Formally, let D norm ≔ { ( x i , y i ) } i = 1 N (1) denote a dataset comprising sensor readings x i ∈ X ⊂ R n and their corresponding ground-truth labels y i ∈ Y = { 1 , … , c } ⊂ N , which indicate whether the system is in a normal, minor fault, severe fault, or another state. This dataset D norm is typically used to train a parameterized classification model C θ ¹ by solving the following optimization problem: min θ ∈ Θ ∑ i = 1 N ℓ y i , y ̂ i est s . t .     y ̂ i est = C θ ( x i ) . (2)

Here, ℓ ( ⋅ ) denotes a loss function, such as the squared error or cross-entropy. Let θ norm ⋆ be the solution of problem Equation 2. Note that θ norm ⋆ will vary depending on the dataset used. Therefore, the training process can be viewed as a mapping from a dataset family F to the optimal parameter θ norm ⋆ , denoted by T : F → Θ .

However, in practical deployments, the fault labels y i are often corrupted due to the following reasons:

• Ambiguity in defining fault boundaries (e.g., gradual degradation processes).

As a result, the dataset may contain noisy labels, i.e., y ̃ i = y i + δ y i ≠ y i with non-negligible probability. That is, the available dataset corresponds to a noisy-labeled version, defined as D noisy ≔ { ( x i , y ̃ i ) } i = 1 N . (3)

Using D noisy for training yields a different model parameter, given by θ ̃ noisy * = T ( D noisy ) , (4) which may result in poor predictive accuracy and weak generalization due to overfitting the noisy labels. Consequently, the resulting classification model is unreliable in safety-critical applications such as system fault detection. The above-described problem and issue are summarized in an intuitive way presented in Figure 1. To address this challenge, it is necessary to propose a robust classification framework that aims to learn accurate decision boundaries despite the presence of label noise.

As shown in Figure 2, instead of directly using the noisy-labeled dataset D noisy for classifier training, this paper introduce a dataset rectification process to filter or clean the data prior to training. This rectification is defined as a mapping R : F → F , which outputs a rectified dataset D rect consisting of estimated clean data points. Let N ̂ rect denote the number of samples in D rect . Importantly, the proposed robust fault diagnosis framework aims not only to optimize the parameter vector θ but also to design the rectification algorithm R , thereby enhancing the robustness of the diagnostic model. The classification (or regression) problem incorporating the rectification algorithm R ( ⋅ ) is formulated as follows: min θ ∈ Θ ∑ i = 1 N ̂ rect ℓ y i , y ̂ i est s . t . y ̂ i est = C θ ( x i ) , x i , y i ∈ D rect , ∀ i = 1 , … , N ̂ rect , D rect = R D noisy . (5)

The following subsections provide detailed explanations of the key components of our proposed framework:

• The construction of the rectification algorithm R using a kernel-based approach, along with a theoretical justification of how this rectification improves the robustness of fault diagnosis;

We summarize the method in the way of giving the algorithm in this subsection. To mitigate the adverse impact of label noise and enhance the reliability of fault diagnosis, we propose a robust classification framework that incorporates a data rectification step prior to model training. The core idea is to leverage kernel density estimation (KDE) (Botev et al., 2010) to identify and discard samples likely to be mislabeled, based on the observation that true labeled data tends to concentrate in high-density regions of the feature space. The procedure of Robust Fault Detection Algorithm is summarized in Algorithm 1.

Algorithm 1

Robust Fault Detection Algorithm.

Require: Noisy labeled dataset D noisy = { ( x i , y ̃ i ) } i = 1 N ;   desired discarding ratio δ ∈ ( 0,1 ) ; kernel function    K e r ( ⋅ ) ; bandwidth h .

To evaluate the effectiveness of the proposed robust battery fault detection algorithm, we compare its performance with several baseline and benchmark methods. In particular, we systematically examine how different levels of kernel-based data cleaning affect the performance of two representative classifiers: Support Vector Machine (SVM) and Extreme Learning Machine (ELM).

The following benchmark algorithms are considered:

• ELM-D: Extreme Learning Machine (ELM) classifier trained directly on the noisy dataset D noisy without data cleaning.

This experimental design enables a comprehensive analysis of the robustness and accuracy gains provided by the proposed data rectification framework across different classification models and varying levels of data cleaning. By comparing the Direct and Clean variants of both ELM and SVM, we can clearly assess the practical benefits of incorporating the rectification step into the battery fault diagnosis pipeline.

This section provides an overview of the performance metric used to evaluate the effectiveness of the proposed model. The selected evaluation metric is the classification accuracy, defined as: α ≔ ∑ k = 1 c N acc , k / ∑ k = 1 c N all , k , (18) where N acc , k denotes the number of correctly classified test samples in class k , and N all , k denotes the total number of test samples in class k .

Figures 3, 4 present the classification accuracy results of ELM and SVM models trained either directly on the noisy dataset or on the rectified dataset obtained using different discarding ratios δ . As shown in Figures 3, 4, both ELM and SVM classifiers exhibit significantly degraded accuracy when trained directly on the noisy dataset, highlighting the detrimental impact of label noise on model performance. In contrast, the proposed kernel-based rectification method substantially improves classification accuracy across both models and under various noise scenarios, demonstrating its effectiveness in mitigating the influence of noisy labels. An important observation is that the choice of discarding ratio δ plays a critical role in achieving optimal performance. In particular, when δ is set close to the true underlying noise rate (e.g., 15 % or 45 % in our experiments), the rectification process is able to remove a majority of mislabeled samples while preserving the informative structure of the clean data, thereby leading to superior classification results. It is important to note that in practical applications, the true noise rate is typically unknown. Therefore, developing adaptive strategies to optimize δ without requiring prior knowledge of the noise level represents an important direction for future research on robust battery fault detection.

In addition to the battery fault detection experiments, the proposed kernel-based rectification method was validated on transformer winding fault detection tasks, with four representative fault types: axial displacement (AD), local buckling (LB), inter-disc short circuit (IDSC), and inter-turn short circuit (ITSC). Figure 5 provides visual evidence of the discriminative oscillating wave signatures used in our fault diagnosis framework. The high-voltage oscillating wave test (OWT) captures these transient responses by applying a damped AC voltage pulse to the transformer winding and recording the resulting oscillation decay profile. These physically interpretable patterns form the basis of the feature vectors processed by our kernel-based rectification framework. The signal preprocessing pipeline, including noise suppression via wavelet thresholding and feature extraction through resonance frequency analysis, follows the methodology established in Wu et al. (2025). Consistent with the battery case study, setting the discarding ratio δ slightly larger than the true label noise rate led to robust performance improvements, which is practically feasible because conservative estimates of labeling quality are usually available. Here, a severe label noise scenario with a 45% noise rate was evaluated to rigorously test the method. As shown in Figure 6, both ELM and SVM classifiers without rectification (ELM-D, SVM-D) suffered major accuracy drops across all fault types under this high noise condition. In contrast, applying the kernel-based rectification with δ = 50 % (ELM-C-50%, SVM-C-50%) recovered high classification accuracy, exceeding 80% in all cases. These results confirm that the rectification approach effectively filters out noisy samples while preserving the core structure of each class, maintaining reliable classification performance consistent with the battery case study. This demonstrates the framework’s applicability across electrochemical energy storage systems in severe label noise scenarios.

In terms of computational cost, the proposed kernel-based rectification framework was implemented in MATLAB on a standard laptop (Intel Core i7 processor), where the average runtime of the KDE-based data cleaning step was measured at approximately 15 ms per dataset partition containing 1,000 samples. Although we have not yet ported the algorithm to an embedded hardware platform, this processing time is well within the capabilities of modern embedded processors, especially considering that fault diagnosis generally operates on time scales of seconds to minutes. This supports our description of the method as computationally lightweight, while acknowledging that future work will further validate its runtime characteristics in actual embedded environments.