Systems with Multi-Components

  • M. Luz Gámiz
  • K. B. Kulasekera
  • Nikolaos Limnios
  • Bo Henry Lindqvist
Part of the Springer Series in Reliability Engineering book series (RELIABILITY)


The main purpose of this chapter is to construct a system structure function based on an observed set of the system output performance and the corresponding performances of its components. To do this, special techniques are proposed by looking at the structure function of a continuous-state system under the scope of a regression model. Multivariate smoothing and isotonic regression methods are adapted to the particular characteristics of the problem at hand.


Structure Function Integrate Square Error Average Square Error Universal Generate Function Symmetric Density Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This chapter is an extension into book-length form of the article Regression analysis of the structure function for reliability evaluation of continuous-state system, originally published in Reliability Engineering and System Safety 95(2), 134–142 (2010).

The authors express their full acknowledgment of the original publication of the paper in the journal cited above, edited by Elsevier.


  1. 1.
    Aven T (1993) On performance measures for multistate monotone systems. Reliab Eng Syst Safety 41:259–266CrossRefGoogle Scholar
  2. 2.
    Ayer M, Brunk HD, Ewing GM, Reid WT, Silverman E (1955) An empirical distribution function for sampling with incomplete information. Ann Math Stat 26:641–647MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing. Holt, Rinehart and Winston, New YorkGoogle Scholar
  4. 4.
    Baxter LA (1984) Continuum structures I. J Appl Prob 21:802–815MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Baxter LA (1986) Continuum structures II. Math Proc Cambridge Philos Soc 99:331–338MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bowman A (1984) An alternative method of cross validation for the smoothing of density estimates. Biometrika 71:353–360CrossRefMathSciNetGoogle Scholar
  7. 7.
    Brunelle RD, Kapur KC (1998) Continuous state system reliability: an interpolation approach. IEEE Trans Reliab 47(2):181–187CrossRefGoogle Scholar
  8. 8.
    Brunelle RD, Kapur KC (1999) Review and classification or reliability measures for multistate and continuum models. IIE Trans 31:1171–1180Google Scholar
  9. 9.
    Burdakow O, Grimwall A, Hussian M (2004) A generalised PAV algorithm for monotonic regression in several variables. COMPSTAT’2004 SymposiumGoogle Scholar
  10. 10.
    Chen SX (1999) Beta kernel estimators for density functions. Comput Stat Data Anal 31:131–145MATHCrossRefGoogle Scholar
  11. 11.
    El Neweihi E, Proschan F, Sethuraman J (1978) Multistate coherent systems. J Appl Prob 15(4):675–688MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Fan J, Gijbels I (1996) Local polynomial modelling and its applications. Monographs on statistics and applied probability. Chapman & Hall, LondonGoogle Scholar
  13. 13.
    Gámiz ML, Martínez-Miranda MD (2010) Regression analysis of the structure function for reliability evaluation of continuous-state system. Reliab Eng Syst 95(2):134–142CrossRefGoogle Scholar
  14. 14.
    González-Manteiga W, Martínez-Miranda MD, Pérez-González A (2004) The choice of smoothing parameter in nonparametric regression through wild bootstrap. Comput Stat Data Anal 47:487–515CrossRefGoogle Scholar
  15. 15.
    Hall P, Hwang LS (2001) Nonparametric kernel regression subject to monotonicity constraints. Ann Stat 29(3):624–647MATHCrossRefGoogle Scholar
  16. 16.
    Härdle W, Müller M, Sperlich S, Werwatz A (2004) Nonparametric and semiparametric Models. Springer, BerlinMATHCrossRefGoogle Scholar
  17. 17.
    Kaymaz I, McMahon C (2005) A response surface method based on weighted regression for structural reliability analysis. Prob Eng Mech 20:11–17CrossRefGoogle Scholar
  18. 18.
    Levitin G (2005) The universal generating function in reliability analysis and optimization. Springer, LondonGoogle Scholar
  19. 19.
    Li JA, Wu Y, Lai KK, Liu K (2005) Reliability estimation and prediction of multi state components and coherent systems. Reliab Eng Syst Safety 88:93–98CrossRefGoogle Scholar
  20. 20.
    Lisnianski A (2001) Estimation of boundary points for continuum-state system reliability measures. Reliab Eng Syst Safety 74:81–88MATHCrossRefGoogle Scholar
  21. 21.
    Lisnianski A, Levitin G (2003) Multi-state system reliability. Assessment, optimization and applications. World Scientific, SingaporeMATHGoogle Scholar
  22. 22.
    Meng FC (2005) Comparing two reliability upper bounds for multistate systems. Reliab Eng Syst Safety 87:31–36CrossRefGoogle Scholar
  23. 23.
    Mukarjee H, Stem S (1994) Feasible nonparametric estimation of multiargument monotone functions. J Am Stat Assoc 89:77–80MATHCrossRefGoogle Scholar
  24. 24.
    Natvig B (1982) Two suggestions of how to define a multistate coherent system. Appl Prob 14:434–455Google Scholar
  25. 25.
    Pourret O, Collet J, Bon JL (1999) Evaluation of the unavailability of a multistate component system using a binary model. Reliab Eng Syst Safety 64:13–17MATHCrossRefGoogle Scholar
  26. 26.
    Rackwitz R (2001) Reliability analysis a review and some perspectives. Struct Safety 23:365–395CrossRefGoogle Scholar
  27. 27.
    Rudemo M (1982) Empirical choice of histograms and kernel density estimators. Scand J Stat 9:65–78MATHMathSciNetGoogle Scholar
  28. 28.
    Ruppert D, Wand MP (1994) Multivariate locally weighted least squares regression. Ann Stat 22:1346–1370MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Sarhan AM (2002) Reliability equivalence with a basic series/parallel system. Appl Math Comput 132:115–133MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Turner R (2009) Iso: functions to perform isotonic regression
  31. 31.
    Wang L, Grandhi RV (1996) Safety index calculations using intervening variables for structural reliability analysis. Comput Struct 59(6):1139–1148MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  • M. Luz Gámiz
    • 1
  • K. B. Kulasekera
    • 2
  • Nikolaos Limnios
    • 3
  • Bo Henry Lindqvist
    • 4
  1. 1.Facultad Ciencias, Depto. Estadistica e Investigacion OperativaUniversidad GranadaGranadaSpain
  2. 2.Department of Mathematical SciencesClemson UniversityClemsonUSA
  3. 3.Centre de Recherches de Royallieu, Laboratoire de Mathématiques AppliquéesUniversité de Technologie de CompiègneCompiègneFrance
  4. 4.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

Personalised recommendations