KICA-DPCA-Based Fault Detection of High-Speed Train Traction Motor Bearings

Abstract

The signals of high-speed train traction motor bearings contain strong noise and exhibit non-linear and non-Gaussian characteristics. To address the aforementioned issues, this paper proposes a method that combines Kernel Independent Component Analysis and Deep Principal Component Analysis (KICA-DPCA) to improve the accuracy of bearing fault detection. Firstly, DPCA is utilized to thoroughly extract fault information from the dataset while simultaneously achieving the purpose of noise reduction. Secondly, KICA is combined to project the data into a high-dimensional feature space and extract independent components, thereby separating the data into two groups following Gaussian and non-Gaussian distributions. Furthermore, the occurrence of bearing faults is determined by evaluating the statistical residuals against the predefined threshold. Finally, the proposed algorithm is validated on both simulation data from the Traction Drive Control System-Fault Injection Benchmark (TDCS-FIB) platform and experimental data from the Case Western Reserve University bearing fault dataset. Comparative tests are conducted using the false alarm rate (FAR) and fault detection rate (FDR) as evaluation metrics, which fully demonstrate the effectiveness and superiority of the proposed method.

1. Introduction

The electric traction system in high-speed Electric Multiple Units (EMUs) constitutes the central power generation unit of the train, with the traction motor emerging as the pivotal component tasked with energy conversion from electrical to mechanical form within this system. Among various traction motor fault modes, bearing faults represent the most prevalent type, constituting approximately 40–50% of all motor fault cases [1]. Failure to detect faults in motor bearings in a timely manner may allow incipient faults to progressively escalate into critical failures. Such a progression not only compromises the operational integrity of the traction system but also poses substantial safety hazards for high-speed EMUs. Consequently, the development of fault detection methodologies for EMU traction motor bearings demonstrates both substantial scientific merit and critical engineering significance.

The fault detection of traction motor bearings presents significant technical challenges, particularly due to the massive sample data volumes generated by monitored components and the demanding real-time performance requirements inherent in detection systems [2].

In recent years, data-driven fault diagnosis has emerged as a prominent research focus in this domain. Notable methodologies in this field include multivariate analysis techniques [3,4], wavelet transform-based approaches [5,6], and clustering analysis frameworks [7], all of which have demonstrated extensive applicability in motor bearing fault diagnosis. Within this spectrum, approaches derived from Principal Component Analysis (PCA) and its enhanced derivatives [8,9] have gained predominant adoption in practical implementations. Ref. [10] utilizes PCA for the dimensionality reduction of multivariate datasets while maintaining the transformed residual subspace’s capacity to capture noise components during detection processes. However, this approach exhibits constrained diagnostic efficacy under high-noise conditions. To overcome these limitations, subsequent approaches leveraging Deep Principal Component Analysis (DPCA) [11,12] have significantly improved multi-level temporal detection capacity through the implementation of multi-layer data decomposition architectures, effectively overcoming PCA’s constraints in extracting latent inter-variable correlations. Fundamentally, these methodologies implement linear transformations within historical dataset spaces, thereby enabling exclusive capability for capturing Gaussian-distributed data characteristics. Their diagnostic efficacy becomes significantly degraded when processing nonlinear and non-Gaussian distributed data patterns [13].

As an enhanced alternative to PCA, Independent Component Analysis (ICA) relaxes the Gaussian distribution assumption for process variables. By extracting higher-order statistical features to derive mutually independent components, ICA preserves more comprehensive data characteristics [14]. This methodological advantage has led to its extensive application in fault diagnosis for industrial engineering systems [15,16]. Building upon this framework, Ref. [17] developed an ICA-PCA hybrid methodology based on the dual-space framework, which fully exploits the complementary advantages of PCA and ICA algorithms. The ICA algorithm utilizes higher-order statistical analysis to provide not only dimensionality reduction through decorrelation (as with PCA) but also ensures the statistical independence of higher-order moments in the extracted components. In comparison, conventional PCA is confined to second-order statistical analysis, generating only uncorrelated orthogonal components. While ICA specializes in processing non-Gaussian signals, PCA achieves optimal performance exclusively with Gaussian-distributed signals [18]. The ICA-PCA hybrid methodology [19,20] substantially improves fault detection efficacy by fully leveraging the synergistic characteristics of PCA and ICA frameworks. Nevertheless, the detection robustness of this integrated approach degrades significantly under intense noise contamination, leading to suboptimal fault detection rates that fall short of practical engineering application standards.

This study proposes a kernel independent component analysis-deep principal component analysis (KICA-DPCA)-based motor bearing fault detection method. The main innovations are summarized as follows: (1) The method leverages the capability of DPCA in multi-level data mining, tackling challenges from high-noise environments and weak fault signals. (2) Unlike conventional ICA-PCA methods, which suffer from error accumulation with nonlinear data, the proposed KICA-DPCA improves detection accuracy for both nonlinear and non-Gaussian data. Comparative experimental results across two benchmark datasets, the Traction Drive Control System Fault Injection Benchmark (TDCS-FIB) simulation platform and the Case Western Reserve University (CWRU) actual bearing fault dataset, demonstrate that the proposed fault detection method achieves superior performance with significantly enhanced comprehensive performance metrics.

2. DPCA

DPCA constitutes an advanced extension of conventional PCA methodology. Its core principle involves implementing hierarchical dimensionality reduction and noise suppression through multi-level PCA decomposition. Assuming the data matrix is $X\in R^{m\times n}$ (where n is the number of variables and m is the number of samples).To eliminate the influence of different data dimensions, the data matrix needs to be standardized to have zero mean and unit variance.

The standardized data matrix

$$X={\displaystyle \frac{X_{•j}-{\overline{X}}_{•j}}{S_j}}$$

(1)

where ${\overline{X}}_{•\mathrm{j}}$ is the mean of the j-th column data, and $S_j$ is its standard deviation.

By computing the covariance matrix of matrix X, one can obtain

$$S={\displaystyle \frac{1}{m-1}}X{X}^T$$

(2)

By performing singular value decomposition (SVD) on the data covariance matrix, the input space can be decomposed into principal component space and residual space. These refined subspaces are subsequently redefined as novel data matrices for recursive decomposition through successive processing stages, ultimately generating multiple orthogonal subspaces with amplified fault signatures.

By performing singular value decomposition on the covariance matrix, one can obtain

$$S=P_1{\Lambda }_{0.1}{P}_1^T$$

(3)

where ${\Lambda }_{0,1}=diag({\lambda }_{0,1},\dots {\lambda }_{0,m})$, ${\Lambda }_{0,1}\in R^{m\times m}$, m represents the number of sensors; ${\lambda }_{0,m}$ denotes the $m-th$ singular value of the covariance matrix S; and $P_1\in R^{m\times m}$ is the matrix of singular value vectors of S.

Based on the cumulative contribution rate method, the number of principal components k is determined, along with the loading matrix $P_{1.1}$ for the principal component space and the loading matrix $P_{1.2}$ for the residual space. Define $P_1=[P_{1.1},P_{1.2}]$ the matrix X can be decomposed into two parts:

$$X=X_{1.1}+X_{1.2}$$

(4)

$$X_{1.1}=P_{1.1}{P}_{1.1}^{T}X,X_{1.2}=\left(I-P_{1.1}{P}_{1.1}^T\right)X$$

(5)

where I denotes the identity matrix.

The decomposed matrices $X_{1.1}$ and $X_{1.2}$ are treated as new data matrices to iteratively repeat the above steps. After j levels of decomposition, $2^j$ mutually exclusive subsets are obtained, where the k-th dataset at the j-th level can be expressed as

$$X_{j,k}=\left\{\begin{array}{cc}{P}_{j,k}{P}_{j,k}^T{X}_{j-1,(k+1)/2}& \mathrm{if}k\left(\mathrm{odd}\right)\\ (I-P_{j,k}{P}_{j,k}^T)X_{j-1,k/2}& \mathrm{if}k\left(\mathrm{even}\right)\end{array}\right.$$

(6)

3. KICA-DPCA-Based Fault Detection Mechanism

The core theoretical foundation of KICA involves incorporating a nonlinear mapping operator $\Phi$ to transform the input data matrix into a reproducing kernel Hilbert space, where conventional ICA is subsequently applied to execute linear decomposition of statistically independent components. Within this architectural framework, this study establishes KICA as the complementary feature subspace to DPCA, thereby systematically compensating for DPCA’s inherent limitations in processing nonlinear system behaviors and non-Gaussian distributed process variables.

3.1. Kernel-Based Data Whitening

In the high-dimensional feature space, the subset $X_{j,k}$ obtained from DPCA decomposition is transformed through a nonlinear mapping as

$${\Phi }_{j,k}=\Phi \left(X_{j,k}\right)$$

(7)

The covariance matrix of the mapped data

$$S_{j,k}^F={\displaystyle \frac{1}{n}}{\Phi }_{j,k}{\Phi }_{j,k}^T$$

(8)

It is evident that the mapped matrices ${\Phi }_{j,k}$ and ${\Phi }_{j,k}{\Phi }_{j,k}^T$ are unknown; thus, the covariance matrix $S_{j,k}^F$ cannot be obtained directly. Since the kernel matrix $K={\Phi }^T\Phi$ is known, $S_{j,k}^F$ can be obtained indirectly through K, while the eigenvalues and eigenvector matrix can also be calculated.

The kernel matrix

$$k_{ir}=\Phi {\left(x_i\right)}^T\Phi \left(x_r\right)=\Phi \left(x_i\right)·\Phi \left(x_r\right)=k\left(x_i,x_r\right)$$

(9)

where $x_i$ and $x_r$ denote the i-th row and r-th column of matrix $X_{j,k}$, respectively.

Select the kernel function $K\left(x_i,x_r\right)=exp\left(-{\displaystyle \frac{{∥x_i-x_r∥}^2}{\sigma }}\right)$ to ensure the mapped matrix ${\Phi }_{j,k}$ has zero mean and unit variance, and center the kernel matrix $K_{j,k}$ as follows:

$${\tilde{K}}_{j,k}=K_{j,k}-I_n{K}_{j,k}-K_{j,k}{I}_n+I_n{K}_{j,k}{I}_n$$

(10)

where $I_n={\displaystyle \frac{1}{n}}{\left[\begin{array}{ccc}1& \dots & 1\\ ⋮& \ddots & ⋮\\ 1& \dots & 1\end{array}\right]}_{n\times n}$.

Scale ${\tilde{K}}_{j,k}$ proportionally within the following range:

$${\overline{K}}_{j,k}={\displaystyle \frac{{\tilde{K}}_{j,k}}{trace\left({\tilde{K}}_{j,k}\right)/n}}$$

(11)

Perform eigenvalue decomposition on ${\overline{K}}_{j,k}$ to extract the top u largest eigenvalues ${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3\ge \dots \ge {\lambda }_u$ and their corresponding eigenvectors $v_1,v_2,\dots ,v_u$, where ${\Lambda }_{j,k}=diag\left({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_u\right)$,$V_{j,k}=\left(v_1,v_2,\dots ,v_u\right)$. Since ${{\overline{\Phi }}^T}_{j,k}{\overline{\Phi }}_{j,k}=n{\overline{K}}_{j,k}$, the eigenvector matrix of the covariance matrix $S_{j,k}^F$ can be further derived as

$$H_{j,k}=n^{-1/2}{\overline{\Phi }}_{j,k}{V}_{j,k}{\Lambda }_{j,k}^{-1/2}$$

(12)

Furthermore, it can be reduced to canonical form as

$$Q_{j,k}=H_{j,k}{\Lambda }_{j,k}^{-1/2}=n^{-1/2}{\overline{\Phi }}_{j,k}{V}_{j,k}{\Lambda }_{j,k}^{-1}$$

(13)

Since the mapping matrix ${\overline{\Phi }}_{j,k}$ can be decomposed via PCA into

$${\overline{\Phi }}_{j,k}=H_{j,k}{Z}_{j,k}$$

(14)

the whitening matrix can be obtained as

$$Z_{j,k}=H_{j,k}^T{\overline{\Phi }}_{j,k}$$

(15)

Reduce to the standard form as

$$Z_{j,k}=Q_{j,k}^T{\overline{\Phi }}_{j,k}$$

(16)

then the vector can be expressed in the following form:

$${\overline{z}}_{j,k}\left(i\right)=Q_{j,k}^T{\overline{\Phi }}_{j,k}\left(i\right)=n^{1/2}{\Lambda }_{j,k}^{-1}{V}_{j,k}^T{\overline{k}}_{j,k}^T\left(i\right)$$

(17)

3.2. Dual-Space Decomposition

Apply the FastICA algorithm to $Z_{j,k}$ to extract independent components and separate them into Gaussian and non-Gaussian parts:

$${\overline{z}}_{j,k}\left(i\right)={\overline{z}}_{j,k}^{NG}\left(i\right)+{\overline{z}}_{j,k}^G\left(i\right)=B_{j,k}{S}_{j,k}+{\overline{z}}_{j,k}^G\left(i\right)$$

(18)

The FastICA algorithm employs kurtosis thresholding to extract non-Gaussian signals, as Gaussian signals are characterized by zero kurtosis. Data rows demonstrating non-convergence during the algorithmic process are systematically excluded, thereby enabling the effective isolation of non-Gaussian signal components. Since normalized data cannot be directly applied in principal component analysis, the matrix ${\overline{z}}_{j,k}^G\left(i\right)$ is processed as follows.

$$Z_{j,k}^G\left(i\right)={\Lambda }_{j,k}^{1/2}{\overline{z}}_{j,k}^G\left(i\right)$$

(19)

Performing PCA on the Gaussian component Z yields:

$${\widehat{z}}_{j,k}^G\left(i\right)=P_{j,k}^G{t}_{j,k}\left(i\right)$$

(20)

After retaining principal independent components, one can obtain

$${\widehat{\overline{z}}}_{j,k}^{NG}\left(i\right)=B_{j,k}^d{S}_{j,k}^d\left(i\right)=B_{j,k}^d{B}_{j,k}^{dT}{\overline{z}}_{j,k}\left(i\right)$$

(21)

Denormalizing ${\widehat{\overline{z}}}_{j,k}^{NG}\left(i\right)$ yields

$${\widehat{z}}_{j,k}^{NG}\left(i\right)={\Lambda }_{j,k}^{1/2}{\widehat{\overline{z}}}_{j,k}^{NG}\left(i\right)$$

(22)

Moreover, following the extraction of independent components, the whitening matrix can be formulated as

$${\widehat{z}}_{j,k}\left(i\right)={\widehat{z}}_{j,k}^{NG}\left(i\right)+{\widehat{z}}_{j,k}^G\left(i\right)$$

(23)

Finally, the statistical measure can be obtained as

$$\begin{array}{cc}\hfill SP{E}_{j,k}\left(i\right)& ={∥e_{j,k}\left(i\right)∥}^2\hfill \\ \hfill & ={∥{\overline{\Phi }}_{j,k}\left(i\right)-{\widehat{\Phi }}_{j,k}\left(i\right)∥}^2\hfill \\ \hfill & ={\overline{\Phi }}_{j,k}^T\left(i\right){\overline{\Phi }}_{j,k}\left(i\right)-2{\overline{\Phi }}_{j,k}^T\left(i\right){\widehat{\Phi }}_{j,k}\left(i\right)+{\widehat{\Phi }}_{j,k}^T\left(i\right){\widehat{\Phi }}_{j,k}\left(i\right)\hfill \\ \hfill & {\displaystyle ={\overline{\Phi }}_{j,k}^T\left(i\right){\overline{\Phi }}_{j,k}\left(i\right)}\hfill \\ \hfill & -2{\overline{\Phi }}_{j,k}^T\left(i\right)n^{-1/2}{\overline{\Phi }}_{j,k}^T\left(i\right)V_{j,k}{\Lambda }_{j,k}^{-1/2}{\widehat{z}}_{j,k}\left(i\right)\hfill \\ \hfill & +{\widehat{z}}_{j,k}^T\left(i\right){\Lambda }_{j,k}^{-1/2}{V}_{j,k}{\overline{\Phi }}_{j,k}^T\left(i\right)n^{-1/2}{n}^{-1/2}{\overline{\Phi }}_{j,k}\left(i\right)V_{j,k}{\Lambda }_{j,k}^{-1/2}\hfill \\ \hfill & =n{\overline{k}}_{j,k}(x_{j,k}\left(i\right),x_{j,k}\left(i\right))\hfill \\ \hfill & -2n^{1/2}{\overline{k}}_{j,k}\left(i\right)V_{j,k}{\Lambda }_{j,k}^{-1/2}{\widehat{z}}_{j,k}\left(i\right)\hfill \\ \hfill & +{\widehat{z}}_{j,k}^T\left(i\right){\Lambda }_{j,k}^{-1/2}{V}_{j,k}^T{\overline{K}}_{j,k}{V}_{j,k}{\Lambda }_{j,k}^{-1/2}{\widehat{z}}_{j,k}\left(i\right)\hfill \end{array}$$

(24)

$$T_{j,k}^2\left(i\right)={\left({\overline{t}}_{j,k}^d\left(i\right)\right)}^T{\overline{t}}_{j,k}^d\left(i\right)={\left(t_{j,k}^d\left(i\right)\right)}^T{\Lambda }_{j,k}^{-1}{t}_{j,k}^d\left(i\right)$$

(25)

where ${\overline{t}}_{j,k}^d\left(i\right)={\Lambda }_{j,k}^{-1/2}{t}_{j,k}^d\left(i\right)$, the $SP{E}_{j,k}\left(i\right)$ and $T_{j,k}^2\left(i\right)$ statistics are used to detect deviations from the model and variations in the principal component space, respectively.

3.3. Threshold Design

$$T_{limit}^2\left(i\right)={\displaystyle \frac{q_{j,k}\left(n^2-1\right)}{n\left(n-q_{j,k}\right)}}{F}_{q_{j,k},n-q_{j,k},\alpha }$$

(26)

where $q_{j,k}$ denotes the number of principal components retained when performing PCA on the Gaussian-distributed data $Z_{j,k}^G\left(i\right)$, n is the sample size; $\alpha$ is the significance level, and $1-\alpha$ is the confidence level. Then, $F_{q_{j,k},n-q_{j,k},\alpha }$ follows an F-distribution with the first degree of freedom being and the second degree of freedom being $n-\alpha$.

Given that the distribution of the statistic remains undetermined, the threshold is derived through the application of kernel density estimation (KDE). For electric traction systems in high-speed trains, the high robustness requirements under engineering practice necessitate representing the nominal limit conditions (fault-free) dataset $\overline{\overline{X}}$ and its corresponding statistics as:

$${\overline{\overline{SPE}}}_{j,k}=\left({\overline{\overline{SPE}}}_{j,k,1},{\overline{\overline{SPE}}}_{j,k,2},\dots ,{\overline{\overline{SPE}}}_{j,k,\overline{\overline{N}}}\right)$$

(27)

where the ${\overline{\overline{SPE}}}_{j,k}$ statistic is calculated from dataset $\overline{\overline{X}}$ collected under maximum load conditions, and its probability density is then estimated using a KDE function (the statistical value corresponding to a probability near 1 is selected as the detection threshold).

The selected kernel density function is

$$f\left({\overline{\overline{SPE}}}_{j,k}\right)={\displaystyle \frac{1}{\overline{\overline{N}}h\sqrt{2\pi }}}\sum _{i=1}^{\overline{\overline{N}}}exp\left\{-{\displaystyle \frac{{\left(SP{E}_x-{\overline{\overline{SPE}}}_{j,k,i}\right)}^2}{2h^2}}\right\}$$

(28)

where $\overline{\overline{N}}$ is the number of elements in the statistical dataset; h is the smoothing parameter; ${\overline{\overline{SPE}}}_{j,k,i}$ denotes the i-th element of the ${\overline{\overline{SPE}}}_{j,k}$ dataset; and $SP{E}_x$ represents the independent variable of the kernel density function.

The probability density function of the statistic is approximated using kernel density estimation, with the threshold selected as the statistical value corresponding to a probability of 0.98, namely, that

$$SP{E}_{limit}\left(i\right)=f_{0.98}^{-1}\left({\overline{\overline{SPE}}}_{j,k}\right)$$

(29)

and whether a fault has occurred is determined based on the following logic:

$$\left\{\begin{array}{cc}SPE\left(i\right)\le SP{E}_{\mathrm{limit}}\left(i\right)\mathrm{and}{T}^2\left(i\right)\le T_{\mathrm{limit}}^2\left(i\right),& \mathrm{normal}\\ \mathrm{otherwise},& \mathrm{fault}\end{array}\right\}$$

(30)

3.4. Evaluation Metrics

This study conducts a comprehensive performance evaluation of the proposed and benchmark methods through the application of two key metrics: the False Alarm Rate (FAR) and Fault Detection Rate (FDR), formally defined according to the following expressions:

$$FAR={\displaystyle \frac{N_p}{N_t}}\times 100\%$$

(31)

$$FDR={\displaystyle \frac{N_v}{N_n}}\times 100\%$$

(32)

where $N_p$ is the count of normal samples erroneously categorized as faulty instances, $N_t$ is the total number of normal samples, $N_v$ is the count of accurately identified faulty samples, and $N_n$ is the total number of faulty samples. Obviously, reduced FAR values coupled with elevated FDR values demonstrate enhanced detection method performance superiority.

4. Simulation Experiments

4.1. TDCS-FIB Simulated Dataset

As shown in Figure 1, Ud denotes the DC-link voltage, Lm is the mutual inductance, KIM and KPM represent the PI control parameters for the first-phase current controller, and Vge signifies the travel speed. The rotor-side inductance and rotor leakage inductance are Lr and Lrl, respectively, and phir is the reference flux. KIT and KPT are the PI parameters for the second-phase current controller. Ts and T correspond to the inverter and rectifier switching frequencies, respectively. The stator-side inductance and stator leakage inductance are Ls and Lsl, respectively, while Rs and Rr denote the stator-side and rotor-side resistances. np stands for the number of pole pairs, and ts is the sampling frequency.

Figure 1 presents the Traction Drive Control System-Fault Injection Benchmark (TDCS-FIB) experimental platform specifically developed for high-speed EMUs by Central South University. Built upon the physical traction drive architecture of CRH2 EMUs, this platform facilitates the precise simulation and controlled injection of diverse electrical fault types. Within this simulation framework, traction motor bearing faults were replicated under controlled conditions (3-s duration with fault injection initiated at the 1-s mark). The generated simulation data (shown as in Figure 2) consists of six variables: stator winding current $I_{sa}$, $I_{sb}$, $I_{sc}$, train speed $S_p$, motor rotation speed $R_s$, and electromagnetic torque $T_e$.

4.2. CWRU Bearing Fault Dataset

The CWRU bearing fault test platform acquires vibration data across multiple operational conditions and fault configurations and is currently established as the most extensively adopted public benchmark dataset for motor bearing fault diagnosis [21]. This experimental analysis specifically focuses on rolling element fault data from bearing components. The rolling element constitutes an essential dynamic component in bearing systems, primarily responsible for transmitting mechanical loads between the inner and outer races. Leveraging the physical and mechanical parameters of CRH2 EMU traction motor bearings documented in Ref. [1], the CWRU dataset with a defect diameter of 0.1778 mm was strategically selected to replicate failure modes induced by excessive wear or foreign particle contamination.

4.3. Experimental Results and Analysis

This study performs comparative experimental analyses employing PCA, ICA-PCA, and KICA-DPCA fault detection methods on both the TDCS-FIB simulation dataset and the CWRU actual bearing fault dataset (shown as in Figure 3, Figure 4, Figure 5 and Figure 6). The efficacy of the proposed approach is quantitatively verified through the application of FAR and FDR evaluation metrics (detailed in

Table 1. Fault detection comparisons based on FDR(%).

	PCA			ICA-PCA			KICA-DPCA
	${\mathit{T}}^{\mathbf{2}}$	SPE		${\mathit{T}}^{\mathbf{2}}$	SPE		${\mathit{T}}_{\mathbf{3},\mathbf{2}}^{\mathbf{2}}$	${\mathit{T}}_{\mathbf{3},\mathbf{3}}^{\mathbf{2}}$	${\mathit{SPE}}_{\mathbf{3},\mathbf{2}}$	${\mathit{SPE}}_{\mathbf{3},\mathbf{3}}$
TDCS-FIB	0	0		2.12	9.11		28.21	30.08	89.76	79.9
CWRU dataset	0	0		0	0		100	100	/	/
Average	0	0		1.06	4.56		64.11	65.04	89.76	79.9

and

Table 2. Fault detection comparisons based on FAR(%).

	PCA			ICA-PCA			KICA-DPCA
	${\mathit{T}}^{\mathbf{2}}$	SPE		${\mathit{T}}^{\mathbf{2}}$	SPE		${\mathit{T}}_{\mathbf{3},\mathbf{2}}^{\mathbf{2}}$	${\mathit{T}}_{\mathbf{3},\mathbf{3}}^{\mathbf{2}}$	${\mathit{SPE}}_{\mathbf{3},\mathbf{2}}$	${\mathit{SPE}}_{\mathbf{3},\mathbf{3}}$
TDCS-FIB	36.25	0		0	0		1.37	1.82	0	0
CWRU dataset	0	0		0	4.35		0	0	/	/
Average	18.13	0		0	2.18		0.68	0.91	0	0

).

Analysis of the TDCS-FIB dataset experimental findings demonstrates that augmenting DPCA decomposition layers generates additional sample subsets and statistical parameters, thereby achieving theoretical enhancement of fault detection capabilities. Nevertheless, this configuration concurrently induces nonlinear escalation in computational complexity and associated computational expenditure. Therefore, this paper selects second-order expansion of the original samples, generating four subset datasets and corresponding eight detection statistics, by comprehensively considering detection accuracy requirements, computational complexity, and system resource allocation constraints. Due to space limitations, this paper only selects the residual matrix of the principal space and the principal component matrix of the residual space, along with their most significantly varying statistics $T_{3,2}^2$, $T_{3,3}^2$, $SP{E}_{3,2}$, and $SP{E}_{3,3}$ to characterize detection performance. This selection strategy demonstrates distinct technical merits in the following critical dimensions:

1. The residual matrix in the principal subspace embodies components orthogonal to principal component projections, which may harbor critical anomaly signatures. The monitoring of these residuals enables identification of abnormal patterns non-conforming to PCA model specifications. The subsequent extraction of principal features from residual subspace facilitates the identification of subtle anomalies undetectable through primary PCA-level analysis.

2. The integration of principal space residuals with the residual space principal component matrix facilitates the holistic monitoring of abnormal data variations. This dual-domain analytical framework enables the concurrent consideration of both dominant PCA-captured features and residual anomalous constituents, thereby producing enhanced fault sensitivity through synergistic information fusion.

The experimental results on the CWRU bearing fault dataset similarly reveal that both PCA-based and ICA-PCA-based fault detection methods demonstrate limited efficacy in bearing fault identification. Given space constraints, Figure 6 exclusively presents the motor bearing fault detection results obtained through the KICA-DPCA framework. The predominant noise signature effectively obscures fault-related information trends, resulting in dimensionality-reduced signals dominated by noise components that render the effective separation of independent components unattainable.

Furthermore, the inherent frequency characteristics of noise interference induce square-wave-like residual fluctuation patterns in SPE statistics. Nevertheless, as per the detection logic defined in Equation (31), the method maintains diagnostic validity provided either SPE or complementary statistical parameters exhibit sensitivity to the bearing fault dataset characteristics.

A systematic comparative analysis of Figure 3, Figure 4, Figure 5 and Figure 6 and Table 1 and Table 2 substantiates that the KICA-DPCA achieves substantial enhancement in bearing fault detection rates, demonstrating comprehensive performance characteristics that significantly outperform conventional PCA and ICA-PCA approaches. Building upon the established theoretical framework, the DPCA component effectively eliminates data redundancy and extraneous noise components while preserving principal dynamic features critical for fault identification through its advanced longitudinal information excavation capabilities. Concurrently, the KICA module adeptly processes nonlinear data characteristics to isolate independent components enriched with multidimensional fault signatures. Furthermore, the kernel-based mapping technique facilitates nonlinear separability by projecting data into high-dimensional feature spaces, thereby achieving further refinement in fault detection accuracy through enhanced separation of non-Gaussian distributed data clusters.

Comparative analysis shows that while the PCA-based method can extract partial fault features from the TDCS-FIB dataset, the weak fault signal residuals still remain sub-threshold, resulting in a 0% detection rate. Furthermore, the $T^2$ statistic erroneously categorizes nominal operational data as faulty, yielding a 36.25% false alarm rate. Regarding the CWRU bearing fault dataset, the PCA-based fault diagnosis framework predominantly isolates noise components rather than fault signatures, consequently exhibiting null diagnostic capability for bearing defects (demonstrating 0% in both false alarm and fault detection rates). Synthetically evaluated, the PCA-based approach exhibits dual deficiencies: ineffective motor bearing fault detection coupled with elevated false positive rates that induce operational misjudgments, thereby substantially constraining its practical applications.

The ICA-PCA-based motor bearing fault detection method partially capitalizes on the complementary subspace synergies between PCA and ICA. Specifically, PCA demonstrates optimal detection performance for Gaussian-distributed data components, whereas ICA, unconstrained by Gaussian distribution prerequisites, provides superior analytical capability for non-Gaussian signal constituents. This subspace fusion consequently enables moderate improvement in fault detection efficacy. Experimental evaluations on the TDCS-FIB dataset revealed that the hybrid method attained a 2.12% fault detection rate through T² statistics and 9.11% via SPE statistics while maintaining zero false alarm occurrences across both metrics. When applied to the CWRU bearing fault dataset under high-noise operational conditions, the ICA-PCA approach demonstrated inadequate feature extraction capability (0% fault detection rate with 4.35% false alarms), highlighting its limitations in complex industrial environments. The experimental results demonstrate that the KICA-DPCA framework achieved a fault detection accuracy of 89.76% with zero false alarms in the testing set, indicating its potential for practical applications.

5. Conclusions

To address the challenges of low fault detection accuracy caused by strong noise interference and nonlinear/non-Gaussian characteristics in high-speed train traction motor bearing signals, this paper proposes a novel bearing fault detection method based on KICA-DPCA. By integrating DPCA’s longitudinal deep information mining capability with KICA as its complementary subspace, the proposed method achieves separate feature extraction from both Gaussian and non-Gaussian distributed data, thereby significantly enhancing fault detection precision. Comparative experiments with PCA and ICA-PCA methods conducted on both the TDCS-FIB simulation dataset and the CWRU actual bearing dataset demonstrate the KICA-DPCA approach’s superior comprehensive detection performance. Future research will focus on advancing early-stage bearing fault detection through the development of an enhanced KICA-DPCA method. The improved time-frequency joint analysis algorithm significantly boosts the identification of incipient fault features, facilitating earlier and more precise fault alarms. This will reliably support equipment predictive maintenance strategies.