The Annals of Applied Statistics

Degradation modeling applied to residual lifetime prediction using functional data analysis

Rensheng R. Zhou, Nicoleta Serban, and Nagi Gebraeel

Full-text: Open access


Sensor-based degradation signals measure the accumulation of damage of an engineering system using sensor technology. Degradation signals can be used to estimate, for example, the distribution of the remaining life of partially degraded systems and/or their components. In this paper we present a nonparametric degradation modeling framework for making inference on the evolution of degradation signals that are observed sparsely or over short intervals of times. Furthermore, an empirical Bayes approach is used to update the stochastic parameters of the degradation model in real-time using training degradation signals for online monitoring of components operating in the field. The primary application of this Bayesian framework is updating the residual lifetime up to a degradation threshold of partially degraded components. We validate our degradation modeling approach using a real-world crack growth data set as well as a case study of simulated degradation signals.

Article information

Ann. Appl. Stat., Volume 5, Number 2B (2011), 1586-1610.

First available in Project Euclid: 13 July 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Condition Monitoring functional principal component analysis nonparametric estimation residual life distribution sparse degradation signal


Zhou, Rensheng R.; Serban, Nicoleta; Gebraeel, Nagi. Degradation modeling applied to residual lifetime prediction using functional data analysis. Ann. Appl. Stat. 5 (2011), no. 2B, 1586--1610. doi:10.1214/10-AOAS448.

Export citation


  • Bogdanoff, J. L. and Kozin, F. (1985). Probabilistic Models of Cummulative Damage. Wiley, New York.
  • Cross, R. J., Makeev, A. and Armanios, E. (2006). A comparison of predictions from probabilistic crack growth models inferred from Virkler’s data. J. ASTM International 3. DOI: 101520/JAI100574.
  • Davison, A. C. and Hinkley, D. V. (1997). Bootstrap Methods and Their Application. Cambridge Series in Statistical and Probabilistic Mathematics 1. Cambridge Univ. Press, Cambridge.
  • Doksum, K. A. and Hoyland, A. (1992). Models for variable-stress accelerated life testing experiments based on Wiener processes and the inverse Gaussian distribution. Technometrics 34 74–82.
  • Efron, B. and Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Monographs on Statistics and Applied Probability 57. Chapman and Hall, New York.
  • Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York.
  • Gebraeel, N. (2006). Sensory updated residual life distribution for components with exponential degradation patterns. IEEE Transactions on Automation Science and Engineering 3 382–393.
  • Gebraeel, N., Lawley, M., Li, R. and Ryan, J. (2005). Residual-life distributions from component degradation signals: A Bayesian approach. IIE Transactions 37 543–557.
  • James, G. M., Hastie, T. J. and Sugar, C. A. (2000). Principal component models for sparse functional data. Biometrika 87 587–602.
  • Karhunen, K. (1947). Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Suomalainen Tiedeakatemia, Finland.
  • Kotulski, Z. A. (1998). On efficiency of identification of a stochastic crack propagation model based on Virkler experimental data. Archives of Mechanics 5 829–847.
  • Liao, C. M. and Tseng, S. T. (2006). Optimal design for step-stress accelerated degradation tests. IEEE Transactions on Reliability 55.
  • Loève, M. (1945). Functions aleatoire de second order. Comptes Rendus Acad. Sci. 220.
  • Lu, C. J. and Meeker, W. Q. (1993). Using degradation measures to estimate a time-to-failure distribution. Technometrics 35 161–174.
  • Müller, H.-G. and Zhang, Y. (2005). Time-varying functional regression for predicting remaining lifetime distributions from longitudinal trajectories. Biometrics 61 1064–1075.
  • Nelson, W. (1990). Accelerated Testing Statistical Models, Test Plans and Data Analysis. Wiley, New York.
  • Padgett, W. J. and Tomlinson, M. A. (2004). Inference from accelerated degradation and failure data based on Gaussian process models. Lifetime Data Anal. 10 191–206.
  • Park, C. and Padgett, W. J. (2006). Stochastic degradation models with several accelerating variables. IEEE Transactions on Reliability 55 379–390.
  • Pettit, L. I. and Young, K. D. S. (1999). Bayesian analysis for inverse Gaussian lifetime data with measures of degradation. J. Statist. Comput. Simulation 63 217–234.
  • Ramsay, J. O. and Silverman, B. W. (1997). Functional Data Analysis. Springer, New York.
  • Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. J. Roy. Statist. Soc. Ser. B 53 233–243.
  • Tseng, S. T. and Peng, C. Y. (2007). Stochastic diffusion modeling of degradation data. J. Data Sci. 5 315–333.
  • Virkler, D. A., Hillberry, B. M. and Goel, P. K. (1979). The statistical nature of fatigue crack propagation. J. Eng. Mater. Technol. 101 148–153.
  • Yao, F., Müller, H.-G. and Wang, J.-L. (2005). Functional data analysis for sparse longitudinal data. J. Amer. Statist. Assoc. 100 577–590.
  • Yu, H. F. and Tseng, S. T. (1998). On-line procedure for terminating an accelerated degradation test. Statist. Sinica 8 207–220.
  • Zhou, R. R., Serban, N. and Gebraeel, N. (2010). Supplement to “Degradation modeling applied to residual lifetime prediction using functional data analysis.” Ann. Appl. Statist. DOI: 10.1214/10-AOAS448SUPP.

Supplemental materials