Robust switching Kalman filter for diagnostics of long-term condition monitoring data in the presence of non-Gaussian noise

IOP Conference Series: Earth and Environmental Science PAPER • OPEN ACCESS Robust switching Kalman filter for diagnostics of long-term condition monitoring data in the presence of non-Gaussian noise To cite this article: Hamid Shiri et al 2023 IOP Conf. Ser.: Earth Environ. Sci. 1189 012007 View the article online for updates and enhancements. You may also like Bearing remaining life prediction method based on ARAD -ELN and multi-stage wiener process Yu Wang, Shujie Liu, Shuai Lv et al. - Real-time identification of performance degradation stages of rolling element bearings by RVCFI Jiadong Meng, Changfeng Yan, Tao Wen et al. - Advancements in bearing health monitoring and remaining useful life prediction: techniques, challenges, and future directions Xinwei Liu, Zongzhen Zhang, Zhuoli Li et al. - This content was downloaded from IP address 139.166.172.157 on 17/11/2025 at 09:48 Content from this work may be used under the terms of theCreativeCommonsAttribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd XXII Conference of PhD Students and Young Scientists IOP Conf. Series: Earth and Environmental Science 1189 (2023) 012007 IOP Publishing doi:10.1088/1755-1315/1189/1/012007 1 Robust switching Kalman filter for diagnostics of long-term condition monitoring data in the presence of non-Gaussian noise Hamid Shiri, Jacek Wodecki, Rados law Zimroz∗ Faculty of Geoengineering, Mining and Geology, Wroclaw University of Science and Technology, Na Grobli 15, 50-421 Wroclaw, Poland E-mail: [email protected] Abstract. Machinery condition prognosis system uses long-term historical data to predict remaining useful life (RUL). One of the critical steps to reach this purpose is to segment long-term data into two or several degradation stages (Healthy, Unhealthy, and Critic stage). Finding changing points between regimes may be a crucial preliminary task for further predicting the degradation process. However, finding the accurate partition into two or more regimes is a challenging task in the actual application when the noise inherent in the observed process is non-Gaussian. Therefore, this paper introduced a robust methodology based on switching Kalman filters to address the problems mentioned. This approach uses multiple dynamic system models to explain different degradation stages, utilizing robust Bayesian estimation. Also, based on this fact, this approach works based on dynamic behavior; a threshold for diagnostics is no longer needed. Ultimately, the proposed approach is applied for the online diagnosis of simulated data sets in the presence of Gaussian and non-Gaussian noise. The result of the applied methodology on simulated data sets proves the method’s efficacy. Diagnostics, condition monitoring, fault detection, robust methods, non-Gaussian noise, threshold, switching Kalman filter 1. Introduction Condition-based maintenance(CBM) has become increasingly popular in the industry with the development of condition monitoring systems. CBM programs are developed to assess the machine’s condition by collecting massive amounts of data during the operation of machines. Using the long-term condition monitoring data is a crucial element in both diagnostics and prognostics. Many methods published in recent years for CBM can be categorized into four main groups [1]: stochastic-based [2, 3, 4, 5, 6, 7, 8], machine learning-based [9, 10, 11, 12], physics or model-based[13, 14], and hybrid methods [15, 16]. Both machine learning and stochastic approaches such as neural networks [12, 17, 18, 19, 20, 21, 22, 23] and hidden Markov models (HMM) [24, 25, 26, 27, 28, 29] 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, model-based methods use a mathematical representation of the degradation process, which needs less training data. XXII Conference of PhD Students and Young Scientists IOP Conf. Series: Earth and Environmental Science 1189 (2023) 012007 IOP Publishing doi:10.1088/1755-1315/1189/1/012007 2 However, it is necessary to have enough knowledge of the degradation process. Also, 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 (critic stage). Unfortunately, we do not know about limit values or the desired lifetime in many cases, especially when the machine is unique. In addition, this task will be more complicated when the machine works in a harsh area, where the inherent noise is a heavy tail. This may occur for several reasons. Wind inflows are one of the possible sources of heavy tail noise in wind turbine [30, 31, 32], the ore falling on devices in the mining environment is another [33, 34]. However, most of the mentioned methods are developed based on the Gaussian noise assumption, which may be less effective in actual application in the face of non-Gaussian noise. Furthermore, our previous work[35] focused on identifying and analyzing the degradation process using the real benchmark data set (available in the prognostics area), which confirms the hypothesis of non-Gaussian assumption noise even in the laboratory data sets. Therefore, in this paper, we tried to introduce a robust method that can address an online diagnostic without using a threshold in the presence of a non-Gaussian noise problem. The robust version of the switching Kalman filter (SKF) is derived to solve these issues by defining multiple dynamic systems to describe a different degradation process using Bayesian estimation. So, a threshold is no longer needed because the proposed methodology applies to dynamic behavior. The paper is structured as follows: after the introduction, the summary of the process is presented. After that, the critical aspects of the processing methods are described in theory. Then, the proposed model is simulated, and the results will be presented. Finally, the conclusions are formed. 2. Theory 2.0.1. Kalman filter(KF) 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 discrete state-space representation of the model is as follows: xt=At−1xt−1+qt yt=Htxt+rt(1) where xtis an actual state at time t,Ais the state-transition model, qtis the process noise, ytis observation of actual state xt,Htis the observation process, which maps the actual state space into the observed space, and rtis the observation noise. The noise terms qtand rtare normally distributed: qt∼ N(0, Qt), rt∼ N(0, Rt). In the literature, several approaches can be employed to extract KF formulation: for instance, Bayesian rule, maximum posterior approaches (MAP), orthogonal principle, weighted least square method (WLS). The quadratic objective function in WLS is defined as follows: J=1 2(yt−Hˆxt)TR−1 t(yt−Hˆxt) + 1 2(ˆxt−Aˆxt−1)TP−1 t|t−1(ˆxt−Aˆxt−1)(2) where Pt|t−1=E[e− te−T], e− t=xt−ˆxt Rt=E[rtrT t], Qt=E[qtqtT](3) Therefore, by solving the Equation: ∂J ∂ˆxt= 0 ,(4) KF can be derived and its formulation is described with the following equations: ˆx− t=Atxt−1(5) XXII Conference of PhD Students and Young Scientists IOP Conf. Series: Earth and Environmental Science 1189 (2023) 012007 IOP Publishing doi:10.1088/1755-1315/1189/1/012007 3 Pt|t−1=AtPt−1|t−1AtT+Qt(6) νt=yt−Htˆx− t(7) Kt= (HTR−1 tH+P−1 t|t−1)−1HTR−1 t(8) xt= ˆx− t+Ktνt(9) Pt|t= (I−KtHt)Pt|t−1(I−KtHt)T+KtRtKtT(10) where νt,Ktand Ptare innovation (or measurement pre-fit residual), Kalman gain, and covariance matrix estimation error, respectively. 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. To address the non-Gaussian problem, Izanloo et al.[36] presented the correntropy filter (CFilter) 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 higherorder measurement signal information. In the following subsection, the maximum correntropy criterion Kalman filter(MCKF) is derived to handle non-Gaussian noise based on the following references: [37, 36, 38, 39] 2.0.2. Maximum correntropy Kalman filter(MCKF) In the following, another form of a cost function based on the maximum correntropy criterion is introduced, see Eq.11, that is more robust in the presence of non-Gaussian noise. Jm=Gσ(∥yt−Hˆxt∥R−1 t) + Gσ(∥ˆxt−Aˆxt−1∥R−1 t|t−1 )(11) where Gσ(xi−yi) = exp(−∥xi−yi∥2 2σ2) and σis kernel size(bandwidth). The MCKF can be driven by minimizing the cost function Eq. 12 . ∂Jm ∂ˆxt= 0 (12) The MCKF equations are given as follows: ˆx− t=Atxt−1(13) Pt|t−1=AtPt−1|t−1AtT+Qt(14) Lt=Gσ(∥yt−Hˆxt∥R−1 t) Gσ(∥ˆxt−Aˆxt−1∥R−1 t|t−1 )(15) νt=yt−Htˆx− t(16) Kt= (LtHTR−1 tH+P−1 t|t−1)−1LtHTR−1 t(17) xt= ˆx− t+Ktνt(18) Pt|t= (I−KtHt)Pt|t−1(I−KtHt)T+KtRtKtT.(19) When a significant outlier appears, the innovation term νtdiverges, but ktcontrols the divergence of the estimator ˆxt. Please see these references [37, 36, 38, 39] for more details about the procedure of driving equations and stability. The most significant limitation of KF and MCKF used for diagnostics and prognostics is that the degradation process must be time-invariant. However, in practical application, the XXII Conference of PhD Students and Young Scientists IOP Conf. Series: Earth and Environmental Science 1189 (2023) 012007 IOP Publishing doi:10.1088/1755-1315/1189/1/012007 4 degradation process is composed of several components that change over time. 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 SKF, also known as the dynamic linear model, is proposed to handle the mentioned issue when the dynamic behavior of a process changes during the time and, if not possible, to use a unique model. However, general SKF has assumed the noise was Gaussian. In the next subsection, we introduce a new version of SKF based on MCKF to address these challenges. 2.0.3. Switching maximum correntropy Kalman filter (SMCKF) In this subsection, we introduce the switching maximum correntropy criterion Kalman filter (SMCKF), which is a robust version of SKF. The SMKF can be described as a dynamic Bayesian network. In each time step, Stas the switching model variable and Xtas the state variable are hidden, which must be figured out from the observation Yt. This may cause numerical problems, especially when the number of the regimes is increased, as argued in [40]. Kevin et al.in [41], developed an approximation approach, namely the generalized pseudoBayesian (GPB) method, to solve this issue. In every time step, the covariance and state estimation from all MCKFs in the last time step is mixed using weights based on the switch variable Si|j tand the transition probabilities Zij, as expressed with the equations 20, 21, respectively. Si|j t=Zij Sit−1 n P i=1 Zij Sit−1(20) Weighted state and covariance estimates are derived as follows: ˜xj t−1= n P i=1 ˜ Si|jxit−1 ˜ Pj t−i= n P i=1 ˜ Si|j t{¯ Pi t−1+ [xi t−1−xj t−1][xi t−1−xj t−1]T(21) Also, by using measurement residuals, the likelihood of MCKF is calculated as Eq.22. Li t=N(vi t; 0,¯ Ct)(22) where ¯ Ct= (LtHTR−1 tH+P−1 t|t−1). In the end, the probability of i-th MCKF model is equal to Eq.23. Si t= Li t( n P i=1 Zij Si t−1) n P i=1 (Li t( n P i=1 Zij Si t−1)) (23) The weighted state and covariance estimates are given by MCKF from Eq.13 to Eq.19 for each filter, which was used before to estimate the predicted state ˜xj t−1and covariance ˜ Pj t−i, see Eq. 21. More details about the procedure of the switching structure and stability of the model are available in [41]. The SMCKF, which is composed of multiple MCKF with different statespace models, is applied to track changes in the degradation process. Then SMCKF is switched between the selected models according to their likelihood calculated from health index(HI). It should be noted that for applying this approach, the degradation model should be selected based on the case’s physics. Based on our primary assumption in this analysis, the degradation process comprises three stages. Zero-, first-, and second-order MCKF are used to simulate healthy, degradation, and critic stage, respectively. According to [42, 43], the actual state, error covariance, and observation matrices representing the polynomial MCKFs are illustrated with the subscripts 1, 2, and 3 XXII Conference of PhD Students and Young Scientists IOP Conf. Series: Earth and Environmental Science 1189 (2023) 012007 IOP Publishing doi:10.1088/1755-1315/1189/1/012007 5 below, showing the zero-, first-, and second-order MCKFs, respectively. Eq. 24 represent state matrices. x1(t) =   x 0 0  , x2(t) =   x ˙x 0  , x3(t) =   x ˙x ¨x  (24) A1(t) =   100 000 000  , A2(t) =   1ts0 010 000  , A3(t) =   1tsts2 2 0 1 ts 0 0 1  (25) where A1,A2, and A3are state transition matrices and tsis discretisation step size. Also, the covariance matrices of the process are defined by the following: Q1(t) = q  100 000 000  , Q2(t) = q  ts3 3 ts2 20 ts2 2ts0 0 0 0  , Q3(t) = q   ts5 20 ts4 8 ts3 6 ts4 8 ts3 3 ts2 2 ts3 6 ts2 2ts   (26) where qis a scalar hyper-parameter (related to the noise) that can describe the uncertainty of the filter in actual application, which can be used for tuning of the SMCKF for a different machine. Observation matrices H1,H2, and H3corresponding to models 1,2,3 are defined as following: H1=H2=H3=  1 0 0  (27) The transition matrix is defined by Eq. 28. Note that there is no possibility of transition to the previous state. Z=  0.998 0.001 0.001 ∼0 0.999 0.001 ∼0∼0 1  (28) The initial model probabilities are set with Eq. 29 ˜ S0=0.9 0.05 0.05 (29) State and covariance estimates have the following form, respectively: x0=  y0 0 0  ,(30) ¯ P0=  100 010 001  (31) 3. Simulation In this section, At first, the long-term data is generated based on the lifetime curve shape following the model known in the literature [35]. this degradation model is developed to explain real degradation of real machine that work in harsh environment such as mining and wind turbine which can be seen the noise with non-Gaussian caractristic. This degradation model consists of 3 regimes: healthy condition, slow degradation, and rapid degradation. The data could be modeled as a mixture of trend and random components. More details about simulated model can find, for example, in [35] . Then the proposed methodology is used to segment data into three regimes in the presence of Gaussian noise and non-Gaussian noise. XXII Conference of PhD Students and Young Scientists IOP Conf. Series: Earth and Environmental Science 1189 (2023) 012007 IOP Publishing doi:10.1088/1755-1315/1189/1/012007 6 3.1. Gaussian Noise In this subsection, the introduced methodology is applied to simulated data set in the presence of non-Gaussian noise with by considering Student’s t distribution with a degree of freedom V= 3. For this case, it assumed Time=1000 and Time =1600 are changing points between the healthy stage and degradation stage (CP1) and the degradation stage and healthy stage (CP2), respectively. Fig. 2 ilistrated the result of the applied SKF and MCSKF. The estimated probability of each state using SKF and MCSKF during the degradation process is presented in panels (b) and (c), respectively. On the panel (b), we can see the probability of the healthy stage (the green line) is starting to reduce from Time=300; after a few fluctuations with the degradation stage (the yellow line), somewhere around Time=880, the degradation state will be the most probable stage which remains on this situation until Time=1446. Behind this time, the critic stage has more probability than the rest stages, but as we can see on panel (b) in Fig.2, the SKF has diverged after Time=1500. Panel (c) presented the result of SMCKF. Also, panels (d) and (e), the most probable stage during the degradation process and changing points extracted by employing SKF and SMCKF, are demonstrated. By comparing these two last panels, as expected, the MCSKF result is closer to the actual changing points while SKF has been significantly affected by non-Gaussian noise and, in the end, is diverged. Figure 1: Detection of the stages in the presence of Gaussian noise, (a) health index (HI), (b) probability of stages performed by SKF, (c) probability of stages performed by SMCKF, (d) most probable stages based on the implementation of SKF, (e) most probable stages based on the implementation of SMCKF. 3.2. Non-Gaussian Noise In this part, the proposed methodology is applied to data generated by the offered model, considering that the noise term has Student’s t distribution with a degree of freedom V= 3. According to the simulation procedure, CP1 and CP2 is equal to Time=1000 and Time=1600, respectively. Fig. 2 shows the result of the proposed methods when noise is non-Gaussian. The estimated probability of each state using SKF and SMCKF during the degradation process is presented in panels (b) and (c), respectively. On the panel (b), we can see the probability of the healthy stage (the green line) is starting to reduce from Time=300; after a few fluctuations with the XXII Conference of PhD Students and Young Scientists IOP Conf. Series: Earth and Environmental Science 1189 (2023) 012007 IOP Publishing doi:10.1088/1755-1315/1189/1/012007 7 degradation stage (the yellow line), somewhere around Time=880, the degradation state will be the most probable stage which remains on this situation until Time=1446. Behind this time, the critic stage has more probability than the rest stages, but as we can see on panel (b) in Fig.2, the SKF has diverged after Time=1500. Panel (c) presented the result of SMCKF. Also, panels (d) and (e), the most probable stage during the degradation process and changing points extracted by employing SKF and SMCKF, are shown. By comparing these two last panels, as expected, the SMCKF result is closer to the actual changing points while SKF has been significantly affected by non-Gaussian noise and, in the end, is diverged. Figure 2: Detection of the stages in the presence of non-Gaussian noise, (a) health index, (b) probability of stages by SKF, (c) probability of stages by SMCKF, (d) most probable stages based on the implementation of SKF, (e) most probable stages based on the implementation of SMCKF. 4. Conclusions This paper introduces a robust version of SKF to address the non-Gaussian noise issue. At first, we extracted an MCKF as a robust version of SKF. We are assuming that the degradation process is composed of three regimes corresponding to the machine life phases: good condition, degradation, and critic stage. Each of these regimes is modeled by MCKF. The SMCKF is introduced to infer from the current observations of the underlying process by calculating which of the three filters has the most significant 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. Also, the proposed methodology was applied to the simulated degradation data set, and the result was presented for two cases in the presence of Gaussian and non-Gaussian noise. The results confirmed the performance of SMCKF in detecting different stages in both Gaussian and non-Gaussian noise in the comparison of SKF. Also, this method works according to the dynamic trend of the degradation process, and the threshold is no longer needed. Funding This work was supported by European Commission via the Marie Sklodowska Curie program through the ETN MOIRA project (GA 955681) - Hamid Shiri XXII Conference of PhD Students and Young Scientists IOP Conf. Series: Earth and Environmental Science 1189 (2023) 012007 IOP Publishing doi:10.1088/1755-1315/1189/1/012007 8 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). 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