Evidential Domain Adaptation for Remaining Useful Life Prediction with Incomplete Degradation

IEEE TRANSA CTIONS ON INSTR UMENT A TION AND MEASUREMENT , V OL. X, NO. X, X X 1 Evidential Domain Adaptation for Remaining Useful Life Prediction with Incomplete De gradation Y ubo Hou , Mohamed Ragab , Y ucheng W ang , Min W u , Senior Member , IEEE, Abdulla Alseiari , Chee-K eong Kwoh , Xiaoli Li, F ellow , IEEE, Zhenghua Chen*, Senior Member , IEEE Abstract —Accurate Remaining Useful Life (R UL) pr ediction without labeled target domain data is a critical challenge, and domain adaptation (D A) has been widely adopted to address it by transferring knowledge fr om a labeled source domain to an unlabeled target domain. Despite its success, existing D A methods struggle significantly when faced with incomplete degradation trajectories in the target domain, particularly due to the absence of late degradation stages. This missing data introduces a key extrapolation challenge. When applied to such incomplete R UL prediction tasks, current D A methods encounter two primary limitations. First, most DA approaches primarily focus on global alignment, which can misaligns late degradation stage in the source domain with early degradation stage in the target domain. Second, due to varying operating conditions in RUL prediction, degradation patterns may differ even within the same degradation stage, resulting in different learned features. As a r esult, even if degradation stages are partially aligned, simple feature matching cannot fully align two domains. T o o vercome these limitations, we propose a novel evidential adaptation approach called EviAdapt, which le verages evidential learning to enhance domain adaptation. The method first segments the source and target domain data into distinct degradation stages based on degradation rate, enabling stage-wise alignment that ensur es samples from corresponding stages are accurately matched. T o address the second limitation, we intr o- duce an evidential uncertainty alignment technique that estimates uncertainty using e vidential lear ning and aligns the uncertainty across matched stages. The effecti veness of EviAdapt is validated through extensive experiments on the C-MAPSS, N-CMAPSS and PHM2010 datasets. Results show that our approach significantly outperforms state-of-the-art methods, demonstrating its potential for tackling incomplete degradation scenarios in RUL prediction. Our code is a vailable via https://github .com/keyplay/EviAdapt. Index T erms —Domain adaptation, remaining useful life pre- diction, e vidential learning, uncertainty , degradation stage. I . I N T RO D U C T I O N Prognostics and health management (PHM) of industrial systems and equipment play a crucial role in enhancing reliability , reducing maintenance costs, and improving safety and operational performance [1]. W ithin the PHM domain, Remaining Useful Life (R UL) prediction is a pi votal task for making well-informed maintenance decisions. Currently , v ari- ous approaches hav e been proposed for RUL prediction, which can be broadly categorized into three types: model-based Y ubo Hou, Y ucheng W ang, Min W u, Xiaoli Li and Zhenghua Chen are with Institute for Infocomm Research (I 2 R), Agency for Science, T echnology and Research (A*ST AR), Singapore. Y ubo Hou, Chee-Keong Kwoh and Xiaoli Li are with School of Computer Science and Engineering, Nanyang T echnological Univ ersity , Singapore. Mohamed Ragab and Abdulla Alseiari are with Propulsion and Space Research Center, T echnology Innov ation Institute, U AE. *Corresponding author: Zhenghua Chen Source Domain T arget Domain Global Alignment Stage Alignment patterns differ in same stage (a) (b) early stage misalignment Fig. 1: Comparison of existing solutions and the proposed solution and the proposed solution for incomplete degrada- tion domain adaptation in R UL prediction. (a) Under global alignment, early degradation in the tar get domain may be incorrectly aligned with late stage degradation in the source domain. (b) The patterns of the two domains may differ in the early stage, where the source domain experiences steady degradation, while the target domain de grades rapidly . approaches, data-dri ven approaches, and hybrid approaches. Specifically , model-based approaches rely on mathematical or physics-based models to describe the degradation behavior of a system, requiring a strong theoretical understanding [2]. Howe ver , as mechanical systems become increasingly com- plex, predicting R UL using model-based methods becomes exceedingly challenging. W ith the gro wing av ailability of data from deployed sensors, data-driven approaches [3], [4], [5] hav e gained popularity for R UL prediction. Despite the promise of data-driv en approaches, their success primarily hinges on the assumption of identically distributed data [6]. Howe ver , gi ven the dynamics of real-world en vi- ronments, models are typically trained under one operating condition and tested under another, leading to significant performance degradation due to domain shift. Furthermore, collecting annotated data in new operating conditions and IEEE TRANSA CTIONS ON INSTR UMENT A TION AND MEASUREMENT , V OL. X, NO. X, X X 2 retraining the model is both impractical and costly . Gi ven these challenges, accurately predicting R ULs under various working conditions with limited labeled data poses significant difficulties. T o address these obstacles, unsupervised domain adaptation (UD A) has emerged a promising technique that facilitates kno wledge transfer from a labeled source domain to a distinct yet related unlabeled target domain [7]. Recently , there has been an increased focus on UDA for R UL prediction task, aiming to learn domain-inv ariant features by reducing domain shift either through adversarial training [8], [9], [10] or minimizing the statistical distance between domains [11], [12], [13]. Although current D A methods have prov en effecti ve in addressing domain shift in R UL prediction, they are typically designed under the assumption that complete run-to-failure data is available in the target domain. Howe ver , in industrial systems, such data is often scarce due to safety concerns, as most systems are not allowed to operate until failure. As a result, the target domain may lack crucial data from the final degradation stage, leaving only data from the early degradation stages av ailable. When existing D A methods are applied in these incomplete settings, they often struggle for two main reasons. First, by not considering the progression of degradation stages, these methods typically achie ve global alignment, leading to misalignment across domains. As illus- trated in Fig. 1 (a), early degradation in the tar get domain may align with late degradation in the source domain, causing misalignment. Second, due to varying operating conditions, degradation patterns of different machines may differ ev en within the same degradation stage [14]. As sho wn in Fig. 1 (b), the patterns of the two domains differ in the early stage, where the source domain experiences steady degradation, while the target domain degrades rapidly . These differing patterns lead to different feature representations. Strict alignment of such features, even when degradation stages are properly aligned, can neg ativ ely impact overall alignment performance. T o address these challenges, we propose a no vel evidential adaptation approach, EviAdapt, for R UL prediction with in- complete degradation data in the tar get domain. T o address the misalignment of degradation stage, we introduce a novel approach that segments both source and target domains into distinct degradation stages based on degradation rate, followed by stage-wise alignment of samples. By aligning samples at same degradation stages, our method ensures accurate degra- dation alignment across domains, addressing a critical limita- tion in existing D A techniques for R UL prediction. T o resolve the issue of strict feature alignment, we propose an eviden- tial uncertainty alignment technique that focuses on aligning uncertainty levels between corresponding degradation stages rather than directly aligning features. Giv en that uncertainty is a second-order statistical equiv alent [15], the consistency in uncertainty lev els across corresponding degradation stages can effecti vely bridge differences between domains, thereby improving the alignment in R UL tasks. Through extensi ve experiments, we have thoroughly ev aluated the performance of our proposed EviAdapt method in accurately predicting the R UL of machines across div erse operating conditions. The main contrib utions of this study are listed as follo ws. • W e propose a stage-wise alignment strategy that aligns sample within the same degradation stage across different domains, effecti vely addressing the misalignment of de- gardation stage of incomplete lifec ycle data in the tar get domain. • W e introduce a novel evidential uncertainty alignment technique that aligns uncertainty le vels between corre- sponding degradation stages. • W e conduct extensi ve e xperiments on the C-MAPSS, N- CMAPSS and PHM2010 datasets to demonstrate that our EviAdapt approach significantly outperforms existing state-of-the-art methods in cross-domain RUL prediction, validating the effecti veness of our strategies in practical scenarios. I I . R E L A T E D W O R K S A. Unsupervised Domain Adaptation for R UL Pr ediction Unsupervised Domain Adaptation (UDA) for RUL predic- tion aims to mitigate the labeling cost by training neural networks to transfer knowledge from a labeled source domain to an unlabeled target domain. Existing UD A methods striv e to achiev e high performance on the tar get domain by minimizing the domain discrepancy . These methods can be cate gorized into two distinct branches, i.e., metric-based methods and adversarial-based methods. Metric-based methods enable networks to learn inv ariant features by enforcing metric constraints. Deep domain con- fusion (DDC) [16] employs the maximum mean discrepancy (MMD) to address the challenge of domain discrepancy . Correlation alignment (CORAL) [17] focused on minimizing the covariance shift between the feature distributions of the source and tar get domains. Adversarial-based methods employ domain discriminator networks to compel the feature e xtractor to acquire represen- tations that are in v ariant across domains. Domain adversarial neural network [10] utilized a rev erse gradient strategy to conduct adv ersarial training for both the domain classifier and the feature extractor . Adversarial domain adaption approach for remaining useful life prediction (AD AR UL) [8] utilized a con v entional GAN loss with flipped labels to learn domain- in v ariant features. Contrastive adversarial domain adaptation (CAD A) [18] incorporated a contrastiv e loss to persev ere target-specific information for R UL prediction. The abov e methods operate under the assumption that the target domain possesses complete run-to-failure data. In reality , acquiring such data in the target domain can be challenging, rendering these methods less effecti ve in prac- tical scenarios. In response to the condition of incomplete target data, Cons D ANN [19] utilized a consistency-based regularization term during alignment to mitigate the negati ve impact of missing information in the incomplete target domain dataset. In [20], the authors proposed a generative adversarial network to generate various types of full-cycle degradation data. Howe ver , the aforementioned methods treat training data as a homogeneous whole and overlook the discrepancies in characteristics across different degradation stages between the source and target domains, potentially leading to negati ve IEEE TRANSA CTIONS ON INSTR UMENT A TION AND MEASUREMENT , V OL. X, NO. X, X X 3 stage 1 weight transfer weight share Source Sample T arget Sample Pseudo Label A verage Quantile Prediction Uncertainty Uncertainty Quantile Prediction Degradation Classification Degradation Classification RUL Label stage 2 stage 1 stage 2 stage 3 Stage-wise Evidential Alignment Fig. 2: An overvie w of our proposed EviAdapt approach. EviAdapt comprises three main components: source encoder E S , target encoder E T and shared predictor R . E S and R are pretrained to learn the R UL distribution and its uncertainty of the source domain using evidential learning. During adaptation, source and target data are segmented into different degradation stages. Eventually , E T is trained to align the uncertainty of the same degradation stages between the source and target domains. transfer ef fects. In [21], the authors proposed an adversarial learning strategy combined with a source-domain instance- weighted degradation fusion scheme for similar degradation lev els. Ho wev er , this method is specifically designed for bearings and lacks generalizability , particularly when applied to aero engine data. Moreov er , all these methods ignore the uncertainty when learning the domain distribution. B. Uncertainty Quantification V arious methods hav e been de veloped to estimate un- certainty , including weights reparametrization [22], [23], dropout [24], and ensembling [25]. Although these methods are ef fecti ve, they are computationally demanding. Evidential learning can estimate uncertainty using single deterministic models. Several works [26], [27], [28] have been proposed for classification tasks. For regression tasks, deep evidential regression [29] was introduced. Ho we ver , a notable limitation is the reliance on Gaussian assumptions, which may restrict its application. While existing methods primarily rely on evidential learning for uncertainty quantification, our approach lev erages uncertainty alignment as a means to reduce the domain gap between source and target. I I I . M E T H O D O L O G Y A. Pr oblem F ormulation W e denote a source domain with N S labeled samples { X i S , y i S } N S i =1 and a target domain with N T unlabeled samples { X i T } N T i =1 , where X i S ∈ R M × L and X i T ∈ R M × L are both multiv ariate time series data consisting of M sensors and L time steps. y i S is the R UL label. W e aim to transfer knowledge from labeled source domain to unlabeled target domain and then improve the performance of R UL prediction on the target. T able I pro vides a summary of the notations employed in this paper . T ABLE I: List of notations. Notation Definition X S /X T source/target data y S source RUL label N S / N T number of source/target samples f S /f T source/target features E S /E T source/target encoder R predictor M number of sensors L sequence length B. Overview The overall structure of our proposed EviAdapt method is illustrated in Figure 2. EviAdapt comprises three main components: source encoder E S , target encoder E T and shared predictor R . First, we pretrain source encoder E S and predictor R to learn the RUL distribution and its uncertainty of the source domain using e vidential learning. Second we segment source and target data into different degradation stages. Eventually , we train tar get encoder E T to align the uncertainty of the same degradation stages between the source and target domains, leveraging the well-trained E S and R to facilitate the process. C. Evidential Pr etraining on Source Domain The first step in our proposed method is to pretrain source encoder and predictor . During this phase, the objectiv e is to train an evidential model using the labeled source domain data to learn the uncertainty of the source domain. Deep evidential regression [29] is a single deterministic forward-pass model that estimates uncertainty under the assumption of Gaussian distribution. This assumption limits the modeling applications. T o overcome this limitation, we emplo y an e vidential Bayesian quantile regression model as the RUL predictor [30]. Specif- ically , a source encoder E S and a predictor R are trained on IEEE TRANSA CTIONS ON INSTR UMENT A TION AND MEASUREMENT , V OL. X, NO. X, X X 4 source data X S . The source encoder extracts features from source data: f S = E S ( X S ) . The predictor estimates the quantile of the RUL v alue and its uncertainty based on the extracted features. Assuming that y S come from a Gaussian distribution pa- rameterized in the form of quantile regression, we place a Gaussian prior on the unkno wn mean and an Inv erse-Gamma prior on the unkno wn variance [30]: y S ∼ N ( µ + τ z , ω σ 2 z ) , µ ∼ N ( γ , σ 2 ν − 1 ) , σ 2 ∼ Γ − 1 ( α, β ) (1) where Γ( · ) is the Gamma function, γ ∈ R , ν > 0 , α > 1 , β > 0 , τ = 1 − 2 q q (1 − q ) and ω = 2 q (1 − q ) are quantile-specific constants from a specified quantile q , and z ∼ Exp  1 β / ( α − 1)  . T ogether , the distributions of µ and σ form the Normal- In v erse-Gamma (NIG) e vidential prior [29]: p ( µ, σ 2 | γ , ν, α, β ) = β α √ ν Γ( α ) √ 2 π σ 2 ( 1 σ 2 ) α +1 exp n − 2 β + ν ( γ − µ ) 2 2 σ 2 o (2) The objectiv e is to infer the parameters ( γ , ν, α, β ) of this evi- dential distribution. By placing the NIG prior on the likelihood parameters, we can deriv e an analytical solution that produces a Student-t predictive distribution. T o learn the parameters of the evidential distribution, we maximize the likelihood of the Student-t distribution. So we minimize the negati ve log- likelihood loss during training: L N LL = 1 2 log ( π ν ) − αl og (4 β (1 + ω z ν )) + ( α + 1 2 ) ( log ( y − γ − τ z ) 2 ν + 4 β (1 + ω z ν )) + log ( Γ( α ) Γ( α + 1 2 ) ) , (3) where z is the mean of z . A tilted loss is used as regularization term to penalize evidence of prediction errors: L R = max ( q ( y − γ ) , ( q − 1)( y − γ ))Φ , (4) where Φ = 2 ν + α + 1 /β is model confidence. Giv en a set of quantile value, the source encoder and the R UL predictor are optimized with the negativ e log-lik elihood loss and tilted loss: L = X q =1 ( L N LL + λ L R ) , (5) W e follow the settings described in [29], [30] and enforce the constraints on ( ν, α, β ) using a softplus activ ation function, with an additional +1 added to α to ensure α > 1 . A linear activ ation function is used for γ . It is worth noting that the R UL predictor estimates the parameters of the NIG distribution for each quantile value q , giv en a set of quantile values [ q 1 , q 2 , . . . , q k ]: ( γ S , ν S , α S , β S ) = R ( f S ) . (6) where γ S = [ γ q 1 S , . . . , γ q k S ] , ν S = [ ν q 1 S , . . . , ν q k S ] , α S = [ α q 1 S , . . . , α q k S ] and β S = [ β q 1 S , . . . , β q k S ] . And R UL value can be estimated by a v erage of γ S . 0 20 40 60 80 100 120 cycle number 0.35 0.40 0.45 0.50 0.55 0.60 health score sluggish moderate accelerated health index estimated trend Fig. 3: Three degradation stages categorized by the health index. D. Stage Se gmentation Giv en the pretrained source model, the focus lies in achiev- ing adaptation upon the unlabeled incomplete target data. Instead of aligning feature, we align uncertainty between source and target domains. By aligning the uncertainty , we can effecti vely align conditional distributions, leading to improved domain alignment. Ho wever , globally aligning uncertainty ov erlooks incomplete target domain data situation, leading to suboptimal alignment. T o address this issue, we propose a stage segmentation to ensure the alignment of corresponding degradation stages during the adaptation phase. Specifically , we classify the degradation of complete source domain and incomplete target domain into different stages based on their degradation speed. 1) Identifying Sour ce De gradation Stages: For the complete source domain, we cate gorize the data into three stages using the a v ailable R UL labels. T o accurately define the boundary for each stage, we calculate an engine’ s “health index” (HI) by forming a linear combination of the key sensor readings, following the methods described in [31], [32]. This curve effecti vely reflects the o verall e volution of the engine’ s health from normal operation to failure. After obtaining the HI, we examine how it changes ov er time for each engine, combined with the R UL information, to observe the varying de gradation rates in dif ferent sections of the lifecycle. Based on the health index, we empirically establish life-cycle ranges of (0, 33%), (33%, 85%), and (85%, 100%) to denote the sluggish, moder- ate, and accelerated degradation stages, respectiv ely [31], [32]. The reasons for choosing these three ranges are as follo ws: • Sluggish stage: The HI curve changes relati vely slowly here, indicating that the engine is in a more “healthy” state with a lo wer degradation rate. • Moderate stage: As operation time increases, the degrada- tion rate be gins to accelerate, though it has not yet entered a high-failure-risk phase. The HI value often shows a marked decline compared to the initial stage. • Accelerated stage: Near failure, the degradation rate in- creases significantly , and the HI curve typically exhibits a rapid drop, indicating a steep rise in failure risk. Figure 3 presents the health index of a single engine across its 120 operational cycles. The full cycle is di vided into three distinct stages: the sluggish stage (cycles 0 to 40, corre- IEEE TRANSA CTIONS ON INSTR UMENT A TION AND MEASUREMENT , V OL. X, NO. X, X X 5 sponding to 0–33%), the moderate stage (cycles 40 to 102, corresponding to 33–85%), and the accelerated stage (cycles 102 to 120, corresponding to 85–100%). The figure clearly shows that the trends in the health inde x vary significantly across these stages, highlighting the dif ferences in degradation behavior . 2) Identifying T ar get De gradation Stag es: For the incom- plete target domain, segmenting into the dif ferent degradation stages becomes a challenge due to the absence of labels. T o address this, we label the target data using the pretrained source model. Due to the incomplete data, it is highly possible that parts of the moderate and fully accelerated degradation stages are missing. Therefore, we classified the data into two stages using pseudo labels. W e empirically determine life- cycle ranges of (0, 70%) and (70%, 100%) to represent the sluggish and moderate stages, respectively . E. Stage-wise Evidential Alignment After segmentation of the degradation stages, we propose a stage-wise e vidential alignment loss L S E A to align uncertainty lev els between corresponding degradation stages rather than directly matching features. The two inputs to this loss are the parameters of the evidential distributions (NIG distrib utions in our case) from the corresponding degradation stages in the source and tar get domains. L S E A = − 2 X n =1 E [ k (( ν S , α S , β S ) n , ( ν T , α T , β T ) n )] . (7) ( ν , α , β ) n represents the parameters of the evidential distri- bution associated with de gradation stage n , where n = 1 corresponds to the sluggish stage and n = 2 to the moderate stage. The k ( · , · ) is a k ernel function to measure the distance between the evidential parameters from the source and the target domain. F . Overall Objective In the EviAdapt algorithm (Algorithm 1), the primary objectiv e is to fine-tune the target encoder E T for the R UL estimation of equipment. As indicated in Line 1, the algorithm begins with the pretraining of the source encoder E S and the predictor R using the source domain data ( X S , y S ) . After pretraining, the tar get domain data X T are passed through the pretrained source encoder E S , and the predictor R generates the pseudo labels ˆ y T (Line 2 and 3). Ne xt, the source domain data X S is segmented into three distinct degradation stages, based on the source labels y S (Line 4). Similarly , the target do- main data X T is segmented into two degradation stages, based on the pseudo labels ˆ y T (Line 5). Then the parameters of the evidential distribution ν S , α S , β S for the segmented source domain data are estimated by the predictor R according to the input samples (Line 6 and 7). During the training iteration, the parameters of the evidential distrib ution ν T , α T , β T for the segmented tar get domain data are estimated (Line 9 and 10). The source and target parameters of the evidential distribution serve as inputs to the stage-wise e vidential alignment loss L S E A , which is designed to align uncertainty lev els between corresponding degradation stages from the source and target domains, thereby training the target encoder E T for better adaptation. (Line 11 and 12). Finally , the well trained target encoder E T can be used in predicting R UL in the tar get domain. Algorithm 1: Our Proposed EviAdapt Input: Source domain: { X S , y S } , T arget domain: X T Output: T rained target encoder E T 1 E S , R ← pr etrain ( X S , y S ) 2 γ T , ν T , α T , β T ← R ( E S ( X T )) 3 ˆ y T ← a verage γ T ov er [ q 1 , q 2 , . . . , q k ] 4 X S 1 , X S 2 , X S 3 ← stage se gmentation for X S based on y S 5 X T 1 , X T 2 ← stage se gmentation for X T based on ˆ y T 6 γ S 1 , ν S 1 , α S 1 , β S 1 ← R ( E S ( X S 1 )) 7 γ S 2 , ν S 2 , α S 2 , β S 2 ← R ( E S ( X S 2 )) 8 while iteration do 9 γ T 1 , ν T 1 , α T 1 , β T 1 ← R ( E T ( X T 1 )) 10 γ T 2 , ν T 2 , α T 2 , β T 2 ← R ( E T ( X T 2 )) 11 loss ← L S E A (( ν S , α S , β S ) n , ( ν T , α T , β T ) n ) 12 θ ( t +1) E T = θ ( t ) E T − lr · ∇ θ E T loss //Update E T by minimizing loss through backpropagation 13 return E T I V . E X P E R I M E N T S A. Data Pr eparation W e employ the C-MAPSS, N-CMAPSS and PHM2010 benchmark datasets to e v aluate the performance of State-of- the-Art methods. • C-MAPSS : This dataset comprises operational data from four different turbofan engines, each functioning under unique operational conditions and exhibiting specific f ault modes, as detailed in T able II. It features readings from 21 sensors placed strate gically to monitor engine health. T o simulate the situation of incomplete target domain data, we excluded the final 40% of the run-to-failure data for each engine in the target training set but kept the full test data, then applied the preprocessing methodology from [18], resulting in a refined dataset that includes data from 14 selected sensors, with labels indicating the engines’ remaining useful life. • N-CMAPSS : This dataset [33] documents the run-to- failure trajectories of turbofan engines. Unlike the C-MAPSS dataset, which is confined to standard cruise phase conditions, N-CMAPSS includes simula- tions of entire flight cycl es—climb, cruise, and descent phases—thereby enhancing the fidelity of degradation modeling. These improv ements make N-CMAPSS better equipped to capture the complex dynamics present in real systems. For our experiments, we utilize datasets DS01, DS02, and DS03, which provide data from 20 channels, as outlined in T able II. Similarly , we excluded the final 40% of the run-to-failure data for each engine in the tar get IEEE TRANSA CTIONS ON INSTR UMENT A TION AND MEASUREMENT , V OL. X, NO. X, X X 6 T ABLE II: Details of benchmark datasets. Dataset C-MAPSS N-CMAPSS PHM2010 Sub-dataset FD001 FD002 FD003 FD004 DS01 DS02 DS03 C1 C4 C6 # Engine units for training 100 260 100 249 6 6 9 N A # Engine units for testing 100 259 100 248 4 3 6 N A # Complete Training samples 17731 48558 21220 56815 4881 5237 5532 69176 70270 69054 # Incomplete Training samples 9438 24607 11893 29434 2918 3131 3302 42162 41505 41432 # T esting samples 100 259 100 248 2717 1240 4225 28108 27671 27622 training set while retaining the full test data. The rest preprocessing follows the methodology described in [34]. • PHM2010 : The PHM2010 dataset [35] pro vides detailed records of cutting tool wear during machining processes. Due to the strong correlation between the R UL of the cutter and wear , and giv en that the dataset labels represent the wear depth, the objectiv e is to predict the wear depth of the cutters after each cut [36]. For our experiments, we leverage datasets C1, C4, and C6, as these records include labels for ev aluation. Each record, representing the continuous use of the same cutter, contains approx- imately 315 cutting instances with 7 sensor channels. Additionally , a sliding window has been employed to preprocess the data. As the data is relativ ely sparse while the dataset is large, a large stride of 1000 has been used to cover the broad range of the whole dataset. Meanwhile, the sliding window of 100 is adopted to reduce computational ov erhead. The data is normalized to scale all features to the range [0, 1]. For domain adaptation tasks, the first 60% of the cutting instances from the records are used as the tar get training set, while the remaining 40% are reserved for testing. B. Experimental Setting All experiments run five times and the average results are shown to prevent the effect of random initialization. For fair comparisons, we adopt the same LSTM [37] feature extractor for proposed method and baseline methods. Due to the variation between different dataset, different number of layers and dif ferent size of hidden states are selected for each dataset. T able III shows the detailed feature extractor parameters for each dataset. Additionally , we set batch size as 256, optimizer as Adam, learning rate as 5e-5 for the target encoder . W e use quantile values [0.25, 0.75] in our proposed method to compare with the state-of-the-art methods. Furthermore, we built and trained our model based on Pytorch and NVIDIA GeForce R TX A4000 GPU. W e adopt root mean square error (RMSE) and Score [18] as e valuation metrics for the C-MAPSS and N-CMAPSS datasets. Notably , as the Score function has parameters specifically designed for turbofan engine datasets and are not suitable for the PHM2010 dataset, only RMSE has been utilized for measurement. The lower the two indicators are, the better the model is. The RMSE metric is defined as follows: RM S E = v u u t 1 N N X i =1 ( y i − b y i ) 2 , (8) where b y i and y i represent the estimated RUL and true RUL respectiv ely . T ABLE III: Parameter setting for the LSTM feature e xtractor . Parameter C-MAPSS N-CMAPSS PHM2010 # of Layers 5 1 5 # of Hidden Size 32 64 32 Dropout 0.5 0.1 0.1 The RMSE metric assigns equal importance to both early and late R UL predictions. Howe ver , in prognostics applica- tions, late R UL predictions have more detrimental conse- quences for the systems. In order to address this concern, the Score metric is employed, which imposes a more se- vere penalty for late R UL predictions. The Score metric is expressed as follows: S cor e i =    e − c y i − y i 13 − 1; b y i < y i , e c y i − y i 10 − 1; b y i > y i , (9) S cor e = N X i =1 S cor e i . (10) C. Comparison with State-of-the-Art Methods W e compare the proposed EviAdapt with a range of state- of-the-art UD A methods, including conv entional domain adap- tation methods like DDC [38], Coral [17], AD AR UL [8], and CADA [18], as well as Cons D ANN [19] which is specifically designed for incomplete domain adaptation in the context of R UL prediction. Further , the results of source only (Source) are also compared. Due to the use of evidential learning for pre-training in the proposed method, the last layer of our predictor dif fers from other domain adaptation methods. Therefore, we present two sets of source-only results. The first set, Source-RMSE, uses a predictor trained with the RMSE loss. The second set, Source-EVI, employs an evidential predictor trained with negati ve log-likelihood loss and tilted loss. T able IV shows the RMSE results and T able V shows the Score results respectively in 12 cross-domain scenarios for R UL prediction on C-MAPSS dataset. From the results, we observe that ev en though Source-EVI performs significantly worse than Source-RMSE, EviAdapt achiev es the best perfor- mance in the 8 scenarios for RMSE and in the 9 scenarios for Score. Notably , EviAdapt improv es the a verage performance by ov er 5% in RMSE and 16% in Score compared to the second-best method. Similarly , T able VI sho ws the RMSE and T able VII shows the Score results respectiv ely in 6 cross-domain scenarios for R UL prediction on N-CMAPSS dataset. From the results, we observe that EviAdapt achiev es the best performance across IEEE TRANSA CTIONS ON INSTR UMENT A TION AND MEASUREMENT , V OL. X, NO. X, X X 7 T ABLE IV: Comparison of the proposed EviAdapt against benchmark approaches on C-MAPSS (RMSE). Note that F1 is short for FD001, and F1 → F2 refers to the scenario where FD001 is the source domain and FD002 is the target domain. Bold indicates the best result, and underline indicates the second-best result. Methods F1 → F2 F1 → F3 F1 → F4 F2 → F1 F2 → F3 F2 → F4 F3 → F1 F3 → F2 F3 → F4 F4 → F1 F4 → F2 F4 → F3 A vg. Source-RMSE 55.35 60.52 55.98 52.29 53.60 54.86 41.94 44.37 40.57 46.09 51.46 49.10 50.51 DDC 40.00 40.06 43.98 38.83 48.24 43.06 41.55 40.65 43.68 40.51 39.61 38.16 41.53 Deep Coral 36.90 41.75 45.21 35.88 41.16 43.85 36.16 37.08 37.78 36.11 36.80 35.96 38.72 AD AR UL 44.73 54.37 36.93 48.41 48.84 49.38 34.19 36.19 39.98 28.05 33.76 37.43 41.02 CAD A 45.24 54.57 38.73 49.60 49.07 48.74 41.23 39.41 40.55 32.22 39.65 38.17 43.10 Cons DANN 28.52 34.65 29.95 22.86 28.77 29.63 21.02 24.32 27.44 21.21 25.40 23.67 26.45 Source-EVI 59.00 65.83 56.42 56.45 57.77 55.30 54.84 55.48 48.53 46.14 51.34 52.43 54.96 EviAdapt 29.95 31.13 33.72 21.64 26.03 28.83 19.19 19.92 28.53 23.67 18.59 20.58 25.00 T ABLE V: Comparison of the proposed EviAdapt against benchmark approaches on C-MAPSS (Score). Note that F1 is short for FD001, and F1 → F2 refers to the scenario where FD001 is the source domain and FD002 is the target domain. Bold indicates the best result, and underline indicates the second-best result. Methods F1 → F2 F1 → F3 F1 → F4 F2 → F1 F2 → F3 F2 → F4 F3 → F1 F3 → F2 F3 → F4 F4 → F1 F4 → F2 F4 → F3 A vg. Source-RMSE 75683 46833 72564 16834 23400 67866 6740 25030 19995 11831 46169 13971 35576 DDC 22208 6368 21318 4930 6514 16958 6140 22422 21397 5939 23050 4238 13457 Deep Coral 11782 6723 18697 2986 6802 16098 3001 11538 9907 3033 11495 3416 8790 AD AR UL 36618 27370 19401 11467 15171 44955 3862 16706 19380 3207 8913 6032 17757 CAD A 37815 27805 24187 13201 15690 47750 6676 22293 19964 4907 13183 6560 20003 Cons DANN 5817 4614 7028 1164 2017 5948 1076 6019 5811 947 4529 1343 3859 Source-EVI 109644 78206 91815 26023 34167 93708 20639 76765 42292 13700 47593 20632 54599 EviAdapt 4976 2435 7456 908 1404 5227 1178 3974 5146 1361 3980 809 3238 T ABLE VI: Comparison of the proposed EviAdapt against benchmark approaches on N-CMAPSS (RMSE). Bold indicates the best result, and underline indicates the second-best result. Methods DS01 → DS02 DS01 → DS03 DS02 → DS01 DS02 → DS03 DS03 → DS01 DS03 → DS02 A vg. Source-RMSE 21.89 28.23 33.29 25.64 40.37 26.50 29.32 DDC 19.23 21.12 21.05 15.86 24.75 17.78 19.96 Deep Coral 14.06 15.75 21.34 16.37 24.42 16.21 18.03 AD AR UL 9.97 13.50 17.51 14.20 14.07 9.18 13.07 CAD A 10.05 14.12 16.07 13.14 18.88 12.90 14.19 Cons DANN 10.03 11.91 16.52 12.96 17.48 10.25 13.19 Source-EVI 23.64 27.89 30.68 24.25 40.23 28.11 29.14 EviAdapt 9.23 9.87 9.69 9.74 9.87 5.81 9.04 T ABLE VII: Comparison of the proposed EviAdapt against benchmark approaches on N-CMAPSS (Score). Bold indicates the best result, and underline indicates the second-best result. Methods DS01 → DS02 DS01 → DS03 DS02 → DS01 DS02 → DS03 DS03 → DS01 DS03 → DS02 A vg. Source-RMSE 7202 58571 62615 49246 118257 11480 51228 DDC 5767 28992 16178 14566 24616 5325 15908 Deep Coral 2880 13444 17037 15810 22697 3804 12613 AD AR UL 1579 11370 11924 11135 5908 1441 7226 CAD A 1632 12260 8568 9814 13335 2459 8011 Cons DANN 1715 8216 9634 9675 10183 1783 6868 Source-EVI 8662 55812 53767 45930 119890 14051 49685 EviAdapt 1515 5742 3804 7553 4205 979 3966 IEEE TRANSA CTIONS ON INSTR UMENT A TION AND MEASUREMENT , V OL. X, NO. X, X X 8 T ABLE VIII: Comparison of the proposed EviAdapt against benchmark approaches on PHM2010 (RMSE). Bold indicates the best result, and underline indicates the second-best result. Methods C1 → C4 C1 → C6 C4 → C1 C4 → C6 C6 → C1 C6 → C4 A vg. Source-RMSE 0.209 0.111 0.311 0.201 0.280 0.312 0.237 DDC 0.176 0.272 0.223 0.340 0.127 0.170 0.218 Deep Coral 0.175 0.194 0.155 0.150 0.116 0.169 0.160 AD AR UL 0.216 0.114 0.313 0.202 0.300 0.323 0.245 CAD A 0.209 0.111 0.311 0.201 0.280 0.312 0.237 Cons DANN 0.173 0.106 0.109 0.118 0.112 0.233 0.142 Source-EVI 0.219 0.112 0.200 0.129 0.259 0.300 0.203 EviAdapt 0.185 0.100 0.105 0.108 0.145 0.192 0.139 T ABLE IX: Ablation study for the proposed EviAdapt on C-MAPSS (RMSE). Alignment Scope Alignment T ype F1 → F2 F1 → F3 F1 → F4 F2 → F1 F2 → F3 F2 → F4 F3 → F1 F3 → F2 F3 → F4 F4 → F1 F4 → F2 F4 → F3 A vg. G F ea 25.47 27.47 34.36 26.11 34.98 36.39 26.87 32.80 33.67 25.42 26.74 25.00 29.61 G U nc 29.61 32.91 36.00 23.81 31.53 31.50 24.35 24.72 26.45 23.08 24.12 23.83 27.66 S U nc 29.95 31.13 33.72 21.64 26.03 28.83 19.19 19.92 28.53 23.67 18.59 20.58 25.00 all scenarios with regards to both RMSE and Score. Notably , EviAdapt improves the average performance by over 30% in RMSE and 42% in Score compared to the second-best method. Moreov er , T able VIII shows the RMSE results in 6 cross- domain scenarios for wear depth prediction on PHM2010 dataset. From the results, we observe that EviAdapt achiev es the best performance in the 3 scenarios. Notably , EviAdapt improv es the av erage performance by over 2% in RMSE compared to the second-best method. These consistent superior performances demonstrate Evi- Adapt’ s ability to align uncertainty of the same degradation stage ef fectiv ely , leading to significant advancements in R UL prediction across dif ferent domains. D. Ablation Study T o validate the contribution of key components, we con- ducted an ablation study on our proposed EviAdapt using the C-MAPSS dataset. W e deriv ed two v ariants of EviAdapt based on different combinations of alignment scope and alignment type. The alignment scope refers to the range o ver which the alignment is applied and includes two kinds: global alignment (“ G ”), which considers the entire dataset, and same degrada- tion stage alignment (“ S ”), which focuses on aligning data within the same de gradation stage. The alignment type refers to the specific aspect of the data being aligned and includes two kinds: alignment by feature (“ F ea ”) and alignment by uncertainty (“ U nc ”). It is worth noting that the combination (“ S ”, “ U nc ”) corresponds to our proposed method. T able IX and T able X presents the comparativ e outcomes between EviAdapt and its variants. Our observations reveal that alignment by uncertainty surpasses alignment by feature, showing an improvement of 65% in terms of average Score. Furthermore, same degradation stage alignment outperforms global alignment, with a maximum improvement of 67% in terms of average Score. These results underscore the effecti ve- ness of stage-wise alignment and alignment by uncertainty . 0 2 4 6 8 10 12 14 16 DS01→DS02 DS01→DS03 DS02→DS01 DS02→DS03 DS03→DS01 DS03→DS02 0.25, 0.5 0.25, 0.75 0.5, 0.75 Fig. 4: The sensiti vity analysis for different set of quantile values on N-CMAPSS (RMSE). E. Sensitivity Analysis W e conducted a sensitivity analysis for different set of quantile values q on N-CMAPSS dataset to in vestigate the impact of evidential learning on the proposed method. Sev eral experiments were carried out using various set of values, including [0.25, 0.5], [0.5, 0.75] and [0.25, 0.75]. Figure 4 and Figure 5 illustrate that the proposed method demonstrates that the optimal set of quantile values varies across different domains. Specifically , the quantiles [0.25, 0.5] yields the best performance for DS01 → DS02 in terms of RMSE and Score, while [0.25, 0.5] achiev es the best performance for DS02 → DS01 and DS02 → DS03 in terms of Score. The impact of the quantile v alues on domain adaptation varies depending on the source and target domains. The potential reason could be attrib uted to different sets of quantiles capture v arying domain information. F . V isualization of F eature Distrib ution T o showcase the effecti veness of the proposed method, we employed t-SNE to visualize the latent features in the reduced-dimensional space both before and after adaptation on the N-CMAPSS dataset for the DS01 → DS03 scenario. As illustrated in Fig.6, prior to adaptation, a considerable IEEE TRANSA CTIONS ON INSTR UMENT A TION AND MEASUREMENT , V OL. X, NO. X, X X 9 T ABLE X: Ablation study for the proposed EviAdapt on C-MAPSS (Score). Alignment Scope Alignment T ype F1 → F2 F1 → F3 F1 → F4 F2 → F1 F2 → F3 F2 → F4 F3 → F1 F3 → F2 F3 → F4 F4 → F1 F4 → F2 F4 → F3 A vg. G F ea 3953 2344 35593 7522 11296 59736 27492 94580 80814 3577 8686 2317 28159 G U nc 6224 2246 8268 3832 7880 29588 6345 26558 15633 891 2608 1093 9264 S U nc 4976 2435 7456 908 1404 5227 1178 3974 5146 1361 3980 809 3238 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 DS01 →DS0 2 DS01 →DS0 3 DS02 → DS01 DS02 →DS0 3 DS03 →DS0 1 DS03 →DS0 2 0.25, 0.5 0.25, 0.75 0.5, 0 .75 Fig. 5: The sensiti vity analysis for different set of quantile values on N-CMAPSS (Score). DS01 DS03 Sour ce Domain T ar get Domain DS01 DS03 Sour ce Domain T ar get Domain Fig. 6: Feature distribution analysis. Up: before adaptation. Down: after adaptation. portion of the source samples are positioned far from the target distribution, emphasizing the domain gap. In contrast, post-adaptation visualization demonstrates that the source and target distributions are closely aligned. Collecti vely , these visualizations clearly indicate that our approach successfully minimizes the disparity between the source and tar get feature distributions. V . C O N C L U S I O N In this paper , we found that most existing domain adaptation methods fail under incomplete target domains. T o address this, we propose a novel approach called EviAdapt for unsupervised domain adaptation in R UL prediction tasks. Unlike pre vious methods that overlook the misalignment of degradation stage and inherent uncertainties in R UL tasks, EviAdapt aligns the uncertainty within the same degradation stage by the proposed stage-wise evidential alignment technique, thereby highlighting the limitations of e xisting methods. Through extensi v e experiments, our results demonstrate the remarkable performance of EviAdapt, surpassing state-of-the-art methods on the C-MAPSS, N-CMAPSS, and PHM2010 datasets, with av erage impro vements of 16%, 42%, and 2%, respecti vely . 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