A Practical Prognostics Method Based on Stepwise Linear Approximation of a Nonlinear Degradation Model

Abstract

Prognostics aims to predict the remaining useful life (RUL) of an in-service system based on its degradation data. Existing methods, such as artificial neural networks (ANNs) and their variations, often face challenges in real-world applications due to their complexity and the lack of sufficient data. In this paper, a practical prognostic method is proposed, based on the stepwise linear approximation of nonlinear degradation behavior, to simplify the prognostic process while significantly reducing computational costs and maintaining high accuracy. The proposed approach is validated using synthetic data generated at different noise levels, with 100 data sets tested at each level, and compared against a typical ANN method. The results demonstrate that the proposed method consistently outperforms the ANN in terms of accuracy and robustness, while remarkably reducing computational time by a factor of 50 to 60, making it a promising solution for real-world applications.

1. Introduction

It is essential to keep a system safe to minimize downtime and avoid catastrophic failures. To achieve this, effective maintenance strategies have been developed, and recent attention has shifted toward predictive maintenance (PdM) that relies on the predicted health condition of a system [1,2,3]. Central to PdM is prognostics, which forecasts a system’s future degradation behavior and remaining useful life (RUL) based on its health monitoring data. Recent advancements in big data analytics, machine learning, and artificial intelligence have significantly influenced the development of prognostic techniques.

Prognostic methods can generally be categorized into physics-based [4] and data-driven [5] approaches, depending on the type of information used. In physics-based approaches, the key element is a physical model that describes degradation behavior over time under various loading conditions. For example, the Paris model [6], which describes crack growth behavior, is a representative physical degradation model. In such approaches, the primary concern is identifying model parameters based on the available data. Common algorithms for estimating these parameters include particle filters [7] and the Bayesian Markov chain Monte Carlo method [8]. Orchard and Vachtsevanos [9] and An et al. [10] employed a particle filter-based framework and a Bayesian method, respectively, to predict crack growth and the RUL. Although physics-based approaches can provide accurate predictions of degradation behavior and RUL with relatively small datasets, recent research on this topic is limited due to the scarcity of physical degradation models. Consequently, there has been growing interest in data-driven approaches.

Data-driven approaches use monitoring data and mathematical models selected based on the characteristics of the data. Since the use of mathematical models is flexible, various algorithms are available, including nonlinear least squares [11], Gaussian processes [12], and artificial neural networks (ANNs) [13]. Among these, ANNs and their variations have been extensively employed for data-driven prognostics. For instance, Asif et al. [14] proposed a data-driven method based on long short-term memory (LSTM) networks, a type of recurrent NN, and validated it on the C-MAPSS dataset for aerospace engines. Using the same dataset, Ensarioğlu et al. [15] proposed a 1D convolutional NN-LSTM model with difference-based features, demonstrating superior performance on the C-MAPSS FD001 dataset. Additionally, Long et al. [16] developed a gated recurrent unit-based method for hydrogen fuel cell RUL prediction, outperforming the back propagation NN and LSTM on the FCLAB datasets.

Despite the advancements in data-driven approaches, the limited availability and quality of degradation data have motivated the development of hybrid approaches. One such approach is physics-informed neural networks (PINNs) [17], which embed physical laws into the training of ANNs as a form of regularization to enhance predictive accuracy. Wang et al. [18] employed a PINN that incorporates attributes affecting battery degradation, based on an empirical degradation equation, for the accurate and stable estimation of battery state of health. Fassi et al. [19] implemented physics-informed machine learning to demonstrate improvements in the precision of RUL prediction, achieving faster learning convergence and better generalizability.

Although it is desirable to develop sophisticated algorithms to improve accuracy with less information, existing prognostic algorithms remain difficult to understand and implement, and their accuracy levels are often unsatisfactory given their complexity. Understanding these algorithms requires knowledge of several mathematical and physics theories, such as Bayesian statistics, probability theory, numerical sampling methods, optimization techniques, and partial differential equations. Moreover, as the complexity of a prediction model increases, so do the computational time and cost. For instance, with one input and one output variable, an ANN with one hidden layer and one hidden node has twice as many parameters to be estimated as a linear regression model. When the number of hidden nodes increases to two, the complexity of the ANN rises to 3.5 times that of the linear regression model. Notably, most ANN variations contain multiple hidden layers and numerous hidden nodes, significantly amplifying their complexity.

In this paper, a practical prognostic method that employs only a linear regression model is presented to achieve significantly lower complexity and computational cost than an ANN while maintaining prediction accuracy comparable to or even better than that of an ANN. Linear regression is a fundamental method based on basic algebra and linear algebra, and it generally requires less specialized knowledge and expertise compared to the theories required for an ANN. Therefore, the proposed method is straightforward to understand and implement, and it significantly reduces the computational burden. As a strategy to improve prediction accuracy using a linear model, a stepwise linear approximation method is developed, and its prediction accuracy and computational efficiency are compared to those of a typical ANN on the same dataset.

This paper is organized as follows: Section 2 briefly introduces linear regression; Section 3 explains the concept of the proposed method, including how to divide the regression steps and predict the RUL; Section 4 compares the results of the proposed method with those of a typical ANN across 100 different datasets; and finally, Section 5 provides the conclusions, while Nomenclature provides a list of notations used in this paper along with their descriptions.

2. Background

Linear regression expresses the linear relationship between input and output by identifying model parameters from the data. In this paper, the linear regression model is represented by the following equation:

$$z=a+bt$$

(1)

where z, t, a, and b represent the degradation behavior (model output), time/cycle, and the model parameters to be estimated, respectively. The two model parameters, a and b, are estimated using the least-squares method [20], which minimizes the sum of squared errors between the degradation data and the model output. The detailed process can be found in reference [20], and the final result for parameter estimation is obtained as follows:

$$\widehat{\mathsf{\theta }}={\left[{\mathbf{X}}^{\mathrm{T}}\mathbf{X}\right]}^{-1}{\mathbf{X}}^{\mathrm{T}}\mathbf{y}$$

(2)

where $\mathbf{X}$ is the design matrix, $\mathbf{y}$ is the vector of measured degradation data, and $\mathsf{\theta }$ is the vector of model parameters. Given $n$ datasets (measured ${\mathbf{y}}_{1:n}$ at ${\mathbf{t}}_{1:n}$), these are expressed as follows:

$$\widehat{\mathsf{\theta }}=\left(\begin{array}{c}\widehat{a}\\ \widehat{b}\end{array}\right), \mathbf{X}=\left[\begin{array}{cc}1& t_1\\ \begin{array}{c}1\\ ⋮\\ 1\end{array}& \begin{array}{c}{t}_2\\ ⋮\\ t_n\end{array}\end{array}\right], \mathbf{y}=\left(\begin{array}{c}{y}_1\\ y_2\\ \begin{array}{c}⋮\\ y_n\end{array}\end{array}\right)$$

(3)

In the proposed method, the nonlinear degradation model is approximated by the linear model in Equation (1), and the model parameters, a and b, are estimated using Equation (2). A detailed explanation of the proposed method is provided in the subsequent section.

3. Proposed Method

Degradation growth generally exhibits exponential behavior, making it natural for a nonlinear model to outperform a linear one in capturing such characteristics. However, if the degradation growth is divided into several stages, a linear model can effectively approximate this behavior. The proposed method is based on this concept, with the key challenge being how to divide the stages and how to ensure the linear model closely approximates the original nonlinear model.

Degradation progress can generally be divided into three stages: minor growth, moderate growth, and accelerated growth [21]. The minor growth stage is considered the normal condition, while meaningful degradation begins in the moderate growth stage. However, even in this stage, many datasets or a physical degradation model are required to predict RUL accurately. Therefore, the proposed method focuses only on the final stage, accelerated growth, using limited datasets and without relying on a physical model. Figure 1 illustrates an example of dividing the degradation stages and predicting in the accelerated growth stage, and the specific method for achieving this is detailed in the following subsection.

3.1. Algorithm for Stepwise Linear Approximation

The proposed method consists of two main steps: Step 1 involves dividing the stages, and Step 2 involves predicting RUL. In Step 1, the goal is to find a data point where a consistent increase in the slope of the linear regression occurs over four consecutive points. This point will be referred to as the stage criterion. Due to noise in the measurement data, the slope may fluctuate between an increase and decrease. A continuous increase in the slope could indicate a change in degradation characteristics, i.e., a stage change. Therefore, finding the stage criterion based on a continuous increase in the slope is the key algorithm of Step 1, which proceeds as follows:

Step 1.

Dividing the stages

$$\begin{array}{ll}1.& \mathrm{Set} \mathrm{the} \mathrm{initial} \mathrm{values}:\\ & -k_0=1, s=0\\ 2.& \mathrm{For} k=k_0+3 \mathrm{to} n:\\ & -\mathrm{Increment} s: s=s+1.\\ & -\mathrm{Estimate} \mathrm{the} \mathrm{model} \mathrm{parameter} b_k \mathrm{from} \mathrm{Equation} \left(1\right) \mathrm{using} {\mathbf{y}}_{k_0:k} \mathrm{a}\mathrm{n}\mathrm{d} {\mathbf{t}}_{k_0:k}.\\ & -\mathrm{Monitor} b_k.\\ & -\mathrm{When} k\ge k_0+7, \mathrm{find} \mathrm{the} \mathrm{first} k\mathrm{that} \mathrm{satisfies} \mathrm{the} \mathrm{following} \mathrm{equation}:\\ & \hfill b_i-b_{i-1}>0 \mathrm{for} i=k-3, k-2, k-1, k\hfill \\ & -\mathrm{If} \mathrm{the} \mathrm{condition} \mathrm{is} \mathrm{satisfied}, \mathrm{stop}, \mathrm{and} \mathrm{set} k_0=k.\\ & -\mathrm{If} s=2, \mathrm{stop} \mathrm{and} \mathrm{move} \mathrm{to} \mathrm{Step}2.\end{array}$$

(4)

where $k$ is the data index, $s$ represents the degradation stages, and $n$ is the total number of data points. To estimate the two unknown parameters in Equation (1) with noisy data, the algorithm starts with four data points (i.e., $k=4$) in the minor growth stage, where $s=1$. The slope at $k=4$, denoted as $b_4$, is estimated using the cumulative data points ${\mathbf{y}}_{1:4}$ and ${\mathbf{t}}_{1:4}$ and is subsequently monitored. The slope $b_k$ is continuously estimated and monitored as the data index is incremented by one.

When $k$ reaches 8, the algorithm can first evaluate the condition term to find the stage criterion. This is because the initial regression is performed at $k=4$, and five slope estimates (${\mathbf{b}}_{4:8}$) are required to check the condition: $b_5-b_4>0$, $b_6-b_5>0$, $b_7-b_6>0$, and $b_8-b_7>0$. The choice of using four consecutive increments is based on experience: using three increments resulted in instability, while using five increments delayed prediction.

This process is repeated until the condition term is satisfied. When the condition is met, $k$ is determined as the stage criterion marking the end of the minor growth stage. The corresponding value of $k$ is then set as $k_0$ for the next moderate growth stage ($s=2$). The entire procedure is repeated to find the stage criterion for $s=2$. Figure 2 illustrates the key concept of how to divide the stages. In Figure 2a, the slopes increase consecutively four times from $b_{13}$ to $b_{17}$, identifying $k=17$ as the stage criterion for the minor growth stage. For the moderate growth stage, with $k_0=17$, the slopes increase consecutively from $b_{23}$ to $b_{27}$, establishing $k=27$ as the stage criterion for the moderate growth stage and marking the end of Step 1.

Once the stages are divided, the remaining process for predicting RUL becomes much simpler. The goal of Step 2 is to predict RUL, and it proceeds as follows:

Step 2.

Predicting RUL

$$\begin{array}{ll}1.& \mathrm{Set} \mathrm{the} \mathrm{parameters}:\\ & -k_0 \mathrm{is} \mathrm{the} \mathrm{last} k \mathrm{at} s=2.\\ & -y_{thres} \mathrm{is} \mathrm{defined} \mathrm{by} \mathsf{a} \mathrm{user}.\\ 2.& \mathrm{For}k=k_0 \mathrm{to} n:\\ & -\mathrm{Estimate} a_k \mathrm{a}\mathrm{n}\mathrm{d} b_k \mathrm{from} \mathrm{Equation} \left(1\right) \mathrm{using} {\mathbf{y}}_{k-2:k} \mathrm{a}\mathrm{n}\mathrm{d} {\mathbf{t}}_{k-2:k}.\\ & -\mathrm{Solve} \mathrm{the} \mathrm{following} \mathrm{equation} \mathrm{for} t_{\mathrm{E}\mathrm{O}\mathrm{L}}:\\ & \hfill y_{thres}=a_k+b_k{t}_{\mathrm{E}\mathrm{O}\mathrm{L}}\hfill \\ & -\mathrm{Obtain} \mathrm{RUL}:\\ & \hfill {\mathrm{R}\mathrm{U}\mathrm{L}}_k=t_{\mathrm{E}\mathrm{O}\mathrm{L}}-t_k\hfill \end{array}$$

(5)

where $y_{thres}$ is the degradation threshold defined by the user. The process is based on linear regression, as in Step 2, but with some key differences: both model parameters, $a_k \mathrm{a}\mathrm{n}\mathrm{d} b_k$, are required, and they are estimated using only three data points, ${\mathbf{y}}_{k-2:k}$ and ${\mathbf{t}}_{k-2:k}$. It was found that using three data points is optimal for capturing the nonlinearity with the linear model in the accelerated growth stage. Using two data points to estimate the two unknown parameters is unstable, while using four data points yields lower accuracy than using three. Once the two parameters are estimated from the most recent three data points, the end of life (EOL)—the time at which degradation reaches the threshold $y_{thres}$—can be calculated using the same linear model as in Equation (1). Consequently, the RUL at $k$ is obtained by subtracting the current time $t_k$ from the EOL.

3.2. Example

The proposed method is explained using a specific example for better understanding, and Step 1 in Equation (4) is applied to divide the stages. First, the minor growth stage ($s=1$) starts with $k_0=1$. Figure 3 shows the regression results for this stage, where the black dots represent degradation data obtained every 100 cycles, and the blue lines depict the linear regression results using these data. Figure 3a shows the first regression result using four data points collected up to 300 cycles ($k=4$), while Figure 3b shows the final regression result in this stage, using 17 data points collected up to 1600 cycles ($k=17$). Fourteen regression results, obtained every 100 cycles from 300 to 1600 cycles, are displayed in Figure 3c. In the proposed method, the criterion for dividing the stages is based on the slope of the regression model, as illustrated in Figure 3d (in the figure, A and B indicate the slopes in Figure 3a and Figure 3b, respectively). As shown, the slope increases consecutively four times from 1200 to 1600 cycles, indicating that 1600 cycles ($k=17$) mark the end of the minor growth stage.

From the results in Figure 2, the moderate growth stage ($s=2$) begins at $k_0=17$, and the first regression is performed at 1900 cycles ($k=20$) after three additional data points are collected (see $k=k_0+3$ in Equation (4)). The first regression result for the moderate growth stage is shown in Figure 4a. In the figure, the gray dots represent data used to divide the minor growth stage, while only the black dots are used to establish the criteria for the newly started moderate growth stage. The regression result using the four data points is shown as the magenta line in the figure. Similarly, Figure 4b shows the final regression result for this stage ($k=27$), using 11 data points collected every 100 cycles from 1600 to 2600 cycles, while Figure 4c presents eight regression results obtained every 100 cycles from 1900 to 2600 cycles. Figure 4d depicts the slope of the regression models during the moderate growth stage, which increases consecutively four times from 2200 to 2600 cycles. With the moderate growth stage ending at 2600 cycles, Step 2 proceeds to predict the RUL of this example.

Step 1 ended at 2600 cycles ($k=27$), and Step 2 in Equation (5) starts with $k_0=27$, assuming $y_{thres}=0.05$. The first prediction in the accelerated growth stage starts at 2600 cycles, and the result is shown in Figure 5a. In the figure, the prediction result, denoted by the red line, is obtained using the most recent three data points, represented by the black dots. The predicted EOL, which is when the red line reaches the green horizontal line ($y_{thres}$), is 3260 cycles. Therefore, the RUL becomes 660 cycles ($=3260-2600$), as indicated by the red diamond marker at 2600 cycles in Figure 5d. For reference, the gray curves in Figure 5a–c represent the true degradation behavior, and the black diagonal line in Figure 5d represents the true RUL. The last prediction in this stage is performed at 3000 cycles ($k=31$), also using the most recent three data points, as shown in Figure 5b. The RUL prediction is indicated by the red diamond marker at 3000 cycles in Figure 5d. The red lines in Figure 5c show five prediction results performed every 100 cycles from 2600 to 3000 cycles. The final result obtained from the proposed method is the RUL prediction for the entire accelerated growth stage, as shown in Figure 5d.

4. Case Study for Comparison

In the previous section, the details of the proposed method were explained. This section presents the validation of the method in terms of computational cost and prediction accuracy. A crack growth problem, based on the Paris model [6], was used to generate degradation data, and an ANN [13] was employed as a typical data-driven prognostic method for comparison.

The simulations presented in this study were performed on a desktop equipped with an AMD Ryzen 9 5900X Processor (3.7 GHz, 12 cores) and 32 GB of RAM. The operating system was Windows 10 x64, and the simulations were executed using MATLAB R2018a.

4.1. Problem Definition

4.1.1. Data Generation

The Paris model, which describes crack growth behavior, is used to compare the RUL prediction results from the proposed method with those from a typical NN. This behavior is expressed by the following equation [6]:

$${\displaystyle \frac{d\alpha }{dN}}=C{\left(\Delta K\right)}^m, \Delta K=\Delta \sigma \sqrt{\pi \alpha }$$

(6)

where $\alpha$ is the crack size (representing the degradation level), $N$ is the number of cycles, $\Delta K$ is the stress intensity factor range related to $\Delta \sigma$ (the stress/loading condition), and $m$ and $C$ are the model parameters to be estimated. Since the information required to compare the two methods is degradation data, the data points, specifically the crack size ${\alpha }_k$ after $N_k$ cycles, are generated by integrating and solving Equation (6) for $\alpha$ as follows:

$${\alpha }_k={\left[N_{k}C\left(1-{\displaystyle \frac{m}{2}}\right){\left(\Delta \sigma \sqrt{\pi }\right)}^m+{\alpha }_1^{1-{\displaystyle \frac{m}{2}}}\right]}^{{\displaystyle \frac{2}{2-m}}}$$

(7)

where ${\alpha }_1$ is the initial crack size. The following parameter values are used to generate the data points: $m=3.8,\mathrm{C}=1.5\times {10}^{-10}, \Delta \sigma =75 \mathrm{M}\mathrm{P}\mathrm{a}$, and ${\alpha }_1=0.01 \mathrm{m}$. These values were selected based on material properties from a representative alloy but are used solely to generate synthetic data for this study. The parameter values themselves are not the focus and need not be considered, as the generated data are intended only for comparing the proposed method with a typical NN.

The black curve in Figure 6 represents the true degradation growth without measurement error, obtained using Equation (7) and the given parameter values, calculated at every 100 cycles. The final data are generated by adding uniformly distributed random noise to the true growth at two different levels to mimic measurement error in practice. The 31 black dots (generated every 100 cycles from 0 to 3000 cycles) in Figure 6a,b represent examples of the final data, with noise levels of ${1\times 10}^{-3} \mathrm{m}$ and ${2\times 10}^{-3} \mathrm{m}$, respectively. Although the noise level remains constant, the generated data may differ with each run due to the inherent randomness of the uniformly distributed noise, leading to variations in the prediction results. Therefore, a total of 100 data sets (with Figure 6 representing one of them) are generated for each noise level to compare the two methods. Lastly, the degradation threshold is assumed to be 0.05 m, represented by the green horizontal line in the figure.

4.1.2. ANN Architecture

ANNs can theoretically model any relationship between input and output by adjusting the number of hidden layers and nodes in these layers and using different combinations of activation functions. This is achieved by minimizing the difference between the model’s output and the actual data using optimization algorithms, such as the Levenberg–Marquardt algorithm, to estimate the weights and biases. A feed-forward neural network [13] with one hidden layer is commonly used to approximate degradation behavior, and the architecture used in this study is shown in Figure 7. The number of input and output nodes in each layer is one, representing time ($t$) as the input and degradation data ($y$) as the output. While the activation functions between layers are a common combination of the tangent sigmoid and pure linear functions, the number of hidden nodes in the hidden layer must be adjusted according to the specific application. Therefore, the optimal number of hidden nodes is determined by testing. The remaining conditions, including the hyperparameters required to implement the ANN, use the default settings in MATLAB, with more detailed information available in reference [22].

4.2. Results

First, 100 data sets with a noise level of ${1\times 10}^{-3} \mathrm{m}$ are used, and the prediction results obtained by applying the proposed method to these data are shown in Figure 8. Figure 8a presents the distribution of the first prediction cycle, denoted as $k_0$ in Step 2 of Equation (5), from 100 data sets, which is caused by the random noise in the data. In the figure, the most frequent first prediction cycle is 2600 cycles, and the majority of the data (82 out of 100) fall within the range of 2000 to 2700 cycles. The red lines in Figure 8b show the RUL predictions from the 100 data sets, while the black diagonal line represents the true RUL. It can be seen that as the cycle increases, the prediction results become closer to the true RUL. The result shown in Figure 5, which was used to explain the proposed method in the previous section, is 1 of these 100 predictions, where the prediction begins at 2600 cycles.

The same data sets are used to predict RUL using the ANN with different numbers of hidden nodes ($n_H$). The prediction is initiated at the same cycle used in Step 2 of the proposed method for each data set (see Figure 8a), and the RUL prediction results for $n_H=1, n_H=2$ and $n_H=3$ are presented in Figure 9a, Figure 9b, and Figure 9c, respectively. While RUL is consistently over-predicted for all data sets when $n_H=1$, the trend of RUL predictions becomes more erratic and fluctuates significantly when $n_H=3$, indicating increased uncertainty in the prediction. In contrast, the results using $n_H=2$ exhibit reduced fluctuation compared to using $n_H=3$ while avoiding the overprediction observed with $n_H=1.$ This balance between stability and accuracy suggests that $n_H=2$ provides the most reliable RUL predictions, and it is now fixed in the ANN model.

As shown in Figure 9a–c, the prediction results from the ANN are erratic, primarily due to variations in the initial settings for weights and biases in each run. To mitigate this uncertainty, the ANN is run 30 times for each of the 100 data sets, and the medians of these 30 results for each data set are presented in Figure 9d. This result is considerably more stable than the single-run outcome shown in Figure 9b. It should be noted that both Figure 9b,d represent results using $n_H=2$.

From this point forward, all results from the ANN will be presented as the median of 30 repeats using $n_H=2$. Figure 10 represents 1 of the 100 predictions shown in Figure 9d, which can be compared with the results from the proposed method in Figure 5. These figures demonstrate that the degradation and RUL predictions from the proposed method are consistently closer to the true values throughout all prediction cycles.

To compare 100 prediction results from the proposed method and the ANN, the relative errors at each cycle are calculated using the following equation:

$$e={\displaystyle \frac{\left|t_{\mathrm{E}\mathrm{O}\mathrm{L}}^{\mathrm{T}}-t_{\mathrm{E}\mathrm{O}\mathrm{L}}^{\mathrm{P}}\right|}{t_{\mathrm{E}\mathrm{O}\mathrm{L}}^{\mathrm{T}}}}\times 100$$

(8)

where $\mathrm{T}$ and $\mathrm{P}$ indicate true and predicted values, respectively. Since the magnitude of errors using RUL becomes sensitive as the remaining cycles decrease, the EOL is used for error calculation instead. The true EOL is 3045 cycles, and the prediction errors from the two methods are shown in Figure 11. In the figure, the solid red line represents the median error across available prediction results at each cycle, while the dashed and dotted lines represent the 5th and 95th percentiles of the error, respectively. It is clearly shown that the error from the proposed method in Figure 11a is much smaller and more precise than that from the ANN in Figure 11b.

The percentiles of the prediction errors in Figure 11 are listed in

Table 1. Comparison of prediction error percentiles for the proposed method and ANN using 100 data sets at a noise level of 1 mm.

Methods (Calculation Time)	Percentile			Error (%)
		2600 Cycles	2700 Cycles	2800 Cycles	2900 Cycles	3000 Cycles
Proposed method (52 s)	5	5.42	2.64	1.63	0.32	0.03
	50	11.20	7.74	4.27	2.16	0.37
	95	20.86	13.66	7.98	4.70	1.33
ANN (2815 s)	5	14.55	7.57	2.55	1.80	0.59
	50	N/A	17.50	7.76	3.71	1.39
	95	N/A	N/A	16.32	5.74	2.79

for cycles ranging from 2600 to 3000. While the proposed method provides results for all percentiles starting at 2600 cycles, the ANN fails to provide predictions at the 2600 and 2700 cycles, indicating that the ANN was unable to predict RUL for many data sets at those points. Additionally, the proposed method shows comparable accuracy at an earlier time step, as indicated by the comparison of percentiles: the results for the $k$-th cycle from the proposed method correspond to those of the $(k+1)$-th cycle from the ANN. Moreover, it is important to highlight that the computational cost for 100 data sets with the proposed method is significantly lower, taking only 52 s compared to 2815 s (approximately 47 min) for the ANN, achieving a 54-fold reduction in computational cost.

From the prediction results in Figure 11 and Table 1, using the data with a noise level of 1 mm in Figure 6a, the proposed method provides more accurate and consistent predictions than those of the ANN while achieving a 54-fold reduction in computational cost. The ANN, by contrast, struggles to provide results for some cycles and percentiles, as indicated by the missing values in Table 1. To assess whether similar results are obtained under a different noise level, a sensitivity analysis is conducted using data with a noise level of 2 mm in Figure 6b. First, the proposed method is applied, and Figure 12 shows the distribution of the first cycle for RUL prediction (Step 2 in Equation (5)). The figure displays a total count of 97 instead of 100, indicating that the proposed method was unable to predict the RUL for 3 out of 100 data sets. This occurs because the large noise in the data causes a delay in the prediction starting point (the algorithm stopped at Step 1 in Equation (4) for these three data sets). The prediction delay is evident when comparing Figure 12 with Figure 8a, as the distribution of the first cycle shifts to the right as the noise level increases. The majority of the data (81 out of 97) fall within the range of 2300 to 3000 cycles in Figure 12, which represents a 300-cycle delay compared to the range of 2000 to 2700 cycles for the data with a noise level of 1 mm in Figure 8a. This shift is a natural phenomenon as the noise level increases, and the proposed method remains robust and effective even at a noise level of 2 mm, compared to the ANN.

Figure 13 and

Table 2. Comparison of prediction error percentiles for the proposed method and ANN using 97 data sets at a noise level of 2 mm.

Methods (Calculation Time)	Percentile			Error (%)
		2600 Cycle	2700 Cycle	2800 Cycle	2900 Cycle	3000 Cycle
Proposed method (34 s)	5	0.61	2.60	0.25	0.09	0.11
	50	12.20	11.05	4.34	1.55	0.81
	95	66.10	33.23	14.52	7.93	2.31
ANN (2145 s)	5	17.76	6.62	5.52	1.60	0.41
	50	N/A	N/A	15.52	5.54	2.18
	95	N/A	N/A	N/A	13.20	3.97

demonstrate the validity of the proposed method by comparing it with the ANN. Figure 13a,b show the RUL prediction results from the proposed method and the ANN, respectively. Although the prediction uncertainty in Figure 13a appears larger than in Figure 13b, this is misleading, as the ANN fails to provide prediction results at the early stage. This is further supported by comparing the prediction errors in Figure 13c,d. The prediction errors using the proposed method in Figure 13c are significantly lower than those using the ANN in Figure 13d. Additionally, it is important to note that the medians of prediction errors from the proposed method remain consistent across two different noise levels, as shown by the solid red (2 mm noise level) and blue (1 mm noise level) lines in Figure 13c. In contrast, the median prediction error from the ANN using a noise level of 2 mm is significantly higher than that using a noise level of 1 mm, as shown in Figure 13d. These results indicate that the proposed method outperforms the ANN in terms of both accuracy and robustness across different noise levels. The specific values of the prediction errors are listed in Table 2. Consistently with the results for a noise level of 1 mm, the computational cost with the proposed method remains significantly lower, taking only 34 s compared to 2145 s for the ANN, achieving a 63-fold reduction in computational cost.

5. Conclusions

This paper introduces a novel method designed to provide a simple yet effective prognostic algorithm, particularly suitable for real-world situations where obtaining multiple sets of degradation data is very difficult. To achieve this, a stepwise linear regression method is proposed to approximate nonlinear degradation behavior. The proposed method demonstrates higher accuracy while reducing computational time by a factor of 50 to 60 compared to that of a typical ANN, which is one of the simplest models among its variations. Additionally, while the median prediction error from the ANN increases significantly when the noise level in the data doubles, the proposed method remains robust and consistent.

Although the prediction performance of the ANN in this study could potentially be improved by fine-tuning hyperparameters related to the training process, the effort required may outweigh the benefits. Furthermore, while the ANN may outperform the proposed method if multiple sets of degradation data are available, obtaining even a single degradation data set is highly challenging in practice. Thus, given the practical constraint of having only a single degradation data set, the proposed method is promising for real-world applications. It offers consistently high accuracy, simplicity of implementation, and most notably, computational efficiency compared to the ANN.

In real-world scenarios, however, degradation data are often obtained as sensor signals (e.g., vibration, current, and voltage) rather than being directly measured. To further enhance the practicality of the proposed method under this condition, a feature extraction step will be considered in future studies.