1

1. Gamma process with noise model
applied on degradation and failure
phenomenon
K. Le Son, A. Barros, M. Fouladirad
Université de Technologie de Troyes
Institut Charles Delaunay, UMR CNRS 6279, France
CM 2011, 20-22 June 2011

2. Context
• Many measurements of systems, components or sensors can provide
degradation informations but may be difficult to analyze.
• The aim is to model degradation phenomena and to estimate the
remaining useful life (RUL) based on degradation measures.
• Probabilistic models applied on the degradation process by using the
stochastic processes open the new research way for prognostic.
Objectives
• Using the prognostic probabilistic approach in the order to compare with
the exciting non-probabilistic methods applied on the 2008 Prognostic
Health Management data.
• Construction of a degradation indicator from the sensors measurements
(2008 Prognostic Health Management (PHM) Challenge data).
• Using a stochastic process to model the deterioration of components
(Remaining Useful Life estimation).
K. Le Son, A. Barros, M. Fouladirad Gamma process with noise model applied on degradation phenomenon 2/16

3. Outline
1 Deterioration model
2 Methodology for Remaining Useful Life Estimation
3 Application to 2008 PHM Challenge data
4 Conclusions

4. Outline
1 Deterioration model
2 Methodology for Remaining Useful Life Estimation
3 Application to 2008 PHM Challenge data
4 Conclusions

5. Deterioration model Methodology Application Conclusions
Deterioration model
Deterioration model construction
• Note that:
Y(Y1 , ..., Yn ) : the observation vector.
X(X1 , ..., Xn ) : the non-observable states of system.
• Our deterioration model :
Yj = f (Xj , ǫj ) = Xj + ǫj
where :
ǫj , j = 1, ..., n : the independent gaussian random variables with
standard deviation σj and mean equals to zero.
Yj , j = 1, ..., n : the degradation indicator observed at time tj .
Xj , j = 1, ..., n : the non-observable state value at time tj .
K. Le Son, A. Barros, M. Fouladirad Gamma process with noise model applied on degradation phenomenon 3/16

6. Deterioration model Methodology Application Conclusions
Deterioration model
Non-stationary Gamma process
• In the order to estimate the RUL of components, the non-observable
states are considered as a non-stationary Gamma process.
• The initial state X0 = 0.
• (Xj )j≥0 is monotone, increasing.
• The Gamma increments δXj = Xj − Xj−1 , j = 1, 2, ..., n are independent
and have the Gamma density as follows:
β v (tj )−v (tj −1 )
fδXj (δ|v (tj )−v (tj−1 ), β) = δ v (tj )−v (tj −1 )−1 e −βδ I(0,∞) (δ)
Γ(v (tj ) − v (tj−1 ))
(1)
∞
Γ(u) = z=0 z u−1 e −z dz : Gamma function for u > 0.
IA (δ) = 1 for δ ∈ A, IA (δ) = 0 for δ ∈ A.
/
Shape function v (t) = αt b and scale parameter β.
K. Le Son, A. Barros, M. Fouladirad Gamma process with noise model applied on degradation phenomenon 4/16

7. Deterioration model Methodology Application Conclusions
Deterioration model
Failure threshold
L
Xn
RU L(tn )
Xi
Γ(v(ti ) − v(tj ), β)
Xj
tj ti tn tL
Figure: Non-stationary Gamma process
K. Le Son, A. Barros, M. Fouladirad Gamma process with noise model applied on degradation phenomenon 5/16

8. Deterioration model Methodology Application Conclusions
Deterioration model
Joint distribution of system state
• For estimating the RUL, the joint conditional density of X figured out the
observation vector Y is calculated as follows:
n g 2 (xj ,Yj )
b b (− )
−βxn α(tj −tj −1 )−1 2σ2
µX/Y (x1 , ..., xn ) = K1 e (xj − xj−1 ) e i |g ′(xj , Yj )|
j=1
(2)
∂g (.,y )
where g ′(., y ) = ∂y
and K1 is the coefficient defined as follows:
n
1 b b g 2 (xj , Yj )
= ... e −βxn (xj −xj−1 )α(tj −tj −1 )−1 e ( − )|g ′(xj , Yj )|dx1 ...dxn
K1 2σ 2
j=1
(3)
• It’s difficult to calculate the coefficient K1 ⇒ We use the Gibbs sampler
algorithm to approximate the conditional density µX/Y .
K. Le Son, A. Barros, M. Fouladirad Gamma process with noise model applied on degradation phenomenon 6/16

9. Deterioration model Methodology Application Conclusions
Deterioration model
Remaining Useful Life estimation
• Remaining Useful Lifetime (RUL) estimation is based on the failure
probability at the next inspection given the n observations Y1 , ..., Yn .
• The distribution function of RUL(tn ) figured out the observations is
defined as follows:
FRUL(tn ) (h) = P(Xtn +h > L|Xn > L, Y1 , ..., Yn ) (4)
= ¯
Fα((tn +h)b −t b ),β (l − x).fL (l ).µXn /Y1 ,...,Yn dldx
n
¯
Fα((tn +h)b −t b ),β : the reliability function of Gamma process with
n
shape function α((tn + h)b − tn ) and scale parameter β.
b
µXn /Y1 ,...,Yn : the conditional density of Xn .
fL (l ) : the density function of the failure threshold.
K. Le Son, A. Barros, M. Fouladirad Gamma process with noise model applied on degradation phenomenon 7/16

10. Outline
1 Deterioration model
2 Methodology for Remaining Useful Life Estimation
3 Application to 2008 PHM Challenge data
4 Conclusions

11. Deterioration model Methodology Application Conclusions
Methodology for RUL estimation
Gibbs algorithm
• Gibbs sampling : a MCMC algorithm used to generate random variables
from a distribution without having to calculate the density.
• A random variable from µX/Y (x1 , ..., xn ) is generated as follows:
Given a starting values set: (z1 0 , ..., zn 0 ).
1 1 1
The next generated values set (z1 , z2 , ..., zn ) is defined as follows:
◮ z 1 is generated from µ1 =µX/Y (z1 |z 0 , ..., z 0 ).
1 Z1 2 n
◮ z 1 is generated from µ1 =µX/Y (zj |z 0 , ..., z 0 , z 0 , z 0 ),
j Zj 1 j−1 j+1 n
j = 2, ..., n − 1.
◮ z 1 is generated from µ1 =µX/Y (zn |z 0 , ..., z 0
n Zn 1 n−1 ).
• Each conditional density µX/Y (xj /x1 , ..., xj−1 , xj+1 , ..., xn ), j = 1, ..., n is
simulated by Gibbs algorithm ⇒ Obtain the output value zjq at q th
q
sequence from µZj .
• Each observations vector Y gives us an approximation vector Z of X.
K. Le Son, A. Barros, M. Fouladirad Gamma process with noise model applied on degradation phenomenon 8/16

12. Deterioration model Methodology Application Conclusions
Methodology for RUL estimation
Parameters estimation by SEM method
• The parameters of model (α, β, b, σ 2 ) are estimated by using the
Stochastic Expectation Maximization (SEM) method on the observations
set (Y1 , ..., Yn ).
• SEM is an iterative algorithm :
q
At q th sequence, the Gibbs outputs (Z1 , ..., Zn ) are used to obtain
q
the sequence parameters (αq , βq , bq , σq ).
Simulating with (αq , βq , bq , σq ) by Gibbs algorithm: the outputs
q+1 q+1
(Z1 , ..., Zn ) are obtained.
And so on, we obtain (αq+1 , βq+1 , bq+1 , σq+1 ), this procedure is
realized until the sequence parameters values are steady.
K. Le Son, A. Barros, M. Fouladirad Gamma process with noise model applied on degradation phenomenon 9/16

13. Deterioration model Methodology Application Conclusions
Methodology for RUL estimation
Failure state
Outputs of Gibbs
Cycle
Figure: The Gibbs outputs trajectories
K. Le Son, A. Barros, M. Fouladirad Gamma process with noise model applied on degradation phenomenon 10/16

14. Deterioration model Methodology Application Conclusions
Methodology for RUL estimation
RUL estimation
• The conditional distribution FRUL(tn ) (h) can be estimated by Gibbs
algorithm as follows:
Q0 +Q
ˆ 1 ¯ q
FRUL(tn ) (h) = Fα((tn +h)b −t b ),β (l − zn ).fL (l )dl (5)
Q n
q=Q0 +1
zn : the output value of Gibbs at the q th sequence.
q
Q0 : number of iterations to get the convergence state.
Q : number of iterations to get an accepted estimation.
• The estimated RUL is considered as the expectation of RUL(tn ):
RULestimated (tn ) = E (RUL(tn )) = ˆ′
h.FRUL(tn ) (h)dh (6)
K. Le Son, A. Barros, M. Fouladirad Gamma process with noise model applied on degradation phenomenon 11/16

15. Deterioration model Methodology Application Conclusions
Methodology for RUL estimation
Failure threshold
L
Zn
Approximated trajectory of X
Degradation
RU L(tn )
Observation vector Y
Cycle tn TL
Figure: Remaining Useful Life Estimation
K. Le Son, A. Barros, M. Fouladirad Gamma process with noise model applied on degradation phenomenon 12/16

16. Outline
1 Deterioration model
2 Methodology for Remaining Useful Life Estimation
3 Application to 2008 PHM Challenge data
4 Conclusions

17. Deterioration model Methodology Application Conclusions
Application to PHM data
2008 PHM Challenge data
3 Operational variables Measurements of 21 sensors
Unit Cycle OP1 OP2 OP3 SM1 SM2 … SM21
1
1 …
T1
...
1
218 …
T218
• Two sub-data set : the training data set and the testing data set.
• The observation set of model Y(Y1(i ) , ..., Yni ) ), i = 1, ..., 218 is built by
(i
using the Principal Component Analysis on the training data set.
• The testing data set is used to estimate the RUL for each testing unit.
K. Le Son, A. Barros, M. Fouladirad Gamma process with noise model applied on degradation phenomenon 13/16

18. Deterioration model Methodology Application Conclusions
Application to PHM data
Estimated parameters of model
0.44 140 1.22
135 1.21
0.42
130 1.2
0.4
125 1.19
α
b
β
0.38 120 1.18
115 1.17
0.36
110 1.16
0.34
105 1.15
0.32 100
0 50 100 150 200 0 50 100 150 200 0 50 100 150 200
Sequence Sequence Sequence
Figure: The estimated sequences parameters
• Q = 200 : all the estimated sequences parameters have a stable behavior.
• Estimated parameters considered as the mean of Q sequence parameters.
α
ˆ ˆ
β ˆ
b σ2
ˆ
Estimated parameter 0.3613 103.07 1.1462 0.3562
Variance 0.0040 0.7055 0.0025 0.0012
K. Le Son, A. Barros, M. Fouladirad Gamma process with noise model applied on degradation phenomenon 14/16

19. Deterioration model Methodology Application Conclusions
Result
Performance criteria
• Applying our method to testing units, an estimated RUL set is obtained.
• Performance criteria : Root mean squared error (RMSE) corresponds with
the error in predicting the number of remaining time cycles:
N
RMSE = (di )2 (7)
i =1
where di = estimated RULi − actual RULi is the prediction error of unit i.
Comparison
• Obtained result RMSE is 517,45.
• The winners in PHM challenge : Non-probabilistic model based on the
neural networks of Peel (2008) and Heimes (2008) obtain respectively the
RMSE = 519, 8 and RMSE = 984.
K. Le Son, A. Barros, M. Fouladirad Gamma process with noise model applied on degradation phenomenon 15/16

20. Outline
1 Deterioration model
2 Methodology for Remaining Useful Life Estimation
3 Application to 2008 PHM Challenge data
4 Conclusions

21. Deterioration model Methodology Application Conclusions
Conclusions
Remarks
• The present paper proposes a prognostic probabilistic method for a
deterioration modeled by a Gamma process with gaussian noise.
• The obtained result on the PHM data are acceptable when we compare
with some other papers using non-probabilistic methods in PHM
conference.
Further works
• Using the RUL distribution in the order to propose a maintenance policy
for industrial systems or components.
K. Le Son, A. Barros, M. Fouladirad Gamma process with noise model applied on degradation phenomenon 16/16

22. Thank you for your attention