# Reliability Analysis Based on a Gamma-Gaussian Deconvolution Degradation Modeling with Measurement Error

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## Abstract

**:**

## 1. Introduction

## 2. Modeling of the Observed Degradation via Gamma Process

## 3. Obtaining the True Degradation Distribution via Deconvolution

## 4. The Effect of the Measurement Error over the First-Passage Time Distributions

## 5. Case Study

#### 5.1. Estimation of Parameters for the Observed Degradation

#### 5.2. Characterization of the Measurement Error and Its Effect

#### 5.3. Comparison of the First-Passage Time Distributions

## 6. Extension for Non-Gaussian Measurement Errors

## 7. Concluding Remarks and Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Wang, Z.; Li, J.; Zhang, Y.; Fu, H.; Liu, C. A novel wiener process model with measurement errors for degradation analysis. Eksploatacja i Niezawodnosc
**2016**, 18, 396. [Google Scholar] [CrossRef] - Ye, Z.S.; Wang, Y.; Tsui, K.L.; Pecht, M. Degradation data analysis using Wiener processes with measurement errors. IEEE Trans. Reliab.
**2013**, 62, 772–780. [Google Scholar] [CrossRef] - Li, J.; Wang, Z.; Liu, X.; Zhang, Y.; Fu, H.; Liu, C. A Wiener process model for accelerated degradation analysis considering measurement errors. Microelectron. Reliab.
**2016**, 65, 8–15. [Google Scholar] [CrossRef] - Si, X.S.; Chen, M.Y.; Wang, W.; Hu, C.H.; Zhou, D.H. Specifying measurement errors for required lifetime estimation performance. Eur. J. Oper. Res.
**2013**, 231, 631–644. [Google Scholar] [CrossRef] - Whitmore, G. Estimating degradation by a Wiener diffusion process subject to measurement error. Lifetime Data Anal.
**1995**, 1, 307–319. [Google Scholar] [CrossRef] - Rabinovich, S.G. Evaluating Measurement Accuracy; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar] [CrossRef]
- Lu, D.; Xie, W.; Pandey, M.D. An efficient method for the estimation of parameters of stochastic gamma process from noisy degradation measurements. Proc. Inst. Mech. Eng. Part J. Risk Reliab.
**2013**, 227, 425–433. [Google Scholar] [CrossRef] - Pulcini, G. A perturbed gamma process with statistically dependent measurement errors. Reliab. Eng. Syst. Saf.
**2016**, 152, 296–306. [Google Scholar] [CrossRef] - Giorgio, M.; Mele, A.; Pulcini, G. A perturbed gamma degradation process with degradation dependent non-Gaussian measurement errors. Appl. Stoch. Model. Bus. Ind.
**2018**, 35, 198–210. [Google Scholar] [CrossRef] - Meister, A. Density estimation with normal measurement error with unknown variance. Stat. Sin.
**2006**, 16, 195–211. [Google Scholar] - Peng, C.Y.; Tseng, S.T. Mis-specification analysis of linear degradation models. IEEE Trans. Reliab.
**2009**, 58, 444–455. [Google Scholar] [CrossRef] - Xie, Y.; Wang, X.; Story, M. Statistical methods of background correction for Illumina BeadArray data. Bioinformatics
**2009**, 25, 751–757. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kallen, M.J.; van Noortwijk, J.M. Optimal maintenance decisions under imperfect inspection. Reliab. Eng. Syst. Saf.
**2005**, 90, 177–185. [Google Scholar] [CrossRef] - Shen, Y.; Shen, L.; Xu, W. A Wiener-based degradation model with logistic distributed measurement errors and remaining useful life estimation. Qual. Reliab. Eng. Int.
**2018**, 34, 1289–1303. [Google Scholar] [CrossRef] - Wang, P.; Tang, Y.; Bae, S.J.; Xu, A. Bayesian Approach for Two-Phase Degradation Data Based on Change-Point Wiener Process With Measurement Errors. IEEE Trans. Reliab.
**2018**, 67, 688–700. [Google Scholar] [CrossRef] - Pan, D.; Lu, S.; Liu, Y.; Yang, W.; Liu, J.B. Degradation Data Analysis Using a Wiener Degradation Model With Three-Source Uncertainties. IEEE Access
**2019**, 7, 37896–37907. [Google Scholar] [CrossRef] - Sun, H.; Pan, J.; Zhang, J.; Cao, D. Non-linear Wiener process–based cutting tool remaining useful life prediction considering measurement variability. Int. J. Adv. Manuf. Technol.
**2020**, 107, 4493–4502. [Google Scholar] [CrossRef] - Tang, S.; Yu, C.; Sun, X.; Fan, H.; Si, X. A Note on Parameters Estimation for Nonlinear Wiener Processes With Measurement Errors. IEEE Access
**2019**, 7, 176756–176766. [Google Scholar] [CrossRef] - Liu, D.; Wang, S. Reliability estimation from lifetime testing data and degradation testing data with measurement error based on evidential variable and Wiener process. Reliab. Eng. Syst. Saf.
**2021**, 205, 107231. [Google Scholar] [CrossRef] - Li, J.; Wang, Z.; Liu, C.; Qiu, M. Stochastic accelerated degradation model involving multiple accelerating variables by considering measurement error. J. Mech. Sci. Technol.
**2019**, 33, 5425–5435. [Google Scholar] [CrossRef] - Sun, B.; Li, Y.; Wang, Z.; Ren, Y.; Feng, Q.; Yang, D. An improved inverse Gaussian process with random effects and measurement errors for RUL prediction of hydraulic piston pump. Measurement
**2021**, 173, 108604. [Google Scholar] [CrossRef] - Chen, X.; Ji, G.; Sun, X.; Li, Z. Inverse Gaussian–based model with measurement errors for degradation analysis. Proc. Inst. Mech. Eng. Part O J. Risk Reliab.
**2019**, 233, 1086–1098. [Google Scholar] [CrossRef] - Hao, S.; Yang, J.; Berenguer, C. Degradation analysis based on an extended inverse Gaussian process model with skew-normal random effects and measurement errors. Reliab. Eng. Syst. Saf.
**2019**, 189, 261–270. [Google Scholar] [CrossRef] [Green Version] - Liu, X.; Wu, Z.; Cui, D.; Guo, B.; Zhang, L. A Modeling Method of Stochastic Parameters’ Inverse Gauss Process Considering Measurement Error under Accelerated Degradation Test. Math. Probl. Eng.
**2019**, 2019, 1–11. [Google Scholar] [CrossRef] - Chen, X.; Sun, X.; Si, X.; Li, G. Remaining Useful Life Prediction Based on an Adaptive Inverse Gaussian Degradation Process With Measurement Errors. IEEE Access
**2020**, 8, 3498–3510. [Google Scholar] [CrossRef] - Plancade, S.; Rozenholc, Y.; Lund, E. Generalization of the normal-exponential model: Exploration of a more accurate parametrisation for the signal distribution on Illumina BeadArrays. BMC Bioinform.
**2012**, 13, 329. [Google Scholar] [CrossRef] [PubMed] - Sarder, P.; Nehorai, A. Deconvolution methods for 3-D fluorescence microscopy images. IEEE Signal Process. Mag.
**2006**, 23, 32–45. [Google Scholar] [CrossRef] - Swedlow, J.R. Quantitative fluorescence microscopy and image deconvolution. Methods Cell Biol.
**2013**, 114, 407–426. [Google Scholar] [CrossRef] - Xu, L.; Ren, J.S.; Liu, C.; Jia, J. Deep convolutional neural network for image deconvolution. Adv. Neural Inf. Process. Syst.
**2014**, 27, 1790–1798. [Google Scholar] - Rodriguez-Picon, L.A.; Perez-Dominguez, L.; Mejia, J.; Perez-Olguin, I.J.; Rodriguez-Borbon, M.I. A Deconvolution Approach for Degradation Modeling With Measurement Error. IEEE Access
**2019**, 7, 143899–143911. [Google Scholar] [CrossRef] - Zinde-Walsh, V. Measurement error and deconvolution in spaces of generalized functions. Econom. Theory
**2014**, 30, 1207–1246. [Google Scholar] [CrossRef] [Green Version] - Wang, X.F.; Wang, B. Deconvolution estimation in measurement error models: The R package decon. J. Stat. Softw.
**2011**, 39. [Google Scholar] [CrossRef] [Green Version] - Neumann, M.H. Deconvolution from panel data with unknown error distribution. J. Multivar. Anal.
**2007**, 98, 1955–1968. [Google Scholar] [CrossRef] [Green Version] - Lu, C.J.; Meeker, W.O. Using degradation measures to estimate a time-to-failure distribution. Technometrics
**1993**, 35, 161–174. [Google Scholar] [CrossRef] - Zhang, M.; Gaudoin, O.; Xie, M. Degradation-based maintenance decision using stochastic filtering for systems under imperfect maintenance. Eur. J. Oper. Res.
**2015**, 245, 531–541. [Google Scholar] [CrossRef] - Xie, M.; Goh, T.; Ranjan, P. Some effective control chart procedures for reliability monitoring. Reliab. Eng. Syst. Saf.
**2002**, 77, 143–150. [Google Scholar] [CrossRef] - Soltan, H. Advances in Control Charts for Reliability. In Proceedings of the 2019 Industrial & Systems Engineering Conference (ISEC), Jeddah, Saudi Arabia, 19–20 January 2019. [Google Scholar] [CrossRef]
- Aslam, M.; Khan, N.; Albassam, M. Control Chart for Failure-Censored Reliability Tests under Uncertainty Environment. Symmetry
**2018**, 10, 690. [Google Scholar] [CrossRef] [Green Version] - Faraz, A.; Saniga, E.M.; Heuchenne, C. Shewhart Control Charts for Monitoring Reliability with Weibull Lifetimes. Qual. Reliab. Eng. Int.
**2014**, 31, 1565–1574. [Google Scholar] [CrossRef] - Singpurwalla, N.D. Survival in Dynamic Environments. Stat. Sci.
**1995**, 10, 86–103. [Google Scholar] [CrossRef] - Van Noortwijk, J. A survey of the application of gamma processes in maintenance. Reliab. Eng. Syst. Saf.
**2009**, 94, 2–21. [Google Scholar] [CrossRef] - Bagdonavicius, V.; Nikulin, M. Accelerated Life Models: Modeling and Statistical Analysis; Chapman and Hall/CRC Press: Boca Raton, FL, USA, 2001. [Google Scholar]
- Bagdonavicius, V.; Nikulin, M.S. Estimation in degradation models with explanatory variables. Lifetime Data Anal.
**2001**, 7, 85–103. [Google Scholar] [CrossRef] [PubMed] - Park, C.; Padgett, W. Accelerated degradation models for failure based on geometric Brownian motion and gamma processes. Lifetime Data Anal.
**2005**, 11, 511–527. [Google Scholar] [CrossRef] - Park, C.; Padgett, W.J. Stochastic degradation models with several accelerating variables. IEEE Trans. Reliab.
**2006**, 55, 379–390. [Google Scholar] [CrossRef] - Brigham, E.O.; Brigham, E.O. The Fast Fourier Transform; Prentice-Hall: Englewood Cliffs, NJ, USA, 1974; Volume 7. [Google Scholar]
- Cooley, J.W.; Tukey, J.W. An algorithm for the machine calculation of complex Fourier series. Math. Comput.
**1965**, 19, 297–301. [Google Scholar] [CrossRef] - Winograd, S. On computing the discrete Fourier transform. Math. Comput.
**1978**, 32, 175–199. [Google Scholar] [CrossRef] - Cooley, J.; Lewis, P.; Welch, P. Application of the fast Fourier transform to computation of Fourier integrals, Fourier series, and convolution integrals. IEEE Trans. Audio Electroacoust.
**1967**, 15, 79–84. [Google Scholar] [CrossRef] - Abate, J.; Whitt, W. The Fourier-series method for inverting transforms of probability distributions. Queueing Syst.
**1992**, 10, 5–87. [Google Scholar] [CrossRef] - Plancade, S.; Whitt, Y. NormalGamma: Normal-Gamma Convolution Model. R Package Version 1.1. 2013. Available online: https://CRAN.R-project.org/package=NormalGamma (accessed on 15 January 2020).
- Rodríguez-Picón, L.A.; Rodríguez-Picón, A.P.; Méndez-González, L.C.; Rodríguez-Borbón, M.I.; Alvarado-Iniesta, A. Degradation modeling based on gamma process models with random effects. Commun. Stat.-Simul. Comput.
**2017**, 47. [Google Scholar] [CrossRef] - Lunn, D.; Spiegelhalter, D.; Thomas, A.; Best, N. The BUGS project: Evolution, critique and future directions. Stat. Med.
**2009**, 28, 3049–3067. [Google Scholar] [CrossRef] [PubMed] - Brooks, S.P.; Gelman, A. General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Stat.
**1998**, 7, 434–455. [Google Scholar] [CrossRef] [Green Version] - Leiva, V. The Birnbaum-Saunders Distribution, 1st ed.; Academic Press: New York, NY, USA, 2016. [Google Scholar]
- The Atomotive Industries Action Group. Measurement Systems Analysis-Reference Manual; The Atomotive Industries Action Group: Troy, MI, USA, 2002. [Google Scholar]

**Figure 1.**Comparison of degradation paths for observed and true degradation. (

**a**) True non-monotone degradation paths in red dotted lines when $\Delta {Z}_{i}\left(\right)open="("\; close=")">{t}_{j}$, (

**b**) true monotone degradation paths in red dotted lines when $\Delta {Z}_{i}\left(\right)open="("\; close=")">{t}_{j}$.

**Figure 2.**Proposed scheme for the optimal reliability analysis of degradation processes with measurement error.

**Figure 6.**Illustration of differences between the observed degradation paths and true degradation paths.

**Figure 8.**The effect of the measurement error illustrated by the comparison of the estimated reliability functions and the Kaplan–Meier estimation with confidence intervals.

**Figure 9.**Comparison of reliability functions for the observed degradation and the true degradation under Gaussian and logistic measurement errors.

Device | Hundred Thousands of Cycles | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |

1 | 0 | 0.01 | 0.03 | 0.055 | 0.107 | 0.165 | 0.183 | 0.2 | 0.26 | 0.302 |

2 | 0 | 0.09 | 0.161 | 0.172 | 0.247 | 0.259 | 0.281 | 0.371 | 0.401 | 0.429 |

3 | 0 | 0.01 | 0.06 | 0.081 | 0.118 | 0.142 | 0.158 | 0.169 | 0.232 | 0.262 |

4 | 0 | 0.016 | 0.076 | 0.087 | 0.104 | 0.127 | 0.198 | 0.208 | 0.218 | 0.258 |

5 | 0 | 0.036 | 0.096 | 0.176 | 0.204 | 0.242 | 0.281 | 0.325 | 0.415 | 0.495 |

6 | 0 | 0.014 | 0.102 | 0.112 | 0.194 | 0.277 | 0.289 | 0.305 | 0.335 | 0.391 |

7 | 0 | 0.037 | 0.064 | 0.078 | 0.096 | 0.124 | 0.164 | 0.234 | 0.254 | 0.326 |

8 | 0 | 0.035 | 0.086 | 0.105 | 0.174 | 0.267 | 0.277 | 0.347 | 0.361 | 0.384 |

9 | 0 | 0.067 | 0.148 | 0.161 | 0.173 | 0.184 | 0.218 | 0.229 | 0.239 | 0.285 |

10 | 0 | 0.025 | 0.052 | 0.064 | 0.076 | 0.151 | 0.187 | 0.205 | 0.222 | 0.262 |

Parameter | Mean | Sd | MC Error | ${\mathit{p}}_{0.025}$ | ${\mathit{p}}_{0.5}$ | ${\mathit{p}}_{0.975}$ |
---|---|---|---|---|---|---|

v | 22.55 | 3.094 | 0.01476 | 16.93 | 22.39 | 29.02 |

u | 0.01664 | 0.002687 | $1.76\times {10}^{-5}$ | 0.0126 | 0.01658 | 0.02309 |

Source | StdDev (SD) | Study Variation (6*SD) | % Study Variation |
---|---|---|---|

Total gage R&R | 0.0006058 | 0.0036347 | 3.83 |

Repeatability | 0.0006058 | 0.0036347 | 3.83 |

Reproducibility | 0.0000000 | 0.0000000 | 0.00 |

Operators | 0.0000000 | 0.0000000 | 0.00 |

Part to Part | 0.0157952 | 0.0947713 | 99.93 |

Total Variation | 0.0158068 | 0.0948409 | 100 |

**Table 4.**Comparison of the mean, variance and CV for the first-passage times of the observed and true degradation.

Mean | Variance | CV | |
---|---|---|---|

Observed | 1.088 | 0.223 | 0.434 |

True | 1.211 | 0.18 | 0.1487 |

${\mathit{I}}_{\mathbf{CV}}\left(\right)open="("\; close=")">{\mathit{T}}_{\mathit{s}},{\mathit{T}}_{\mathit{o}}$ | ${\mathit{I}}_{\mathbf{Var}}\left(\right)open="("\; close=")">{\mathit{T}}_{\mathit{s}},{\mathit{T}}_{\mathit{o}}$ | ${\mathit{I}}_{\mathit{E}}\left(\right)open="("\; close=")">{\mathit{T}}_{\mathit{s}},{\mathit{T}}_{\mathit{o}}$ | ${\mathit{I}}_{{\mathit{z}}_{5}}\left(\right)open="("\; close=")">{\mathit{T}}_{\mathit{s}},{\mathit{T}}_{\mathit{o}}$ | |
---|---|---|---|---|

Index | 0.6574 | 0.1919 | 0.1136 | 0.2309 |

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**MDPI and ACS Style**

Rodríguez-Picón, L.A.; Méndez-González, L.C.; Romero-López, R.; Pérez-Olguín, I.J.C.; Rodríguez-Borbón, M.I.; Valles-Rosales, D.J.
Reliability Analysis Based on a Gamma-Gaussian Deconvolution Degradation Modeling with Measurement Error. *Appl. Sci.* **2021**, *11*, 4133.
https://doi.org/10.3390/app11094133

**AMA Style**

Rodríguez-Picón LA, Méndez-González LC, Romero-López R, Pérez-Olguín IJC, Rodríguez-Borbón MI, Valles-Rosales DJ.
Reliability Analysis Based on a Gamma-Gaussian Deconvolution Degradation Modeling with Measurement Error. *Applied Sciences*. 2021; 11(9):4133.
https://doi.org/10.3390/app11094133

**Chicago/Turabian Style**

Rodríguez-Picón, Luis Alberto, Luis Carlos Méndez-González, Roberto Romero-López, Iván J. C. Pérez-Olguín, Manuel Iván Rodríguez-Borbón, and Delia Julieta Valles-Rosales.
2021. "Reliability Analysis Based on a Gamma-Gaussian Deconvolution Degradation Modeling with Measurement Error" *Applied Sciences* 11, no. 9: 4133.
https://doi.org/10.3390/app11094133