initial 
|
Optimized 
|
Initial belief distribution | Optimized belief distribution |
|---|---|---|---|---|---|
| 1 | VL∧VL | 0.8 | 1.3728 | {0.80, 0.20, 0.00, 0.00, 0.00} | {0.03, 0.03, 0.61, 0.30, 0.03} |
| 2 | VL∧L | 0.8 | 1.0443 | {0.00, 0.80, 0.20, 0.00, 0.00} | {0.00, 0.34, 0.60, 0.04, 0.02} |
| 3 | VL∧M | 0.8 | 0.118 | {0.00, 0.00, 0.80, 0.20, 0.00} | {0.13, 0.42, 0.00, 0.42, 0.03} |
| 4 | VL∧H | 0.8 | 1.1397 | {0.00, 0.00, 0.00, 0.80, 0.20} | {0.58, 0.17, 0.00, 0.26, 0.00} |
| 5 | VL∧VH | 0.8 | 0.9745 | {0.20, 0.00, 0.00, 0.00, 0.80} | {0.55, 0.04, 0.17, 0.04, 0.20} |
| 6 | L∧VL | 0.8 | 1.3005 | {0.80, 0.20, 0.00, 0.00, 0.00} | {0.74, 0.00, 0.00, 0.00, 0.25} |
| 7 | L∧L | 0.8 | 0.6295 | {0.00, 0.80, 0.20, 0.00, 0.00} | {0.06, 0.00, 0.18, 0.07, 0.68} |
| 8 | L∧M | 0.8 | 0.4267 | {0.00, 0.00, 0.80, 0.20, 0.00} | {0.08, 0.08, 0.59, 0.08, 0.17} |
| 9 | L∧H | 0.8 | 0.364 | {0.00, 0.00, 0.00, 0.80, 0.20} | {0.00, 0.00, 0.00, 0.37, 0.63} |
| 10 | L∧VH | 0.8 | 0.6742 | {0.20, 0.00, 0.00, 0.00, 0.80} | {0.08, 0.08, 0.08, 0.21, 0.55} |
| 11 | M∧VL | 0.8 | 1.2511 | {0.80, 0.20, 0.00, 0.00, 0.00} | {0.71, 0.00, 0.00, 0.29, 0.00} |
| 12 | M∧L | 0.8 | 1.543 | {0.00, 0.80, 0.20, 0.00, 0.00} | {0.20, 0.00, 0.44, 0.36, 0.00} |
| 13 | M∧M | 0.8 | 0.8408 | {0.00, 0.00, 0.80, 0.20, 0.00} | {0.00, 0.00, 0.25, 0.00, 0.75} |
| 14 | M∧H | 0.8 | 0.7037 | {0.00, 0.00, 0.00, 0.80, 0.20} | {0.00, 0.52, 0.00, 0.31, 0.18} |
| 15 | M∧VH | 0.8 | 1.1678 | {0.20, 0.00, 0.00, 0.00, 0.80} | {0.38, 0.00, 0.00, 0.13, 0.49} |
| 16 | H∧VL | 0.8 | 0.1506 | {0.80, 0.20, 0.00, 0.00, 0.00} | {0.10, 0.43, 0.02, 0.02, 0.44} |
| 17 | H∧L | 0.8 | 1.7327 | {0.00, 0.80, 0.20, 0.00, 0.00} | {0.01, 0.41, 0.52, 0.01, 0.05} |
| 18 | H∧M | 0.8 | 0.694 | {0.00, 0.00, 0.80, 0.20, 0.00} | {0.00, 0.00, 0.00, 0.73, 0.27} |
| 19 | H∧H | 0.8 | 0.6559 | {0.00, 0.00, 0.00, 0.80, 0.20} | {0.24, 0.24, 0.34, 0.00, 0.19} |
| 20 | H∧VH | 0.8 | 0.6272 | {0.20, 0.00, 0.00, 0.00, 0.80} | {0.00, 0.64, 0.22, 0.00, 0.14} |
| 21 | VH∧VL | 0.8 | 0.6696 | {0.80, 0.20, 0.00, 0.00, 0.00} | {0.26, 0.00, 0.42, 0.33, 0.00} |
| 22 | VH∧L | 0.8 | 0.1555 | {0.00, 0.80, 0.20, 0.00, 0.00} | {0.05, 0.44, 0.22, 0.05, 0.23} |
| 23 | VH∧M | 0.8 | 0.2341 | {0.00, 0.00, 0.80, 0.20, 0.00} | {0.14, 0.57, 0.16, 0.11, 0.02} |
| 24 | VH∧H | 0.8 | 0.1189 | {0.00, 0.00, 0.00, 0.80, 0.20} | {0.15, 0.33, 0.16, 0.21, 0.15} |
| 25 | VH∧VH | 0.8 | 0.9567 | {0.20, 0.00, 0.00, 0.00, 0.80} | {0.57, 0.01, 0.33, 0.09, 0.01} |
Note: The representative submodel corresponds to the submodel selected for the main mechanism analysis.
Author contributions
B.L. conceived the study, developed the methodology, implemented the software, and wrote the main manuscript text. B.L. and N.L. performed the validation and curated the data. B.L. and J.W. conducted the formal analysis. B.L. and Z.X. carried out the investigation. B.L. prepared all figures. S.Z. and Y.L. reviewed and edited the manuscript. Y.L. supervised the project and acquired the funding. All authors reviewed the manuscript.
Funding
This research was funded by the Basic Research Fund for Provincial Undergraduate Universities of Heilongjiang Province (Grant No. 2025-KYYWF-ZR0118), the Key Project of Higher Education Teaching Reform Research of Heilongjiang Province (Grant No. SJGZB2024160), and the Key Projects of Higher Education Teaching Reform Research of Harbin Normal University (Grant Nos. XJGYJSZ202401 and XJGZ202514).
Data availability
The datasets used in this study are publicly available. The primary experimental bearing dataset can be accessed through the Case Western Reserve University Bearing Data Center at: https://engineering.case.edu/bearingdatacenter/download-data-file. The gear fault dataset used for the generalization experiment is available on Figshare at: https://figshare.com/articles/Gear_Fault_Data/6127874/1.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Boyu Liu and Ning Li contributed equally to this work.
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Associated Data
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Data Availability Statement
The datasets used in this study are publicly available. The primary experimental bearing dataset can be accessed through the Case Western Reserve University Bearing Data Center at: https://engineering.case.edu/bearingdatacenter/download-data-file. The gear fault dataset used for the generalization experiment is available on Figshare at: https://figshare.com/articles/Gear_Fault_Data/6127874/1.