Using long-term condition monitoring data with non-Gaussian noise for online diagnostics

Using long-term condition monitoring data with non-Gaussian noise for online diagnostics Hamid Shiri ∗a, Pawel Zimroza, Jacek Wodeckia, Agnieszka Wyłomańskab, Radosław Zimroza, Krzysztof Szabatc aFaculty of Geoengineering, Mining and Geology, Wroclaw University of Science and Technology, Na Grobli 15, 50-421 Wroclaw, Poland bFaculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wyspianskiego 27, 50-370 Wroclaw, Poland cFaculty of Electrical Engineering, Department of Electrical Machines, Drives and Measurements, Wroclaw University of Science and Technology, Wyspianskiego 27, 50-370 Wroclaw, Poland Abstract The number of timely diagnoses based on condition monitoring data is increasing with the growing usage of monitoring systems. In most of the methods used in these systems, a pre-established fault detection threshold is needed, while there are no specific limit values or thresholds in many cases, especially when the machine is unique. Also, in most actual applications, due to the kind of process and harsh environment, the noise inherent in the observed process exhibits non-Gaussian characteristics, making it a challenging task for diagnostics based on condition monitoring (CM) data. Therefore, this paper introduced a robust methodology based on the switching maximum correntropy Kalman filter (SMCKF) to address the mentioned problems (threshold and online diagnostics in the presence of non-Gaussian noise by using CM data). This approach uses multiple dynamic system models to explain different degradation stages, utilizing robust Bayesian estimation. As this approach is based on dynamic behavior, a threshold for diagnostics is no longer needed. Ultimately, the proposed approach is applied to the online diagnosis of simulated and actual data sets. The results of both simulated and real data sets prove the method’s efficacy. Keywords: diagnostics, condition monitoring, fault detection, Kalman filter, robust methods, non-Gaussian noise, dynamic linear model. 1. Introduction Condition-based maintenance (CBM) has become increasingly popular in the industry with the development of condition monitoring systems. CBM software is developed to assess the machine’s condition by collecting massive amounts of data during the operation of machines. Using long-term condition monitoring data is crucial in diagnostics and prognostics. Many methods published in recent years for CBM can be categorized into four main groups [1]: stochastic-based [2–4], machine learning-based [5–7], physics modelbased [8,9], and hybrid methods [10,11]. Machine learning approach such as neural networks [12–19], have strong potential to be used for diagnostics and prognostics areas. Nevertheless, they need a large number of data for the model to be trained. Furthermore, by changing the domain of working or changing the machine, they need to retrain again, which is often not possible in actual application. On the other hand, physics model-based methods use a mathematical representation of the degradation process based on the failure mechanisms or the damage principle [1], which needs less training data. However, for some complex mechanical systems, it is hard to understand the physics of damage, which significantly restricts the range of ∗Corresponding author. Email address: [email protected] (Hamid Shiri ∗) Preprint submitted to Mechanical Systems and Signal Processing November 12, 2025 application of mechanism modeling-based approaches. Compared to the previous methods, stochastic-based approaches (also named empirical model-based approaches) do not need a large number of degradation data and mechanical knowledge of equipment (physics or mechanical principles). Statistical model-based approaches work according to establishing statistical models based on empirical knowledge and generally try to model a degradation process using stochastic process models under a probabilistic method. Also, the stochastic-based model has an enormous potential to consider degradation uncertainty. Most of the mentioned methods work based on a pre-established fault detection threshold that is often provided by the manufacturer; a threshold corresponding to a change from ”Good Condition” (healthy stage) to ”Warning” (degradation stage) and from ”Warning” to ”Alarm” stage (critical stage). Unfortunately, we do not know about limit values or the desired lifetime in many cases, especially when the machine is unique. Also, it should be noted that most of these thresholds are introduced by manufacturing industries, and they are defined based on particular working and environmental conditions standards that may not be valid when the conditions change. In addition, this task will be more complicated when the machine works in a harsh area, where the inherent noise exhibit non-Gaussian characteristics. This may occur for several reasons. Wind inflows are one of the possible sources of impulsive noise in wind turbine [20–22], the falling ore on devices in the mining environment is another source that can be mentioned [23,24]. The Kalman filter (KF) is a well-known, powerful tool in engineering applications such as diagnostics [25– 29] and prognostics [30–33]. However, the KF gives the optimal solution when the process and measurement noise have a Gaussian distribution, whereas if noise has a different distribution, KF produces a sub-optimal solution. It is because only the second-order measurement information is used. Many methods have been developed to handle this drawback of the KF. For example, some researchers tried to develop filters for the system by considering its random impulsive behavior producing outliers in the data. In this family of methods, noise distribution is assumed to have non-Gaussian characteristics, which is fulfilled for distributions such as Student’s t-distribution, stable distributions, etc. [34,35]. However, finding analytic solutions for these distributions is complicated and needs high computational effort. Therefore, using them in practical applications is more complicated than using KF. Another approach that tries to handle non-Gaussian noise is its approximation with a Gaussian mixture distribution [36–39]. The high computational cost is the main drawback of these approaches. The third family of methods that tries to handle non-Gaussian noise is sequential Monte Carlo (SMC) sampling, which has the potential to approximate any distribution [40]. Particle filter (PF) is the most famous example [37,41]. The unscented Kalman filter (UKF) is similar to the PF but uses different sampling techniques [42]. As with the previous methods, the computational cost is the biggest drawback of this approach. To reduce the aforementioned disadvantages, Izanloo et al. [43] presented the correntropy filter (C-Filter) for state estimation and indicated that the C-Filter has better performance than KF in the presence of non-Gaussian noise, thanks to using correntropy criterion, which utilizes higher-order measurement signal information. Also, a new version of KF based maximum correntropy criterion was developed [43–46], which is robust in the presence of noise with non-Gaussian characteristics. However, the most significant limitation of KF and maximum correntropy criterion Kalman filter (MCKF) used for diagnostics and prognostics is that the degradation process must be time-invariant. However, in practical application, the degradation process is composed of several components that change over time, see Section 2.1. Therefore, using a single model to describe the whole process would make incorrect state estimations and cause divergence or fluctuation. In lieu of this, the switching Kalman filter (SKF) is proposed to handle the mentioned issue when the dynamic behavior of a process changes during the time and, if not possible, to use an unique model. The SKF is frequently employed to track multiple moving targets [47], econometric time series [48] and meteorologic observations [49]. Likewise the SKF is used in maintenance areas; for instance, to detect sensor and actuator faults [50]. In [51], SKF is employed to find changing points in nonlinear stochastic systems. In [33,52] SKF is used to diagnostics and prognosis of bearing. However, most of these methods assumed the noise distribution is Gaussian. The use of standard SKF for online condition monitoring in most actual applications in the presence of non-Gaussian noise may be problematic. Furthermore, our previous work [53] focused on identifying and analyzing the degradation process using the real benchmark data sets (available in the prognostics area), which confirms the hypothesis of noise with non-Gaussian characteristics even in the laboratory data sets. To address the above problems, this 2 paper introduced a novel, robust approach for online diagnostics without using a threshold in the presence of impulsive noise based on SKF and MCKF. The proposed method improves the accuracy and effectiveness of online diagnostics based on SKF, particularly in harsh environments, when the noise inherent in the observed process exhibits non-Gaussian characteristics. The main contributions of this paper are as follows: •A model of health index (HI) data was proposed as three segments sequence with non-Gaussian noise (Student’s t-distributed) to describe the degradation process, which can be used to simulate the artificial data set. •Switching maximum correntropy Kalman filter regime is derived, which is a robust version of SKF in the presence of noise with non-Gaussian characteristics. •An online diagnostics method based on SMCKF is proposed, and extensive experiments are carried out on the simulated data set in the presence of Gaussian and non-Gaussian noise and FEMTO and wind turbine datasets to verify its effectiveness. The paper is structured as follows: after the introduction, in Section 2, the critical aspects of the processing methods are described in theory. Then, in Section 3, the results for simulated data are presented. Finally, the results of applying the proposed approach to two benchmark data sets are presented in Section 4with an indication of all intermediate steps. Conclusions are formed in Section 5. 2. Methodology The data is acquired using sensors and goes to the SCADA system. After preprocessing in the SCADA system, the SCADA data is used as a HI for diagnostics and prognosis (SCADA data is usually a statistic or frequency domain feature of the sensor’s data that is gathered in a specific duration of time). At the end, by using the SMCKF, the HI is segmented into three stages (healthy, degradation, and critical). More details about the algorithm are presented in the following subsections. 2.1. Long term data modeling In the PHM community, one widespread assumption is to consider the HI value following specific trends during the degradation process. Therefore, based on the application, the researchers use several conventional models to describe the global trend of the degradation process, see Fig. 1. For instance, employing a linear trend is a common assumption to describe the drilling degradation process [1,54], see panel (b) in Fig. 1. On the other hand, the exponential degradation is used to express the battery degradation process [55–58]. For the degradation process of other elements such as bearings, gears, and shafts, it is popular to use models [59,60] like in (d), (e), (f) panels in Fig. 1. In this work, it is assumed that the degradation trend is divided into three stages (see panel (f) in Fig. 1). Considering some assumptions, some analogies between this model and the three steps of crack growth can be made. Continuous damage and fracture mechanics are investigated in detail using physics-based modeling, considering the dependence on the current stage [61]. Stage 1 considers there is no degradation; it can be compared to the healthy area or the initiation of a crack in the microscopic domain; the length of this stage is approximately unpredictable. Stage 2 represents linear degradation growth with the potential to compose different kinds of Gaussian and non-Gaussian noise plus fluctuations in deterministic trend (self-healing phenomena [54]); this step can be compared to the fatigue crack propagation. The length of this stage is varied. It depends on many parameters, such as load, temperature, etc. Stage 3 is a rapid exponential (or polynomial) degradation growth corresponding to a fast fracture area; see Fig. 2. The length of this stage is short. Such a model is frequently used in the literature [59,60]. To bring this model closer to reality, a model based on the following assumptions is presented. In our previous paper [53], we demonstrate that the degradation process data are non-homogeneous (their characteristics change in time), may include some random components, and may exhibit non-Gaussian (heavy-tailed) behavior. The nature of the deterministic component and the scale of the data is different for different stages. However, a transition might exist for random components’ distribution - from Gaussian to strongly non-Gaussian. In Fig. 3, we demonstrate the above characteristics and their correspondence with 3 the degradation process. Moreover, in Table 1, we show the main characteristics of the data corresponding to different stages that result from the preliminary analysis of the actual data also, for more details about the model, we refer to our previous work [53]. Figure 1: Different types of HI variations and degradation models: (a) good condition (constant trend), (b) good to gradual wear (linear trend), (c) exponential trend, (d) good to accelerated wear (linear trend), (e) good to accelerated wear (exponential trend), (f) three stages model (good, linear progress and exponential progress of degradation) [53]. (a) (b) Figure 2: Multi stage degradation model, (a) three steps fatigue crack growth [62], (b) three stages degradation model [52]. 4 Figure 3: The proposed long-term degradation model with three stages [53]. Table 1: Main characteristics of the data for three stages indicated in Fig. 3[53]. stage 1 stage 2 stage 3 Trend constant linear exponential or polynomial Scale nearly constant linearly growing exp.or polynomial growing Autodependence of the noise white / colored white / colored white / colored Distribution of the noise Gaussian /non-Gaussian Gaussian /non-Gaussian Gaussian/ non-Gaussian 2.2. Diagnostics In this subsection, we focus on introducing the methodology for online diagnostics of the machines with long-term condition monitoring data. 2.2.1. Kalman filter The KF is a kind of Bayesian filter used for recursive state estimation of a dynamic system by minimizing the mean squared error in the presence of process and measurement noise. The EKF works based on follows: xt=Atxt−1+qt, yt=Htxt+mt,(1) where xtis the actual state at time tand ytis its observation, the matrix Atdescribes the state-transition model, qtis the process noise, Htis the observation matrix, which maps the actual state space into the observed space, and mtis the observation noise. The noise terms qtand mtare assumed to be normally distributed: qt∼ N(0, Qt),mt∼ N(0, Mt). In the literature, several approaches can be employed to extract KF formulation: for instance, Bayesian rule, maximum posterior approaches (MAP), orthogonal principle, and weighted least square method (WLS). To describe the state estimation procedure given the observations ytwe need to introduce the notation of ˆxt|tand ˆxt|t−1being the posterior and the prior estimates of the state xtgiven the observations till the time tand t−1, respectively. The cost function in WLS, defined as a sum of two squared quadratic norms [43], which combines the measurement (first term) and the prior estimate (second term) in finding the optimal posterior estimate of the xt, is given as follows: J=1 2∥yt−Htxt∥2 M−1 t+1 2 xt−Atˆxt−1|t−1  2 P−1 t|t−1 ,(2) where −1in the superscript denotes matrix inversion, quadratic norm of a vector udefined with a weights matrix Wis the quadratic form obtained with the formula (denoting matrix transposition with Tin the 5 superscript): ∥u∥W=√uTWu (3) and Pt|t−1is the prior estimate covariance matrix: Pt|t−1=E[(xt−ˆxt|t−1)(xt−ˆxt|t−1)T].(4) The posterior estimate covariance matrix Pt|tis expressed in an analogous way as above: Pt|t=E[(xt−ˆxt|t)(xt−ˆxt|t)T].(5) The KF formulation can be derived by minimizing J from Eq. (2) with respect to xt, namely the posterior estimator of the state of the model is defined as: ˆxt|t= argmin xt{J}.(6) At the same time the covariance matrix is also estimated. The resulting recursive procedure is described with the following equations which are usually divided into two steps, namely the prediction step: ˆxt|t−1=Atˆxt−1|t−1, ˆ Pt|t−1=Atˆ Pt−1|t−1AtT+Qt,(7) and the update step: ˆxt|t= ˆxt|t−1+Kt(yt−Htˆxt|t−1), ˆ Pt|t= (I−KtHt)ˆ Pt|t−1(I−KtHt)T+KtMtKtT.(8) The term Ktin the above equations is called Kalman gain and for the standard KF its optimal value is given by the formula: Kt=ˆ Pt|t−1HT t(Htˆ Pt|t−1HT t+Mt)−1.(9) As we mentioned before, the KF gives the optimal solution when both the process noise and the measurement noise have a Gaussian distribution. For other distributions, the KF produces a sub-optimal solution, which is because only the second-order measurement information is used. In the following subsection, the MCKF is derived to handle non-Gaussian noise based on the following references [43–46]. 2.2.2. Maximum correntropy Kalman filter In the following, for the model expressed with Eq. (1) another form of an objective function is introduced [43]. Based on the maximum correntropy criterion it has the form: JMC =Gσ(∥yt−Htxt∥M−1 t) + Gσ( xt−Atˆxt−1|t−1 P−1 t|t−1 ),(10) where Gσ(u) = exp(−u2 2σ2)and σis kernel size. This function used instead of a previous cost function makes the estimation more robust in the presence of non-Gaussian noise, which is the case when the real observation process does not necessarily fulfill the assumptions of the KF model. The formulation of MCKF can be derived by finding xtthat maximizes the objective function JMC given by Eq. (10). It should be emphasized that in the derivation based on WLS to obtain the solution given by a closed formula an approximation is made, namely the posterior estimate is replaced at one place with the prior estimate, i.e. ˆxt|t≈ˆxt|t−1(please see [43] for details). The resulting predict and update steps are driven by the same equations as for the standard KF, see Eq. (7) and (8) but the optimal Kalman gain for the MCKF is given with the formula: Kt=ˆ Pt|t−1λtHT t(Htˆ Pt|t−1λtHT t+Mt)−1,(11) 6 where λt=Gσ yt−Htˆxt|t−1 M−1 t.(12) In the state estimation procedure using MCKF, when a significant outlier appears, the term yt−Htˆxt|t−1 in Eq. (8) (usually called innovation or pre-fit residual) diverges, but Ktcontrols the divergence of the estimator ˆxt|t. Please see references [43–46] for more details about the procedure of driving equations and stability. To illustrate the performance of MCKF we consider a two-state dynamic system as follows: xt=Atxt−1+qt, yt=Htxt+mt+wt, At=1 ∆t 0 1 , Ht=1 0 ,(13) where ∆tis the discretisation step size and for this case we consider ∆t= 0.1. The initial parameters of estimator are ˆx0|0= [ 0 0 ]T,ˆ P0|0= [ 6.2e −08 1.2e −06; 1.2−06 2.5−05 ]. The noise terms qtand mtare normally distributed: qt∼ N([0; 0],[1 4(∆t)4,1 2(∆t)3;1 2(∆t)3,(∆t)2]),mt∼ N(0,10). The additional term wtis a non-Gaussian noise generated as a series of of independent random variables from Student’s t-distribution with number of degrees of freedom ν= 2.8(please note that this term is not included in the standard KF model). The simulation results are plotted in Fig. 4. Panel (a) is presented an actual state by a red line, and the observation of the real state includes heavy-tailed noise. As we can see on panel (b) of Fig. 4, in the presence of impulsive noise, MCKF could properly track the mean state, while standard KF estimation was affected by heavy-tailed distributed noise and therefore could not follow the mean state with comparable accuracy. Figure 4: State estimation in the presence of non-Gaussian noise, (a) measurement signal and real state, (b) results of state estimation with KF and MCKF. 2.2.3. Switching maximum correntropy Kalman filter In this subsection, we introduce the switching maximum correntropy criterion Kalman filter, which is a robust version of SKF. The SMCKF can be described as a dynamic Bayesian network in which during the lifetime of the system we switch between different KF or MCKF models. see Fig. 5. The additional process 7 stcalled switching variable is introduced. Its state space is finite, i.e. consists of kvalues: 1,2, . . . , k. If st=j, the degradation process is in the j-th of ksubsequent stages and the process xtis in this stage modelled with the j-th of kMCKF models. In each time step both the switching model variable stand the state variable xtare hidden and thus must be figured out from the observation yt. This may cause numerical problems, especially when the number of the stages is increased, as argued in [63]. Kevin et al.in [43,64], developed an approximation approach, namely the generalized pseudo-Bayesian (GPB) method, to solve this issue. Figure 5: Dynamic Bayesian network. Our goal in the segmentation through degradation stages is to evaluate, based on the measurement y1:t={y1, . . . , yt}, the probabilities of the model being in a specific stage at each time twhich directly corresponds to the values of πj t=P(st=j|y1:t)(14) for j= 1, . . . , k. Note that the above probabilities are the ones filtered with the SMCKF based on measurement y1:t. For modelling the process stwe use a discrete time Markov chain, thus the non-filtered probabilities can be expressed with a recursive formula. Namely denoting the transition probabilities of st as zij =P(st=j|st−1=i)(assumed to be time-invariant) we have: P(st=j) = k X i=1 P(st−1=i)·zij.(15) The framework we use is similar to the one used in [43] and is based on the assumption that in general all parameters of the model, i.e. At,Qt,Htand Mtbut also the form of state vector xt, may explicitly depend on the current value of st. In contrary to [43] we assume that in each of 1to k-th stages the current HI needs to be estimated using MCKF instead of KF. Thus, the Kalman gain is calculated using Eq. (11) which contains the coefficient λt. Likewise the covariance matrix of the innovation term ϵt=yt−Htˆxt|t−1 (given the measurement till time t) contains the same coefficient λt, namely we obtain ϵt∼ N(0, Ct), where Ctis given by the formula: Ct=Htˆ Pt|t−1λtHT t+Mt.(16) As we noted before, the parameters of the degradation model depend on stdescribing the current stage. The estimates ˆxt|tand ˆ Pt|tobtained by repeating iterations of SMCKF would be therefore different for each possible series of {s1, . . . , st}. To reduce computational complexity of the algorithm (avoid iterating MCKF-s through all possible trajectories of st), we limit our procedure to iterate only through possible values of the switching variable in the current and previous time, i.e. stand st−1. Therefore we assume the following form of the conditional probability density function: P(ϵt=u|st−1=i, st=j, y1:t) = g0,Ct(u),(17) 8 where gµ,Σ(u)is multivariate Gaussian probability density function of the random variable with mean vector µand covariance matrix Σ. Using the Bayes’ theorem conditioning on ϵtwe obtain: P(st−1=i, st=j|ϵt, y1:t−1) = g0,Ct(ϵt)·P(st−1=i, st=j|y1:t−1) Pk l=1 Pk m=1 g0,Ct(ϵt)·P(st−1=m, st=l|y1:t−1),(18) which holds for each pair of indices iand j. Summing these probabilities with respect to i(and using the definition of transition probabilities), we receive a recursive formula for the filtered probability of the j-th MCKF model: πj t=P(st=j|ϵt, y1:t−1) = k X i=1 P(st−1=i, st=j|ϵt, y1:t−1) = Pk i=1 g0,Ct(ϵt)·πi t−1·zij Pk l=1 Pk m=1 g0,Ct(ϵt)·πl t−1·zlm .(19) We can apply this formula to calculate πj t-s in each step of the SMCKF. To make the estimation procedure feasible at time iteration after prediction and update steps (described in the previous subsections) we add a approximation step which merges the estimates removing the dependence on previous values of the switching variable, thus we only need to store in the memory the estimates ˆxt|tand ˆ Pt|tfor k×kpairs of possible values of st−1and st. For the estimates calculated given the specific realization of the switching variable process let us introduce the following notation: ˆxt|t|st−1=i,st=jstands for the posterior estimate of xtobtained for st−1=iand st=j. The approximated value of the posterior estimate ˆxt|tgiven stis evaluated taking a weighted mean of the state estimates obtained with each of the MCKS-s, i.e. starting with each of kpossible values of st−1. Similar formula is applied to covariance matrices of the estimates. The two approximation formulas are following: ˜xt|t|st=j= k X i=1 wij tˆxt|t|st−1=i,st=j,(20) ˜ Pt|t|st=j= k X i=1 wij t(ˆ Pt|t|st−1=i,st=j+rrT),(21) where r= ˜xt|t|st=j−ˆxt|t|st−1=i,st=j. Using Bayes’ theorem and applying the results of Eq. (18) and (19) we obtain the proper weights: wij t=P(st−1=i|st=j, ϵt, y1:t−1) = P(st−1=i, st=j|ϵt, y1:t−1) P(st=j|ϵt, y1:t−1)=g0,Ct(ϵt)·πi t−1·zij Pk l=1 g0,Ct(ϵt)·πl t−1·zlj .(22) In this way kestimates, after multiplying to k×kposterior estimates in prediction-update steps, reduce back to kvalues in the approximation step. The approximated values of the state estimate and covariance matrix (which overwrite the posterior) are then used as an input to the prediction step (obtaining new priors) in the next time iteration of the SMCKF. More theoretical details about the procedure of the switching structure (in the case of SKF which is analogous) and stability of the model are available in [64]. The SMCKF, which is composed of multiple MCKF-s with different state-space models, is applied to track changes in the degradation process. Then SMCKF is switched between the selected models according to their likelihood (Gaussian distribution of the innovation term) calculated from HI. Thanks to the use of the maximum correntropy criterion the procedure is more robust than in the case of SKF. It should be noted that for applying this approach, the degradation model should be selected based on the case’s physics, as discussed in Section 2.1. Based on our primary assumption, the degradation process comprises three stages. Three MCKF models are used to simulate healthy, degradation, and critical stage. According to [52,65] the polynomial MCKFs of zero-, first-, and second-order are illustrated below, respectively. Using the superscripts to denote the 9 Figure 10: Specificity analysis, (a) health stage, (b) degradation stage, (c) critical stage. 4. Real data analysis This section evaluates the proposed methodology for available real data sets. These data are typically employed as benchmark data sets for different papers and competitions and have specific behavior corresponding to noise properties and deterministic trends. In the following, essential information about objects, experiments, and data will be recalled, and appropriate references are provided. 4.1. FEMTO data set The FEMTO data set comprises 17 historical runs to failure of bearing degradation using the PRONOSTIA platform see Fig. 11. Franche-Comté Electronics Mechanics Thermal Science and Optics–Sciences and Technologies institute recorded and published this data set as PHM challenge 2012. The frequency resolution of the data set is 25600 Hz. But, the length of the signal is 0.1 second, so it is not enough to apply most conventional frequency analyses and spectrograms. More information about this data set can be found in [66]. This data set has been employed in many publications for HI construction [67–74], segmentation of degradation process [75–79], and RUL prediction [80–85]. Figure 11: PRONOSTIA platform [66]. 16 In this paper, the bearing 1−1is selected as a case study from the FEMTO data set, which is composed of a few historical vibration data. Each of the mentioned sets acquired 0.1 seconds vibration signal with a 25.6 kHz sampling rate; see panel (a) in Fig. 12. This process is repeated every 10 seconds. Also, in this paper, the RMS of each vibration set is used as HI; see panel (b) in Fig. 12. (a) (b) Figure 12: FEMTO data set, (a) raw bearing run-to-failure vibration signals, (b) HI. (RMS of the raw signal) 4.2. Wind turbine data set This data set is collected from a wind turbine see Fig. 13. The sensor has been mounted on a high-speed bearing shaft of the 2.2MW power wind turbine. This data set is acquired for more than 50 days using two different ways. In the first approach, raw vibration signal is acquired for 6 seconds with 100k sampling frequency per day for +50 days, see panel (a) of Fig. 14. For the second version, the bearing inner race energy is calculated every 10 min for +50 days see panel (b) of Fig. 14. Then for more information about the methodology for calculating inner race energy, see [86]. At the end, the inner race-bearing fault occurred, which was proved by inspection. The bearing type used during the test is 32222 −J2SKF. It should be mentioned that this data set has been used for prognosis by several papers, see, e.g., [86–89]. Figure 13: Wind turbine data set platform (top) and photo of the damage. For the application of the proposed approach to the wind turbine data set, inner race energy is selected as a health index. As can be seen in panel (b) of Fig. 14 this data set includes numerous outliers. Thus, we consider it as a trend with non-Gaussian noise [53]. In addition, a few fluctuations have appeared on HI that may correspond to the variation of the load or some phenomena like self-healing. This specific behavior in real cases significantly challenges the diagnostic and prediction of RUL. 17 (a) (b) Figure 14: Wind turbine data set, (a) raw bearing run-to-failure vibration signals, (b) HI (inner race energy of the raw signal). 4.3. FEMTO diagnostics result In Fig. 15 the SMCKF, and SKF results on FEMTO data set is demonstrated. Panels (b) and (c) are shown the probability of the stages (i.e. π1 t, π2 t, π3 t) during the degradation process based on SKF and SMCKF, respectively. Also, the most probable stage detected and changing points between stages by SKF and SMCKF are illustrated in panels (d) and (e), respectively. By comparing the results of SKF and SMCKF in the two last panels, it can be seen that the results more or less are same even though on the panel (d) that SKF, we can see fluctuation around Time=1300. Also, in both methods, the critical stage is detected on Time=2411, which seems to be identified earlier than the actual changing point. Figure 15: Segmentation of the FEMTO data set, (a) HI, (b) probability of stage by SKF, (c) probability of stage by SMCKF, (d) most probable stages based on the implementation of SKF, (e) most probable stages based on the implementation of SMCKF. 18 4.4. Wind turbine diagnostics result In Fig. 16, the SMCKF and SKF results from the wind turbine data set are demonstrated. Panels (b) and (c) show the probability of stages during the degradation process based on SKF and SMCKF, respectively. Also, the most probable stage detected and changing points between stages by SKF and SMCKF are illustrated in panels (d) and (e), respectively. By comparing the results of SKF and SMCKF in the two last panels, it can be seen that both methods detected 18-Mar as the first changing point while the second changing point is identified on 9-Apr by using SKF, and this point is observed on 21-Apr by employing SMCKF. Figure 16: Segmentation of the Wind turbine data set, (a) HI, (b) probability of stage by SKF, (c) probability of stage by SMCKF, (d) most probable stages based on the implementation of SKF, (e) most probable stages based on the implementation of SMCKF. As discussed, this data set has two versions; another version is collected and composed of a raw vibration signal recorded for six seconds daily with a 100k sampling rate, see panel (a) in Fig. 17. So it means it has an excellent potential to apply various methods of frequency analysis to do condition monitoring. The envelope analysis is applied for each vibration set, and the results are sequentially plotted in panel (b) of Fig. 17. As shown in panel (b) in Fig. 17 after 18 March, harmonic frequency appeared, which means the bearing went to the degradation stage. Also, after 22 April, the amplitude of harmonic frequency is dramatically increased, which can be considered CP2. By comparing the SMCKF and SKF results to the envelope analysis, it can be found that CP1 and CP2, identified by SMCKF (CP1=18 March and CP2 =22 April), has the closest value to the CP1 and CP2 detected by visual check on envelope analysis. 19 (a) (b) Figure 17: Changing points detection via envelope analysis for Wind turbine, (a) raw bearing run-to-failure vibration signals, (b) waterfall plot of envelope spectrum calculated for available raw vibration data. Table 2demonstrates the computational cost of both methods for real data sets. The proposed method is implemented with Matlab 2021b, and the hardware property of the system for this implementation are as follows: Processor: Intel (R) Core (TM) i7-10750H CPU @ 2.60GHz 2.59 GHz and Ram: 32.0 GB. As shown in Table 2, the computational cost of SKF and SMCKF are low. However, the SKF is a faster method than SMCKF. Table 2: Computational cost SKF SMCKF FEMTO data set 0.358424 (s) 1.052233 (s) Wind turbine data set 0.524977 (s) 1.658114 (s) Furthermore, in this paper, we selected the kernel size σby using trial and error, which is a drawback of the proposed methodology that, in the future, we will try to deal with this problem. 5. Conclusions This paper introduces a robust methodology for online diagnostics of long-term condition monitoring data in the presence of non-Gaussian noise. The long-term data describes the degradation process. Several approaches assume a model of such degradation. Therefore, a general look at different degradation processes was carried out. Due to this, it is assumed that degradation can evolve over time. It comprises three stages corresponding to the machine life phases: healthy, degradation, and critical stages, see Fig. 3. In actual applications, it is essential to identify when a machine’s condition changes from a healthy stage to a degradation stage (warning) and the degradation stage to a critical stage (alarm). It is the base of additional analyses like prognostics. Furthermore, researchers focus on the last segment. However, they don’t mention how they have found the division between stages 2 and 3 and whether this detected point coincides with the true start time of the rapid development of the fault in the machine. Each of these stages is modeled by MCKF, being a robust version of the Kalman filter. The SMCKF is introduced to infer from the current observations of the underlying process by calculating which of the three filters has the biggest probability in each time step. This method can provide more information to the decision-maker by providing probabilistic information about each degradation stage over time. Furthermore, this method works according to the dynamic trend of the degradation process, and the threshold is no longer needed. Also, the proposed method’s computational cost is low, which has massive potential to employ it for actual applications. Moreover, it provides a possibility to start the prediction of RUL from the beginning of the critical stage (which has advantages). The biggest drawback of this approach is that the measurement 20 noise part of the processes has been tuned manually. In the future work, the solution to address it will be investigated in detail. Acknowledgments The authors (Hamid Shiri) gratefully acknowledge the European Commission for its support of the Marie Sklodowska Curie program through the ETN MOIRA project (GA 955681). Project no. 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