Modelling and Prognostics of system degradation using Variance Marwa Belhaj Salem, Estelle Deloux, Mitra Fouladirad

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Marwa Belhaj Salem, Estelle Deloux, Mitra Fouladirad. Modelling and Prognostics of system degrada- tion using Variance Gamma process. 30th European Safety and Reliability Conference and 15th Prob- abilistic Safety Assessment and Management Conference (ESREL2020 PSAM15), Nov 2020, Venice, Italy. ￿hal-03181697￿

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Marwa Belhaj Salem ICD-M2S, University of Technology of Troyes, France. E-mail: [email protected]

Estelle Deloux ICD-M2S, University of Technology of Troyes, France. E-mail: [email protected]

Mitra Fouladirad ICD-M2S, University of Technology of Troyes, France. E-mail: [email protected]

Nowadays estimation of Remaining Useful Life (RUL) is essential to the prognostics and health management of high-priced systems. Initially, the properties of the VG process and estimation of the parameters are presented. Analytical approximation is also discussed and due to its complexity, a simulated method is proposed. Finally, the simulation method and the goodness of fit tests are used to predict the system life.

Keywords: Variance Gamma, Stochastic Models, Degradation Processes, Parameter Estimation, Remaining Useful Life, Prognostics.

1. Introduction are good candidates for mod- elling degradation. As indicated by Ye and Xie Reliability and safety of systems are the most im- (2015), the stochastic nature of these processes is portant aspect confronting by the industries these capable of modelling the unexplained randomness days. The history of failure data of systems is one of degradation over time due to non-observed en- of the most significant informations required for vironmental factors or unknown effects. degradation modelling and estimation of the Re- Gamma process is proven to be an effective tool maining Useful Life. Most aging failures could be in stochastic modelling of monotonic and gradual attributed to some underlying degradation mecha- degradation. It is suitable for modelling gradual nism under which the damage accumulates over damage that accumulates over time in a sequence time. The failure of the product is determined of tiny increments such as wear, fatigue, corro- depending on a predefined deterministic value of sion, crack growth, etc. (van Noortwijk (2009), the failure provided by the product manufacturer, Zhang et al. (2015), Pulcini (2016), Rodríguez- i.e. failure threshold. The recovery of real failure Picón et al. (2016), Yan et al. (2016)). In real data from the industries or laboratories is chal- applications, one can observe that various sources lenging due to the strict confidentiality policies of variations affect the characteristic of a product’s that lead to a significant demand for a high reli- degradation process. Due to some non-observable ability degradation model. The degradation mod- effects, the degradation of a product can have elling techniques such as product design, testing, a non-monotonous variation. A simple gamma lifetime prediction, maintenance, cost planning, process cannot capture such variations Rodríguez- etc. were widely used in reliability engineering. Picón et al. (2016). Since degradation models play a significant role Similarly, the has been widely in reliability analysis, they are considered as the used in non-monotonic degradation (Ye and Xie basis of product analysis and related decision- (2015), Karatzas and Shreve (1998), Øksendal making. (2003)) and in structural reliability, they are suit- Current researches indicate a growing interest able to show alternatingly the increase and the in the application of degradation models in the decrease of the resistance of the structure. From a estimation of reliability. It also indicates that physical point of view, the degradation can be seen substantial progress has been made in the appli- as an additive superposition of a large number cation of degradation models in various indus- of small external effects. Thus, the degradation trial sectors (Gorjian et al. (2010), Wang et al. increment can be normally distributed because of (2018)). Two main classes of degradation models the . In this regard, this pro- are general path models and stochastic models cess is a good model for the degradation. Despite Meeker et al. (2014). Stochastic processes such the fact that Wiener processes have been used to as Wiener process, Gamma process and Inverse model many degradation phenomena, they are not

Proceedings of the 30th European Safety and Reliability Conference and the 15th Probabilistic Safety Assessment and Management Conference Edited by Piero Baraldi, Francesco Di Maio and Enrico Zio Copyright c ESREL2020-PSAM15 Organizers.Published by Research Publishing, Singapore. ISBN: 978-981-14-8593-0; doi:10.3850/978-981-14-8593-0 2886 Proceedings of the 30th European Safety and Reliability Conference and the 15th Probabilistic Safety Assessment and Management Conference 2887 suitable for modelling processes of monotonous 2. Variance Gamma Process degradation, such as wear or cumulative damage. As mentioned before, the shape of the degradation The Black & Scholes (B & S) model is a non- paths is crucial to identify the degradation model. linear and a non-monotonic process. One of its In this study, the hydraulic pump is mentioned advantages comes from the analytical properties as an example and motivation. In such case, of the continuous time which the state of the system (presented by Variance allows for a simple computation of its formula. Gamma) represent the difference between the in- TheB&Smodel has been increasingly criticized coming sources (presented by a positive gamma) and the weaknesses emerge from an unrealistic and the leakage (presented by a negative gamma). set of assumptions such as constant volatility, i.e. The result path demonstrates a non-monotonic and implied data’s volatility does not tend to vary a non-linear behaviour which can be replicated which is not realistic. This will cause difficul- using Variance Gamma process. In this section, ties in precise estimation and prediction. The the Variance Gamma process and its properties are B & S model has its limitations when it comes presented. to accurately predicting the model (Manaster and Koehler (1982), Karoui et al. (1998), Arriojas et al. (2007)). There are plenty of data that cannot 2.1. Variance Gamma as time changed be fitted using conventional Wiener, Gamma and Brownian motion B & S processes. Due to this, another called Variance Gamma which can better Here, the Variance Gamma (VG) process is intro- replicate the non-monotonic degradation path is duced as an extension of Brownian motion which proposed here. can be obtained by evaluating wiener process at Variance Gamma process is initially introduced random times defined by a gamma process. In in the financial analysis and it was brought in as other terms, the time in the Brownian motion an extension of the Brownian motion. The first will be replaced with a gamma process (Geman complete presentation of the model in its simpli- et al. (2001)). Let B be a brownian motion with fied symmetric form is explained in Madan and positive parameters, the drift θ and volatility σ and Seneta (1990) and has been considered in Madan a standard Brownian motion W(t). It is defined as: and Milne (1991) and Madan et al. (1998). Some B(t; θ, σ)=θt + σW(t) (1) of its properties were discussed in Madan and Seneta (1987) and empirical comparisons with The gamma process of independent gamma dis- other models were presented. This stochastic tributed increments on a time interval (t, t+h), process contains two more parameters compared γ(t; μ, ν) with μ as the mean rate and ν as the to the geometric Brownian motion, which help in variance rate was considered. The VG process is the control of the and the . This defined as: Lévy process can be written as a Brownian motion X(t; σ, ν, θ)=B(γ(t; μ, ν),θ,σ) (2) evaluated at random times. The Variance Gamma process can be obtained by replacing the time The density function of VG process at a time t in the Brownian motion with a Gamma process. can be expressed as a normal density function con- The selection of the degradation model is crucial ditional on the realisation of the time change given and it is determined by the shape of the degra- by the gamma distribution (Madan and Seneta dation paths. Due to its two presentations, VG (1990)). can be used to model any kind of non-monotonic The VG process has four parameters: σ the degradation phenomenon as far as the increments volatility of the Brownian motion, ν the variance follow the VG distribution. rate of the gamma time changes and θ the drift In this study, the Variance Gamma process will in the Brownian motion with drift and μ the mean be defined in Section 2 and the estimation of the rate. The process offers two additional dimensions Variance Gamma parameters will be carried out of control on the distribution over and above that in Section 3. A comparative study will be per- of the volatility. The parameters θ and ν control formed between Variance Gamma and other pro- the skewness and kurtusis respectively. The den- cesses such as Wiener, Black & Scholes in fitting sity function for the VG process at time t can be simulated data. An analytical approximation of given by this expression (Scott et al. (2011)): the distribution of the failure time (FT) will also 2 ( ( − ) 2) ( ; 2 )= exp√ θ x μ /σ be presented and discussed. A simulation-based f x μ, σ ,θ,ν 1 σ 2Π( )ν θ(ν) method will be proposed to obtain the failure time  ν   ν− 1 √ (3) | − | 2 | − | 2 2 + 2 distribution. It will be fitted to different distribu- √ x ν x ν σ ν θ Kν− 1 2 tions and goodness of fit measures will be applied 2θ2ν + σ2 2 σ in Section 4. Finally, the simulated method and the goodness of fit tests will be used to establish where x ∈ R \ μ, Kν (.) is the modified Bessel the system life forecasting in Section 5. function of the second kind with index ν, μ is the location parameter, σ2 is the scale parameter, Proceedings of the 30th European Safety and Reliability Conference and the 15th Probabilistic Safety Assessment and Management Conference 2888

θ is the skewness parameter and ν is the shape defined as the process which starts at zero, has parameter (Abramowitz (1972)). independent and stationary increments. The in- When x = μ, two cases need to be distin- (VG) − (VG) ≤ 1 2 crement Xs+t √ Xs follows the Variance guished: when ν / , the density is singular ( ) at μ and when ν>1/2 the density reduces to: Gamma VG σ t, ν/t, tθ . VG process can also be defined as a difference of two independent 2 θ(ν − 1/2) f(x; μ, σ ,γ,ν)=  gamma processes as below: σ 2Π(1/ν)ν θ(ν)  ν− 1 (4) 2 2 2 √ σ 2 2 (VG) (1) (2) 2σ ν + θ Xt = Gt − Gt The characteristic function of VG process is given by:     t (1) (1) 1 ν where G = Gt ,t≥ 0 is an indepen- ( )= [ (iuX(t))]= (5) φX(t) u E e σ2ν 2 1 − iθνu +( 2 )u dent gamma process with a shape parameter (2) a=C and scale parameter b=M, whereas G = A sample path of VG was plotted using the param- (2) eters VG(σ=0.5, θ=0.5, ν=0.5, μ=0) in Figure 1. Gt ,t≥ 0 is an independent gamma process It can be observed that the path consists of many with parameters the same shape parameter a=C small jumps as expected. and the scale parameter b=G with 1 C = ν > 0 ,

−1 1 2 2 1 2 1 G = 4 θ ν + 2 σ ν − 2 θν > 0 ,

−1 1 2 2 1 2 1 M = 4 θ ν + 2 σ ν + 2 θν > 0 .

3. Properties: Parameteric estimation and Failure Time distribution 3.1. Parameteric estimation The parametric estimation of the VG process has been the subject of several research papers re- cently. Cervellera and Tucci (2017) confirms that, it is impossible to replicate the estimation ob- tained in Madan et al. (1998). In order to validate Fig. 1. Sample path of VG(σ=0.5, θ=0.5, ν=0.5, μ=0) this latter, the investigation of the computational problems related to finding the maximum likeli- hood estimator of the parameters was performed. Both R, MATLAB and a non-standard optimiza- 2.2. Variance Gamma process as the tion software such as Ezgrad were used. The complexity of log-likelihood function is due to the difference of two Gamma processes presence of many local optima and Bessel func- As mentioned by Madan et al. (1998), the VG tion of the second kind. A new algorithm (Bee process can be written as a difference of two in- et al. (2018)) was developed based on Nitithum- creasing independent gamma process since it is a bundit and Chan (2015) corresponding multivari- process of finite variation. The first represents the ate versions, for maximum likelihood estimation increases in the process and the second represents (MLE) of the univariate VG distribution and re- the decrease in the process. The process can be sults were compared with the results obtained written as the law of the characteristic function of from R Variance Gamma (Scott et al. (2018)) and VG(σ, ν, θ): the ghyp packages (Luethi et al. (2013)). The distribution of the model is flexible enough 1 2 2 −1/ν ΦVG(u; σ, ν, θ)=(1− iuθν + σ νu ) (6) to accommodate skewness and leptokurtosis but 2 the obtention of MLE is difficult because the prob- This distribution is infinitely divisible and the ability density function (PDF) is not in closed (VG) form and is unbounded for small values of the (VG) = ≥ 0 VG process X Xt ,t can be shape parameter. The log-likelihood function is Proceedings of the 30th European Safety and Reliability Conference and the 15th Probabilistic Safety Assessment and Management Conference 2889 given as follows: of data to analyse the robustness of the algorithms. T The estimation will be performed with 1000 and 2 ( − ) ( )=T ( )+ x μ θ − 10000 number of samples having a sample length L μ, θ, σ, ν 2 log Π 2 t=1 σ √ of 10000 using different values of parameters T T 2 2 + 2 | − | VG(σ=0.5, θ=0.5, ν=0.5, μ=0), VG(σ=1, θ=1, (Γ( ) )+ ( ( σ θ x μ ))+ log a θ log Kν−0.5 2 (7) t=1 t=1 σ ν=1, μ=0) and VG(σ=3, θ=1.5, ν=2, μ=0). The obtained RMSE (Root Mean Square Error) results T 1 1 ( − )( (| − |− (2 2 + 2))) were presented in Table 1. It is noted that the esti- μ 2 log x μ 2log σ θ t=1 mation of the VG parameters or the log-likelihood This equation explains why the researchers are function is affected by the addition or removal confronting difficulties in the treatment and the of observations (Cervellera and Tucci (2017)). It calculation of the MLE. also shows enormous changes in the estimated The parametric estimation of the VG in this parameter values which proves the impact of the paper is performed using Variance Gamma and number of data on the estimation. However, the ghyp package in R. In Variance Gamma pack- estimation of the parameter ν using this package age, the VgFit function allows the user to employ did not show big improvement while the number the Nelder-Mead, the BFGS method (Broyden- of data increases, and this is due to the presence Fletcher-Goldfarb-Shanno) or a Newton-type al- of the Bessel function in the log-likelihood ex- gorithm while the fit.VGuv function of the ghyp pression. This is also related to the computational complexity in the estimation of the loglikelihood package is based on the Nelder–Mead algorithm. =05 Both packages use the Nelder–Mead algorithm as when ν . . implemented in the optim R function. BFGS is The results of the RMSE using the same data also implemented in optim, whereas the Newton- and same number of samples show that the ghyp type algorithm is from the nlm (Non-Linear Min- package is considered to be more efficient in imization) function. the estimation comparing to the Variance Gamma package. It is essential to mention that the es- timation depends only on the data and on the Table 1. RMSE results from fit.Vguv and VGfit methods starting values. Both methods are efficient, but Parameters ghyp VGfit ghyp package provides good estimation results even though it shows some sensitivity when the VG(σ=0.5, θ=0.5, ν=0.5, μ=0) =05 N = 1000, n= 100 value of ν . . As the work in this study was μ 0.003859279 0.013340216 performed using the simulated data, the estimation σ 0.0005072501 0.0637157550 results were precise. In the case of the estimation θ 0.004379053 0.033325612 of real data, it is essential to verify the results ν 0.4397658 0.8360905 using statistical tests before the beginning of the N = 10000, n= 100 prognostics. μ 0.0002849596 0.0059760528 σ 4.451899e-05 5.009331e-02 θ 0.0003302026 0.0238633483 3.2. Failure Time ν 0.44141475 0.5795967 VG(σ=1, θ=1, ν=1, μ=0) The system is considered to have failed if the level N = 1000, n= 100 of degradation reaches the threshold referred to as μ 0.001631385 0.004462029 the failure threshold. In order to avoid failure and σ 0.001749834 0.131206113 an undesirable period of inactivity, it is of great θ 0.004413952 0.352547189 ν interest to model the evolution of the degradation 0.008584044 0.365553041 and to be able to predict the failure. N = 10000, n= 100 Considering a process (Xt)t≥0 with initial μ 0.0002031116 0.0002094777 = σ 0.0002094777 0.0001527683 value X0 x0, the failure time is defined to be the θ 0.0004655316 0.0004709406 first crossing time of the failure threshold denoted ν 0.00054044 0.00105044 by L and it is defined to be the first hitting time VG(σ=3, θ=1.5, ν=2, μ=0) which can be written as follow: N = 1000, n= 100 ∗ =inf{ ≥ 0; ≥ } μ 1.652700e-05 1.951251e-05 tL t Xt L (8) σ 0.1771473 0.4843569 θ It could be easy to obtain the distributional 0.1508169 0.7965047 ∗ ν 0.1394611 0.2002643 properties of tL,if(Xt)t≥0 is a Lévy subordinator N = 10000, n= 100 or a time changed Brownian motion with a path μ 2.826034e-06 7.711633e-06 continuous process as the underlying time process σ 0.1761572 0.3642330 (Borodin and Salminen (2012)). But in the case θ 0.1301680 0.6865433 ν 0.1403962 0.08342571 of VG process, the underlying time process is a gamma process which is not path continuous, the situation will become more complicated and These two methods are tested on different sets challenging. This subject made a lot of discussion Proceedings of the 30th European Safety and Reliability Conference and the 15th Probabilistic Safety Assessment and Management Conference 2890 among the researchers. In the articles of Hurd 4. Goodness of fit (2009) and Hurd and Kuznetsov (2009), it is men- As mentioned before, a non-monotonic and a tioned that the usual first hitting time cannot be non-linear process is recommended. Wiener, studied. Thus, the author proposed to calculate B & S and VG processes are used to model the first hitting time of the second kind which can the degrdataion, data calibration and their ade- be defined as the first time when the time change quacy is compared. The adequacy of the pro- process (gamma process) exceeds the failure time cess is performed with four goodness of fit tests: of the Brownian motion (B). ∗ Kolmogorov-test (KS), Chis-square test (chisq), Let’s consider T to be the the first pas- ∗ Anderson-test (AD), Cramer-test (Cramer). The sage time of B which is defined as T = inf { : + + ( ) ≥ 0} goodness of fit typically summarizes the dissim- T x0 θt σW t . And for a ilarity between observed values and the values TCBM process like VG process t = G , the X B t expected under the model in question. first passage time of the second kind of Xt is ∗ =inf{ : ≥ ∗} The three processes are tested in order to deter- defined as t t Gt T (Hurd and mine the most flexible process which can cover all Kuznetsov (2009)). Based on this definition, one the fluctuations and changes of the system based can notice that the first hitting time of the second on simulated data. Data are simulated respectively kind shares certain properties with the typical first from VG, Wiener, B & S. The goodness of fit test hitting time and can be implemented in a similar results are presented in Table 2. The probability way. The advantage of this new concept is that of error (significance level) is set as value =005 it can be calculated efficiently in many situations p . where the normal failure time cannot be com- i.e. confidence of 95% and the results are analysed puted. For this purpose, it is essential to introduce and presented in Table 2. the equation 9 and equation 10 which are used to In this study, data are generated from the three calculate the failure time distribution of second different processes (VG, Wiener and B & S) and kind given by the equation 11. The structure the parameters are estimated. Later, the data are ∗ regenerated and the goodness of fit tests is used to function p1(x0; s, x1) for general LSBMs associ- ∗ verify to which model these data fit the most. ated with failure time tL is introduced using this formula: Table 2. Results of goodness of fit : VG, Wiener, B & S −s/ν ∗ exp(β(x1 − x0)) (1 + iνz) ( 0; 1)=  Data p1 x s, x 2 2Πν R β − 2iz   2 2 Process KS Chisq Cramer AD exp(−x0 β − 2iz) × exp(|x1| β − 2iz)   (9) Data from VG process 2 2 Ei −|x1| (α + β − 2iz) − exp(−|x1| β − 2iz) VG 0.2408 0.2419 0.2255 0.2466  −| | ( − 2 − 2 ) Wiener 0.00692 0.026457 5.967e-03 0.003193 Ei x1 α β iz B.S 4.218e-04 0.0247 0.023113 0.006120  Data from Wiener process 2 where α = 2/ν + β and Ei(x) presents the VG 0.00692 0.3876 0.2517 0.3165 function of the exponential integral. And in the Wiener 0.2439 0.086457 5.714 e-03 0.1045 particular case when s =0: B.S 9.158e-04 6.643e-03 5.967e-03 0.003193 Data fromB&Sprocess

∗ exp(β(x1 − x0) − α(x0 + |x1|)) VG 0.2452 0.4309 0.3750 0.7362 p1(x0;0,x1)= (10) ν(x0 + |x1|) Wiener 4.71e-05 3.94e-06 6.59e-08 4.51e-06 B.S 0.375 0.2183 0.1632 0.0853 The failure time distribution of the second type can be estimated by iterating the following for- It is noted that, the goodness of fit tests applied mula 2 to 3 times and it can be written as: on the data generated from the VG process re- 0 jected both Wiener and B & S processes. In the ∗ ∗ pi (x0; s)= p1(x0;0,x1)dx+ case of data generated from Wiener process, VG −∞ was accepted with 95% confidence according to +∞ s (11) ∗ ∗ the results of chisq, Cramer and AD tests but it dy dup1(x0; u, y)pi−1(y; s − u) i ≥ 2 0 0 was rejected by KS test. On the other hand, B & S is rejected by all the four tests. According to It is important to have the numerical calculation the goodness of fit tests, the VG process can fit of the failure time distribution and in order to to data originated from B & S process, however achieve that it was crucial to reproduce the work the Wiener process was rejected by all the tests. It presented in Hurd (2009). The equations used can be concluded that the VG is the most flexible are very complex which made their numerical model able to fit to the different type of data computation so complicated. To overcome this, Table 2. a simulation method will be used in order to ap- The data were generated from the three differ- proximate the distribution of the failure time. ent processes (VG, Wiener, BS) with the same Proceedings of the 30th European Safety and Reliability Conference and the 15th Probabilistic Safety Assessment and Management Conference 2891

number of samples N=10000. Consequentely, the operates under well-defined conditions from a assumption that the size of the samples will lead to well-defined date that ends with a failure of the reject the null hypothesis for all the processes can system. The main objective of the prognostics not be taken in consideration, since they accepted is to provide information for good decisions as the results for VG process and another process in well as risk measures for the monitored systems. each scenario. To assure a better prognostics, different type of It can be mentioned that VG process is able to metrics must be considered. The main one is the fit several types of data generated from different RUL (Remaining Useful Life) or the residual time processes due to its flexibility. It is due to its before failure like TTF (Time To Failure). The representation as a random change of the time in accuracy in the prediction of the RUL or TTF a Brownian motion, following a gamma process. of the defected components is significant to the This time change allows to reflect the random prognostics of the system and it will assist the speedups and slowdowns in real time. operators to replace the components at the appro- priate time. The failure of the system is signaled by the passage of the degradation level through 5. Prognostics on simulated data its predefined failure threshold. The failure time It is always valuable to evaluate a parametric defined in the equation 8 is used to give a more model of the FT distribution since it provides the analytical definition of the RUL. In this case, the possibility to integrate the evolution of the state calculation of the failure time is performed condi- of the system at the inspection time. In other tionally to the state of the system at the present word, the expression of the FT distribution will be time (time of the last inspection ti), which can updated at every inspection based on the collected introduce the RUL as the first exceeding time of information. First, the histogram of the FT real- indicator X(t) to the threshold L follows: isation is studied graphically in order to analyse ∗ tt := inf {h ≥ 0; X(ti + h) ≥ L | X(ti)

Fig. 2. Histogram of FT of (a) first, (b) second and (c) third threshold satisfactory, when the first passage time is located prognostics results showed that in most of the in the confidence interval. Same work was also cases VG was able to provide a good prognostics reproduced to the same chosen path at time T = about the system. Although this work was based tinsp and the new location of the FT were obtained on simulated data, it should be possible to extend and compared to the new histogram. This work the approach to more complex real data collected was reproduced with n number of samples and from complex systems. The model accuracy can similar satisfactory results were found. Based on be further improved by using deeper research on these results, one can affirm that the real FT is each parameter effect. always located in the confidence interval, which can lead to affirm that VG can offers a good prognostics. Acknowledgement The authors would like to acknowledge the valu- able financial support of the European Regional 6. Conclusion Development Fund (FEDER) and the Departmen- In this paper, the VG process is proposed as a tal Council of Aube, France during this research. degradation model. An analysis of its parame- ters and properties is carried out in order to jus- References tify its use in degradation modelling. The esti- mation of the VG parameters is obtained using Abramowitz, M. (1972). Elementary analytical two R packages and the complexity of the Log- methods. Handbook of Mathematical Func- likelihood function is discussed. The evaluation tions with Formulas, Graphs, and Mathemati- of the two packages is insured by the calculation cal Tables, 9–63. of the RMSE. As a degradation model, it was Ahmadzadeh, F. and J. Lundberg (2014). Re- important to study the distribution of its failure maining useful life estimation. International time. Due to the difficulty in determining the dis- Journal of System Assurance Engineering and tribution of FT analytically, the simulation method Management 5(4), 461–474. is proposed as an alternative. The data generated Arriojas, M., Y. Hu, S.-E. Mohammed, and G. Pap from different processes: VG, Wiener, B&Sare (2007). A delayed black and scholes formula. used to calibrate the choice of the model and based Stochastic Analysis and Applications 25(2), on a comparative study, most of the goodness of 471–492. fit tests accepted VG as the appropriate model. Bee, M., M. M. Dickson, and F. Santi (2018). Since it is important to find the distribution of the Likelihood-based risk estimation for variance- FT, simulations are used to define it and goodness gamma models. Statistical Methods & Appli- of fit tests is used to fit it to some distributions. cations 27(1), 69–89. Unfortunately, all the distributions used were not Borodin, A. N. and P. Salminen (2012). Hand- able to fit according to all goodness of fit tests book of Brownian motion-facts and formulae. to the distribution of the FT. The concept of FT Birkhäuser. is introduced into the definition of RUL which Cervellera, G. P. and M. P. Tucci (2017). A note is important in suggesting the prognostics. The on the estimation of a gamma-variance process: Proceedings of the 30th European Safety and Reliability Conference and the 15th Probabilistic Safety Assessment and Management Conference 2893

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