A Hybrid Approach Based on a Windowed-EMD Temporal Convolution–Reallocation Network and Physical Kalman Filtering for Bearing Remaining Useful Life Estimation

Abstract

Rolling bearings are one of the core components of industrial equipment. Owing to the rapid development of deep learning methods, a multitude of data-driven remaining useful life (RUL) estimation approaches have been proposed recently. However, several challenges persist in existing methods: the limited accuracy of traditional data-driven models, instability in sequence prediction, and poor adaptability to diverse operational environments. To address these issues, we propose a novel prognostics approach integrating three key components: time-intrinsic mode functions-derived feature representation (TIR) sequences, a one-dimensional temporal feature convolution–reallocation network (TFCR) with a flexible configuration scheme, and a physics-based Kalman filtering method. The approach first converts denoised signals into TIR-sequences using windowed empirical mode decomposition (EMD). The TFCR network then extracts hidden high-dimensional features from these sequences and maps them to the initial RUL. Finally, physics-based Kalman filtering is applied to enhance prediction stability and enforce physical constraints, producing refined RUL estimates. The experimental results based on the XJTU-SY dataset show the superiority of the proposed approach and further prove the feasibility of this method in bearing RUL estimation.

1. Introduction

The Internet of Things (IoT) connects various devices and objects to the network, and its applications drive the transformation of multiple industries through intelligent data collection, analysis, and processing [1]. Among these applications, the Industrial Internet of Things (IIoT) uses IoT technology within the industrial sector to enhance industrial intelligence [2,3]. Prognostic and health management (PHM) is a part of this advancement, which uses cloud and edge computing patterns to more effectively maintain equipment health [4,5]. This cloud-edge calculation pattern in PHM, which rapidly processes historical and local data, can prevent failures and improve the production efficiency of industrial equipment [1,6,7]. As a crucial aspect of the prognostics methods, remaining useful life (RUL) estimation contributes significantly to the PHM of industrial parts [8]. For example, rolling bearings are core components of industrial rotary equipment [9]. Their operating status directly affects the overall performance and production efficiency of the equipment. The accurate estimation for rolling bearings can reduce unexpected downtime and maintenance costs and enhance equipment reliability [10,11]. Typical methods for predicting RUL mainly include two categories [12]:

• Model-based methods: These methods are based on physical models and use mathematical equations to describe degradation mechanisms such as fatigue, wear, or corrosion [13]. However, these methods are unsuitable for prognostics with modeling complexity and limited generalizability [14].
• Data-driven methods: These approaches utilize statistical techniques or machine learning algorithms, e.g., the Markov process, Kalman filter, Wiener process, or particle filter, to model degradation processes [15]. While effective, large amounts of monitoring information render it difficult to capture complex mapping using statistical distributions [16].

Recently, deep learning has effectively addressed this problem by automatically extracting features to reveal hidden degradation information [17,18]. Furthermore, they can be deployed across the source domain without prior knowledge or assumptions, rendering them more suitable for complex systems and domains [19]. Firstly, the effectiveness of the estimation heavily depends on the quality of the features [15], which can be divided into three main approaches:

• Direct approach: In this approach, the model employs an end-to-end prediction strategy by integrating the feature extraction step directly into the model framework. For instance, Xia et al. [20] propose a Long Short-Term Memory (LSTM) neural network to provide prognostics. This method segments time sequence data through sliding windows and applies the LSTM model to extract features and predict the RUL for each window. Similarly, Wang et al. [21] introduced a Deep Separable Convolutional Neural Network (DSCN) approach to process overlapping sampling windows for RUL prediction directly. The above methods use a model to segment the raw signal data and obtain a large number of windows mapped to RUL, solving specific nonlinear problems. The model must be capable of dynamically adjusting to large volumes of windowed data, which increases the computation complexity of training.
• Hybrid approach: This approach can flexibly process physically meaningful features based on the characteristics of the signal, and then evaluate HI, in a total of two steps. For example, Chen et al. [22] propose a hybrid method to provide prognostics that combine five bandpass energy values from the frequency spectrum with an attention-based recurrent neural network (RNN) for further prediction. Similarly, Guo et al. [23] propose a health indicator (RNN-HI), which includes correlation similarity features and classical time-domain features based on recurrent neural networks. Moreover, Zhou et al. [24] also introduce a hybrid approach, transformer (TRM), to perform RUL tasks based on 14 time-domain features. Furthermore, Cao et al. [25] propose TCN-RSA, which employs causal dilated convolutions to capture long-term dependencies and extract high-level features from the time-frequency domain, incorporating residual self-attention mechanisms to obtain feature contributions at different time steps during bearing degradation. Additionally, Liu et al. [26] propose a hybrid framework that combines temporal convolutional networks (TCNs) and LSTM networks with a Convolutional Block Attention Module (CBAM) for multi-dimensional feature weighting. In short, the above methods have been proven to be effective in prognostics. This can reduce the parameters and increase calculation speed, but it may face challenges when it comes to dealing with high interference noise and complicated mapping over long distances [27].
• Physics-informed hybrid approach: This emerging paradigm integrates physical degradation laws and domain knowledge into data-driven models to enhance prediction reliability and interpretability. Specifically, Lu et al. [28] integrate physical consistency constraints into the LSTM loss function to ensure predictions comply with monotonic degradation laws. Similarly, Yang et al. [29] utilize dynamic adaptive IDFT frequency domain blocks with multi-state memory units for physics-constrained time–frequency fusion prediction. Furthermore, Hu et al. [30] combine separable convolutional feature extraction with physics-informed neural networks, learning implicit physical degradation patterns while incorporating physics constraint losses. Collectively, these methods address the black-box nature of data-driven approaches by incorporating physical principles, thereby enhancing model interpretability and prediction reliability. Nevertheless, they may face challenges in modeling complex real-world physical relationships and require domain expertise for proper constraint formulation.

Secondly, the model architecture substantially impacts prognostic accuracy. Fixed input dimensions and model structures can reduce prediction effectiveness, limiting suitability for practical engineering applications. For instance, Yin et al. [8] emphasize that effective feature extraction and appropriate model parameter settings can significantly enhance prediction performance.

Thirdly, complex architectures with numerous parameters are typically required to capture temporal and nonlinear degradation information. However, this complexity can result in high latency and poor performance when deployed on cloud and edge platforms [31].

An effective prediction depends on well-informed input for deep-learning-based approaches [15]. Among signal processing methods, Empirical Mode Decomposition (EMD) [32] is particularly suitable for processing complex non-stationary and nonlinear signals. Meanwhile, this method has been demonstrated to effectively enhance input information for prediction tasks [33,34,35]. As part of hybrid feature engineering, EMD can adaptively decompose the raw signal into multiple time–frequency-scale intrinsic modal functions (IMFs) across different time–frequency scales, thereby mitigating the influence of noise on the original signal. This advantage effectively solves the problem of hybrid engineering mentioned above. Moreover, in recent times, one-dimensional convolution neural networks (1D-CNN) have demonstrated considerable potential in processing long sequences [36]. The convolution mode inherently has the feature of parameter sharing to reduce the number of parameters. Additionally, it can further reduce the parameters by adjusting the convolution category, kernel sizes, and the number of channels [37,38]. In a typical 1D-CNN architecture, the number of channels gradually increases as the network depth increases, while the time dimension, namely the feature map, is gradually reduced through pooling operations [21,39,40]. However, reducing the time dimension resolution through pooling can lead to the loss of important temporal information, which is critical in tasks such as RUL prediction. In this regard, a Physics-informed hybrid bearing RUL estimation method is proposed based on the windowed EMD feature engineering method, an effective multi-model flexible configuration strategy, and one-dimensional temporal convolution neural network models, and the novel method provides a robust and accurate solution for bearing RUL estimation. It primarily comprises three aspects:

• Vibration signals are monitored and analyzed, providing comprehensive degradation information. The feature engineering method is an adjustable windowed EMD time-domain feature extraction method, and its purpose is to convert the denoised vibration signals into TIR-sequences as the input. Moreover, experimental validation demonstrates that this method is feasible for characteristic degradation.
• A one-dimensional temporal convolution model (TFCR) is combined to extract hidden features from TIR-sequences and construct RUL. Specifically, the proposed model employs a causal convolution architecture in the time dimension to preserve temporal information. In parallel, a bottleneck top–down design is used in the channel dimension, with channel weight reallocation at the final stage. It can ensure that channel compression does not lead to a significant loss of critical features. Experimental results also demonstrate that it can address bearing life prediction tasks with high accuracy and efficiency.
• The TFCR model combines flexible configuration to select the most appropriate model, thereby adapting to complex working conditions with different data distributions. Then, TFCR is combined with physics-based Kalman filtering to improve estimation stability.

The remainder of this paper is organized as follows. Firstly, Section 2 presents the proposed TIR-sequences, and Section 3 presents the proposed specific model components of TFCR, and the architecture of each layer. In addition, Section 4 introduces a flexible configuration scheme and a physically constrained Kalman filter approach for physically constraining, smoothing, and suppressing outliers in the RUL series extracted by TFCR. Finally, the validation experiments are shown in Section 5, in which a comparison of related studies is provided, while Section 6 discuss limitations and future research directions.

2. The TIR Sequences for the Temporal Degradation Model

In this section, the rolling bearing temporal degradation model is introduced. Subsequently, the process and method for obtaining TIR sequences are presented, which serve as the input for the TFCR model discussed in Section 3.

2.1. The Temporal Degradation Model

Rolling bearings inevitably undergo a nonlinear degradation process until failure. However, the inherent complexity of this process poses significant challenges for accurate predictive modeling. The temporal degradation model simplifies this nonlinearity by approximating it with a linearly decreasing number of remaining sampling cycles.

As illustrated in Figure 1, the circular symbols $n_1,n_2,\dots ,n_d$ represent feature vectors extracted during sampling duration, with the horizontal axis depicting the time series of the bearing degradation process. Together, these elements establish the temporal framework for mapping features to their corresponding degradation stages.

2.2. The TIR Sequences Extraction Method Based on Windowed EMD

The TIR sequence based on windowed EMD feature engineering contains three key steps:

• Step 1: Divide the vibration signal at each time node into adjustable, non-overlapping time windows, and apply EMD to decompose the denoised signal into intrinsic mode functions (IMFs) and residual function (RES).
• Step 2: Extract time-domain features from the IMFs and RES within each window, and concatenate them to form a feature vector representing that time node.
• Step 3: Arrange these feature vectors chronologically to construct a complete TIR-sequence with temporal and feature dimensions.

It should be noted that the original signal has been denoised before applying EMD, because EMD is mainly used for signal decomposition rather than noise reduction in this approach. Figure 1 shows the complete flowchart of these steps. (TIR-sequences incorporate features extracted from both IMFs and residual components obtained through EMD decomposition, providing a comprehensive representation of the signal characteristics across time windows). The feature dimension is calculated as follows:

$$n_d=n_i\times n_f\times n_w\times n_s$$

(1)

where $n_d$ refers to the dimensions extracted, $n_i$ represents the number obtained after the EMD process, $n_f$ is the number of time-domain features from [24] that have been verified as degradation-sensitive features, $n_w$ denotes the number of adjustable windows into which the sampling time is divided, and $n_s$ signifies the number of sensors involved.

Unlike traditional EMD, which continues decomposition until the residual becomes monotonic, this approach uses a fixed number of decomposition layers (m) based on empirical analysis of bearing signal characteristics. This modification prevents over-decomposition while ensuring consistent feature dimensionality across all time windows. Ultimately, Equation (2) shows the final decomposition with m modes:

$$x\left(t\right)=\sum _{i=1}^{m}IM{F}_i\left(t\right)+r_m\left(t\right)$$

(2)

Regarding feature processing, the cumulative feature transformation method proposed in [24] was initially considered to enhance monotonicity and trend characteristics through point-wise accumulation and scaling operations. However, two issues were identified when applying this method after TIR sequences extraction: (1) noise amplification in the transformed features, and (2) severe overfitting during model training. Consequently, we discarded the cumulative feature transformation method and used the raw features directly extracted from IMFs and residuals.

3. The TFCR Model Architecture

This section introduces the proposed patchwork model-TFCR, which consists of temporal feature compression layers and channel weight redistribution layers and can be categorized into large and light models.

3.1. Temporal Feature Compression Layer

The temporal feature compression layers use a causal convolution architecture [36], implementing masked padding in the temporal dimension to form a unidirectional structure, which effectively addresses the limitations of conventional convolutions in capturing temporal dependencies, as shown in Figure 2 and Figure 3. The architecture maintains the temporal dimension across layers while applying compression to the channel, thereby reducing the parameters and achieving dimensionality reduction. Simultaneously, compared with layer a, layer b expands the receptive field of the temporal dimension by employing dilated convolution kernels. The convolution Equations (3) and (4) of these layers are as follows:

$$y(t,c)=\sum _{m=1}^M\sum _{k=0}^{K-1}{w}_{k,m,c}·x(t-d·k,m)$$

(3)

$$x(t-d·k,m)=\left\{\begin{array}{cc}x(t-d·k,m)\hfill & t-d·k\ge 0\hfill \\ \mathrm{masked}\hfill & t-d·k<0\hfill \end{array}\right.$$

(4)

where $w_{k,m,c}$ refers to the kernel weight, associated with temporal index k, input channel m, and output channel c; $y(t,c)$ is the output at time node t and channel c; $x(t-d·k,m)$ is the value of the input at time node $t-d·k$ and channel m, which is masked when $t-d·k<0$ (typically set to zero). d is the dilation coefficient that controls the expansion of the receptive field.

3.2. Channel Weight Redistribution Layer

The channel weight redistribution layer dynamically adjusts feature importance. Figure 4 shows its architecture with a global average pooling layer and a feature redistribution block. In contrast to SEblock [41], which uses fully connected layers for dimensionality reduction, the layer in the figure employs $1\times 1$ convolution without channel scaling, preserving all feature information. The output is obtained by element-wise multiplication with the generated channel weights. The mathematical formulation of this layer is given by Equation (5):

$$\mathrm{FRL}{\left(x\right)}_{t,c}=\sigma \left(w_2\delta \left(w_1{\displaystyle \frac{1}{T}}\sum _{i=1}^T{x}_{i,c}\right)\right)·x_{t,c}$$

(5)

where $x_{t,c}$ denotes the input at time step t and channel c, and T represents the total time steps. Matrices $w_1$ and $w_2$ are learnable parameters. The ReLU activation function $\delta$ introduces nonlinearity, while the Sigmoid function $\sigma$ constrains the channel weights to the range $(0,1)$. These weights selectively emphasize important channel features while suppressing less relevant ones.

4. The Implementation of RUL Estimation

This section integrates the padded and normalized TIR sequences with the proposed TFCR models using a flexible scheme strategy to generate initial RUL predictions. Additionally, physics-based Kalman filtering is introduced to refine these initial RUL estimates.

4.1. Data Preparation and Flexible TFCR Scheme

To enhance model convergence and performance, TIR sequences undergo normalization, scaling the data to a suitable range compatible with the output of sigmoid activation function (0–1). Additionally, padding is applied at the beginning of the sequences to ensure alignment between input and output sample counts, as shown in Equations (6)–(8).

$${\tilde{x}}_{f,t}={\displaystyle \frac{x_{f,t}-x_{f,\min }}{x_{f,\max }-x_{f,\min }}}$$

(6)

$$\left[\left.P\right|X\right]=\left[\left.\begin{array}{ccc}{p}_1& \dots & p_{l-1}\\ p_1& \dots & p_{l-1}\\ ⋮& \ddots & ⋮\\ p_1& \dots & p_{l-1}\end{array}\right|\begin{array}{ccc}{x}_{1,1}& \dots & x_{1,t}\\ x_{2,1}& \dots & x_{2,t}\\ ⋮& \ddots & ⋮\\ x_{f,1}& \dots & x_{f,t}\end{array}\right]$$

(7)

$$\left|P\right|=l-1$$

(8)

The above equations represent the preprocessing steps applied to the TIR sequences. Here, f represents the feature dimension, t denotes the temporal dimension, and l is the input sequence length for the model. Equation (6) performs feature-wise Min-Max normalization, where $x_{f,t}$ is the original value of feature f at time step t, and $x_{f,\min }$ and $x_{f,\max }$ are the minimum and maximum values of feature f across all time steps. Equations (7) and (8) illustrate the zero-padding process, where matrix P with length $l-1$ is prepended to the original data matrix X to ensure temporal alignment when using a stride of 1.

The flexible scheme is based on a dynamic adjustment strategy, specifically adjusting the feature dimension of the TIR-sequences according to the window size $n_w$ and m. As described in Section 2, the accepted dimension and parameters inside TFCR also gradually reduce to adapt to changes in input. For example,

Table 1. TIR sequences input and network configurations.

Scheme	${\mathbf{n}}_{\mathbf{i}}$	${\mathbf{n}}_{\mathbf{w}}$	${\mathbf{n}}_{\mathbf{s}}$	Channel	Layer	Dilation	Blocks
Large 1	6	8	2	1056 → 512	a	1	2
					b	2	2
					c	1	1
Large 2	5	8	2	880 → 220	a	1	2
					b	2	2
					c	1	1
Light 3	5	4	2	440 → 160	a	1	1
					b	2	2
						4	1
					c	1	1
Light 4	5	2	2	220 → 60	a	1	1
					b	2	2
						4	1
					c	1	1

shows four distinct schemes in this approach with their corresponding parameters. These schemes can be divided into large and light models. In particular, these schemes represent only a subset of possible configurations.

4.2. Physics-Based Kalman Filtering for RUL Refinement

The original network outputs often exhibit fluctuations, particularly in long sequence outputs, which are misinterpreted by the filter as observation noise. Thence, trend windows $F_t$ are employed to compute the averaged RUL values as observation inputs for the Kalman filter. Simultaneously, all trend values within the smooth window $F_s$ are compared against an anomaly threshold $C_o$ using Z-score (9). When anomalies are detected (i.e., $Z_k>C_o$), the system adaptively increases the observation variance to reduce the Kalman gain, while decreasing the trust parameter to make the filtering results closer to historical states (10). Finally, the adaptive filtering output is generated through a trust-weighted combination (11), and RUL physical constraints are applied to ensure the rationality of predictions.

$$Z_k={\displaystyle \frac{|z_k^t-MD\left(H_k\right)|}{MAD\left(H_k\right)\times 1.4826}}$$

(9)

$${\alpha }_k=C_t\times {\left(1+min\left({\displaystyle \frac{Z_k}{C_o}},3.0\right)\right)}^{-1},R_k=C_m\times \left(1+min\left({\displaystyle \frac{Z_k}{C_o}},3.0\right)\right)$$

(10)

$${\widehat{x}}_k={\alpha }_k·{\widehat{x}}_k^{raw}+(1-{\alpha }_k)·{\widehat{x}}_{k-1}$$

(11)

where $z_k^t$ represents the trend value at time k obtained by averaging DL predictions within the trend window, $H_k$ denotes the historical trend values within the smooth window, ${\widehat{x}}_k^{raw}$ is the standard Kalman filter output, and ${\widehat{x}}_{k-1}$ is the previous filtering result. The scaling factor of 1.4826 and saturation threshold of 3.0 are adopted from robust statistical practice to provide meaningful parameter ranges, with all final parameter values optimized through a systematic grid search on validation data rather than theoretical derivation.

5. Experimental Evaluation of the Proposed RUL Estimation Approach

5.1. Introduction to the XJTU-SY Dataset

The XJTU-SY bearing dataset contains vibration signals from bearings with various failure modes (inner/outer ring wear and fracture and cage fracture). Data was collected at a 25.6 kHz sampling frequency with dual accelerometers until failure occurred (when amplitude reached 10× normal levels).

Table 2. Dataset split for training, validation, and testing sets.

Condition			Train	Validation	Test
No.	Load (kN)	Speed (rpm)
1	12	2250	$1_1$, $1_2$, $1_3$, $1_4$	Synthetic	$1_5$
2	11	2500	$2_1$, $2_2$, $2_3$, $2_4$		$2_5$
3	10	2400	$3_1$, $3_2$, $3_3$		$3_4$

shows the classification of training sets and test sets in this paper.

5.2. Training Strategy in Experiment

The training uses the AdamW optimizer with MSE loss, variable learning rates, and early stopping. Training runs 100–800 epochs depending on dataset size, with the learning rate starting at $1\times {10}^{-4}$ ($1\times {10}^{-5}$ for condition 3) and reducing by 50% after 60 epochs without improvement. Regularization includes weight decay ($1\times {10}^{-5}$, $5\times {10}^{-5}$), gradient clipping (0.8, 1.0, 2.0), and Kaiming initialization [42] for temporal layers. The implementation was conducted using PyTorch 2.1.0 on an Intel i7-12700 CPU (Intel Corp., Santa Clara, CA, USA) and an NVIDIA RTX 3070 Ti GPU (NVIDIA Corp., Santa Clara, CA, USA).

The synthetic time-series data preserve temporal characteristics based on the linear degradation model from Section 2. The synthetic component generates new RUL values with small perturbations, then uses linear interpolation to map RUL to features. Three perturbations ensure uniqueness: (1) horizontal perturbations based on inter-bearing variation, (2) vertical perturbations proportional to feature standard deviation, and (3) degradation perturbations across degradation stages. Synthetic samples use 50–70% of the original bearing data length. To validate the quality of the synthetic data, a comprehensive similarity analysis was performed between training, testing, and validation datasets using working conditions 1 and 2 as representative examples. Figure 5 presents the comparative analysis of dataset similarity on normalized features using multiple metrics, including statistical moments, quantiles, distribution shapes, KS-statistic-based similarity scores, and correlation structures, with averaged similarity scores for evaluation.

$$y=y_1+{\displaystyle \frac{x-x_1}{x_2-x_1}}(y_2-y_1)+{\Delta }_{\mathrm{horizontal},j}+{\Delta }_{\mathrm{vertical},j}+{\Delta }_{\mathrm{Degradation},j}$$

(12)

where x is the newly generated RUL value, $x_1$ and $x_2$ are adjacent RUL values from original data ($x_1\le x\le x_2$), $y_1$ and $y_2$ are corresponding feature values. The three perturbations are as follows: ${\Delta }_{\mathrm{horizontal},j}\sim \mathcal{N}(0,{\sigma }_{\mathrm{inter},j})$ represents inter-bearing differences, where ${\sigma }_{\mathrm{inter},j}$ is the standard deviation of feature j means across all training bearings. This fixed offset simulates the differences in characteristics between bearings. ${\Delta }_{\mathrm{vertical},j}\sim \mathcal{N}(0,{\sigma }_{\mathrm{feature},j}·r_{\mathrm{noise}})$ represents measurement variations, where ${\sigma }_{\mathrm{feature},j}$ is the internal variability of the selected bearing for feature j. ${\Delta }_{\mathrm{Degradation},j}\sim \mathcal{N}(0,{\sigma }_{\mathrm{feature},j}·{\displaystyle \frac{RU{L}_{\max }-RU{L}_{\mathrm{current}}}{RU{L}_{\max }-RU{L}_{\min }}})$ represents degradation-dependent noise that increases as RUL decreases, simulating signal instability near failure.

5.3. TFCR–Kalman Filter for RUL Estimation of TIR Sequences

Table 3. Performance comparison of models.

Scheme	Rmse			Avg.	Score			Avg.	Time			Size	Best Length
	C1	C2	C3		C1	C2	C3		C1	C2	C3		C1	C2	C3
1	0.073	0.081	0.112	0.088	0.81	0.53	0.12	0.49	0.105	0.076	0.056	24.89	18	40	40
2	0.084	0.107	0.095	0.095	0.82	0.40	0.14	0.45	0.121	0.104	0.054	10.77	20	56	48
3	0.091	0.102	0.114	0.102	0.73	0.38	0.15	0.42	0.092	0.065	0.041	3.80	20	40	40
4	0.101	0.105	0.132	0.113	0.70	0.42	0.11	0.40	0.065	0.058	0.029	0.98	16	48	40

shows the average experimental results of various models at their optimal sequence lengths on the test set. The evaluation metrics include Rmse, Score (as defined by equations in the PHM 2012 Challenge), Time (average processing time per sample in ms), and Size (in MB). Although the models demonstrate similar overall average prediction results, they each excel differently under their respective advantageous conditions and scenarios. Figure 6, Figure 7 and Figure 8 show the prediction curves of each model under different conditions, where the black line represents the actual RUL values.

The objective function adapts to sequence length: short sequences prioritize MSE accuracy ($F_t=1$), while long sequences minimize center-deviation penalties with distance-based penalty scaling.

Table 4. Kalman filter configuration and performance summary.

Parameter	Scheme 1	Scheme 2	Scheme 3	Scheme 4
Core Parameters (C1, C2, C3)
Process Variance ($C_p$)	0.05, 0.05, 0.001	0.05, 0.0005, 0.0005	0.02, 0.001, 0.005	0.05, 0.02, 0.0005
Meas. Variance ($C_m$)	0.0001, 0.0001, 0.005	0.0001, 0.0001, 0.02	0.0001, 0.05, 0.02	0.0001, 0.0005, 0.05
Trust Factor ($C_t$)	0.95, 0.6, 0.85	0.9, 0.6, 0.9	0.8, 0.6, 0.95	0.6, 0.6, 0.98
Outlier Sensitivity ($C_o$)	5.0, 5.0, 4.0	5.5, 2.5, 3.0	2.0, 3.0, 3.5	2.5, 3.0, 2.5
Filter Parameters (C1, C2, C3)
Smoothing Window ($F_s$)	15, 3, 19	15, 3, 19	3, 2, 13	9, 2, 19
Decay Factor ($F_d$)	0.998, 0.9995, 0.9999	0.998, 0.9998, 0.9995	0.995, 0.9998, 0.9995	0.999, 0.998, 0.9995
Trend Window ($F_t$)	0, 4, 5	0, 3, 11	0, 2, 11	0, 3, 11
Performance Changes (C1, C2, C3)
RMSE (↓)	5.9, 12.6, 14.3	4.3, 8.2, 30.3	15.4, 19.5, 29.2	24.3, 6.5, 13.5
Score (↑)	2.5, 14.0, 9.9	0.8, 7.9, 13.4	7.2, 9.3, 3.7	7.5, 15.4, 14.6
MAE (↓)	5.6, 11.9, 10.4	5.1, 3.2, 29.5	15.1, 29.4, 19.9	25.3, 9.2, 8.2

presents the optimal parameters identified through grid search and the corresponding ablation study results for Kalman filter configuration and performance evaluation.

The proposed model is evaluated in terms of computational speed and accuracy.

Table 5. Computational speed comparison.

Model	Time Per Sample/Length (ms)
GRU	0.0101
LSTM	0.0120
Transformer	0.0153
TFCR	0.0067

presents a comparison of the efficiency of the computation between the proposed TFCR model and established sequential models from the NLP domain, including GRU [43], LSTM [44], and Transformer [45], under identical experimental conditions (as specified in Table 1). The average computational time is normalized by sequence length to enable a fair comparison across different input sizes.

Table 6. Computational performance comparison.

Model	RMSE
DSCN	0.0739
RCNN	0.0803
RVM	0.1082
MTCN	0.0732
Transformer	0.0701
PGLSTM	0.0866
TCN-RSA	0.0699
Proposed TFCR 1	0.0884
Proposed TFCR 2	0.0953
Proposed TFCR 3	0.1023
Proposed TFCR 4	0.1135
Proposed TFCR-KF 1	0.0767
Proposed TFCR-KF 2	0.0827
Proposed TFCR-KF 3	0.0894
Proposed TFCR-KF 4	0.0946

compares the precision of the proposed model with that of other studies on the same dataset, demonstrating its effectiveness.

5.4. Result

The experimental results demonstrate that the proposed TFCR method successfully maps TIR sequences to RUL, with the physical Kalman filtering integration providing consistent performance improvements across all variants. The four TFCR configurations achieved baseline RMSE values of 0.0884, 0.0953, 0.1023, and 0.1135, respectively. After physical Kalman filter integration, these improved to 0.0767, 0.0827, 0.0894, and 0.0946, representing RMSE reductions of 13.24%, 13.22%, 12.61%, and 16.65%, respectively. Similarly, MAE improvements ranged from 9.3% to 21.4% (9.3%, 12.6%, 21.4%, and 14.2% for scheme 1–4), while prediction scores enhanced by 8.8%, 7.4%, 6.7%, and 12.5%, respectively.

6. Discussion

The experimental validation confirms the feasibility of the proposed hybrid method. Despite our best-performing method (TFCR-KF 1: 0.0767) lagging behind the leading approach, our hybrid method provides superior interpretability and ensures RUL physical consistency.

More importantly, our method exhibits significant limitations. First, the TIR sequences extraction process requires extensive data preprocessing, substantially increasing system complexity. Second, our TFCR network faces bottlenecks in feature representation and RUL mapping capabilities, particularly in frequency domain processing. While our approach employs windowed EMD with temporal feature extraction, leading methods utilize more sophisticated frequency analysis techniques. For instance, advanced approaches like TCN-RSA employ marginal spectrum analysis from the Hilbert–Huang Transform, which provides richer frequency domain representations compared to our basic windowed EMD approach. Similarly, other state-of-the-art methods incorporate advanced spectral techniques such as adaptive frequency decomposition, multi-scale wavelet analysis, and frequency-specific attention mechanisms, whereas our method relies on relatively simple temporal feature extraction from EMD components. Third, while Kalman filtering provides physical constraints, it also introduces additional parameter tuning requirements that may affect the method’s generalization capability.

Inspired by the foundational role of pre-training in large language models, future work could explore using TFCR as a temporal autoencoder component to perform pre-training dimensionality reduction on high-dimensional TIR sequences, which could then serve as input for downstream time-series models (such as TimesNet, Informer, etc.) to further enhance performance and efficiency.