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Methodology and Computing in Applied Probability

, Volume 14, Issue 3, pp 649–684 | Cite as

State-of-the-Art in Sequential Change-Point Detection

  • Aleksey S. Polunchenko
  • Alexander G. Tartakovsky
Article

Abstract

We provide an overview of the state-of-the-art in the area of sequential change-point detection assuming discrete time and known pre- and post-change distributions. The overview spans over all major formulations of the underlying optimization problem, namely, Bayesian, generalized Bayesian, and minimax. We pay particular attention to the latest advances in each. Also, we link together the generalized Bayesian problem with multi-cyclic disorder detection in a stationary regime when the change occurs at a distant time horizon. We conclude with two case studies to illustrate the cutting edge of the field at work.

Keywords

CUSUM chart Quickest change detection Sequential analysis Sequential change-point detection Shiryaev’s procedure Shiryaev–Roberts procedure Shiryaev–Roberts–Pollak procedure Shiryaev–Roberts–r procedure 

AMS 2000 Subject Classifications

62L10 60G40 62C10 62C20 

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References

  1. Atkinson K, Han W (2009) Theoretical numerical analysis: a functional analysis framework. In: Texts in applied mathematics, vol 39, 3rd edn. Springer doi: 10.1007/978-1-4419-0458-4
  2. Basseville M, Nikiforov IV (1993) Detection of abrupt changes: theory and application. Prentice Hall, Englewood CliffsGoogle Scholar
  3. Brodsky BE, Darkhovsky BS (1993) Nonparameteric methods in change point problems. In: Mathematics and its applications, vol 243. Kluwer Academic PublishersGoogle Scholar
  4. Feinberg EA, Shiryaev AN (2006) Quickest detection of drift change for Brownian motion in generalized Bayesian and minimax settings. Stat Decis 24(4):445–470. doi: 10.1524/stnd.2006.24.4.445 MathSciNetzbMATHCrossRefGoogle Scholar
  5. Ferguson TS (1967) Mathematical statistics—a decision theoretic approach. Academic Press, New YorkzbMATHGoogle Scholar
  6. Fuh CD (2003) SPRT and CUSUM in hidden Markov models. Ann Stat 31(3):942–977. doi: 10.1214/aos/1056562468 MathSciNetzbMATHCrossRefGoogle Scholar
  7. Fuh CD (2004) Asymptotic operating characteristics of an optimal change point detection in hidden Markov models. Ann Stat 32(5):2305–2339. doi: 10.1214/009053604000000580 MathSciNetzbMATHCrossRefGoogle Scholar
  8. Girschick MA, Rubin H (1952) A Bayes approach to a quality control model. Ann Math Stat 23(1):114–125. doi: 10.1214/aoms/1177729489 CrossRefGoogle Scholar
  9. Harris TE (1963) The theory of branching processes. Springer-Verlag, BerlinzbMATHGoogle Scholar
  10. Kesten H (1973) Random difference equations and renewal theory for products of random matrices. Acta Math. 131(1):207–248. doi: 10.1007/BF02392040 MathSciNetzbMATHCrossRefGoogle Scholar
  11. Lai TL (1995) Sequential changepoint detection in quality control and dynamical systems. J R Stat Soc B 57(4):613–658zbMATHGoogle Scholar
  12. Lai TL (1998) Information bounds and quick detection of parameter changes in stochastic systems. IEEE Trans Inf Theory 44:2917–2929. doi: 10.1109/18.737522 zbMATHCrossRefGoogle Scholar
  13. Lorden G (1971) Procedures for reacting to a change in distribution. Ann Math Stat 42(6):1897–1908MathSciNetzbMATHCrossRefGoogle Scholar
  14. Mevorach Y, Pollak M (1991) A small sample size comparison of the Cusum and the Shiryayev-Roberts approaches to changepoint detection. Am J Math Manage Sci 11:277–298zbMATHGoogle Scholar
  15. Moustakides GV (1986) Optimal stopping times for detecting changes in distributions. Ann Stat 14(4):1379–1387MathSciNetzbMATHCrossRefGoogle Scholar
  16. Moustakides GV (2008) Sequential change detection revisited. Ann Stat 36(2):787–807. doi: 10.1214/009053607000000938 MathSciNetzbMATHCrossRefGoogle Scholar
  17. Moustakides GV, Polunchenko AS, Tartakovsky AG (2011) A numerical approach to performance analysis of quickest change-point detection procedures. Stat Sin 21(2):571–596MathSciNetzbMATHCrossRefGoogle Scholar
  18. Page ES (1954) Continuous inspection schemes. Biometrika 41(1):100–115MathSciNetzbMATHGoogle Scholar
  19. Pollak M (1985) Optimal detection of a change in distribution. Ann Stat 13(1):206–227MathSciNetzbMATHCrossRefGoogle Scholar
  20. Pollak M (1987) Average run lengths of an optimal method of detecting a change in distribution. Ann Stat 15(2):749–779MathSciNetzbMATHCrossRefGoogle Scholar
  21. Pollak M, Siegmund D (1986) Convergence of quasi-stationary to stationary distributions for stochastically monotone Markov processes. J Appl Probab 23(1):215–220MathSciNetzbMATHCrossRefGoogle Scholar
  22. Pollak M, Tartakovsky AG (2009a) Asymptotic exponentiality of the distribution of first exit times for a class of Markov processes with applications to quickest change detection. Theory Probab Appl 53(3):430–442MathSciNetzbMATHCrossRefGoogle Scholar
  23. Pollak M, Tartakovsky AG (2009b) Optimality properties of the Shiryaev-Roberts procedure. Stat Sin 19:1729–1739MathSciNetzbMATHGoogle Scholar
  24. Polunchenko AS, Tartakovsky AG (2010) On optimality of the Shiryaev-Roberts procedure for detecting a change in distribution. Ann Stat 36(6):3445–3457. doi: 10.1214/09-AOS775 MathSciNetCrossRefGoogle Scholar
  25. Poor HV, Hadjiliadis O (2008) Quickest detection. Cambridge University PressGoogle Scholar
  26. Ritov Y (1990) Decision theoretic optimality of the CUSUM procedure. Ann Stat 18(3):1464–1469MathSciNetzbMATHCrossRefGoogle Scholar
  27. Roberts S (1966) A comparison of some control chart procedures. Technometrics 8(3):411–430MathSciNetCrossRefGoogle Scholar
  28. Shewhart WA (1931) Economic control of quality of manufactured product. D. Van Nostrand Company, Inc., New YorkGoogle Scholar
  29. Shiryaev AN (1961) The problem of the most rapid detection of a disturbance in a stationary process. Sov Math Dokl 2:795–799zbMATHGoogle Scholar
  30. Shiryaev AN (1963) On optimum methods in quickest detection problems. Theory Probab Appl 8(1):22–46. doi: 10.1137/1108002 zbMATHCrossRefGoogle Scholar
  31. Shiryaev AN (1978) Optimal stopping rules. Springer-Verlag, New YorkzbMATHGoogle Scholar
  32. Shiryaev AN (2006) From “disorder” to nonlinear filtering and martingale theory. In: Bolibruch A, Osipov Y, Sinai Y (eds) Mathematical events of the twentieth century. Springer Berlin Heidelberg, pp 371–397, doi: 10.1007/3-540-29462-7_18
  33. Shiryaev AN (2009) On the stochastic models and optimal methods in the quickest detection problems. Theory Probab Appl 53(3):385–401. doi: 10.1137/S0040585X97983717 MathSciNetzbMATHCrossRefGoogle Scholar
  34. Shiryaev AN (2010) Quickest detection problems: fifty years later. Seq Anal 29:345–385. doi: 10.1080/07474946.2010520580 MathSciNetzbMATHCrossRefGoogle Scholar
  35. Siegmund D (1985) Sequential analysis: tests and confidence intervals. Springer Series in Statistics. Springer-Verlag, New YorkzbMATHGoogle Scholar
  36. Springer MD, Thompson WE (1970) The distribution of products of Beta, Gamma and Gaussian random variables. SIAM J Appl math 18(4):721–737. doi: 10.1137/0118065 MathSciNetzbMATHCrossRefGoogle Scholar
  37. Tartakovsky AG (1991) Sequential methods in the theory of information systems. Radio & Communications, Moscow, RussiaGoogle Scholar
  38. Tartakovsky AG (2005) Asymptotic performance of a multichart CUSUM test under false alarm probability constraint. In: Proceedings of the 2005 IEEE Conference on Decision and Control, vol 44, pp 320–325Google Scholar
  39. Tartakovsky AG (2008) Discussion on “Is average run length to false alarm always an informative criterion?” by Yajun Mei. Seq Anal 27(4):396–405. doi: 10.1080/07474940802446046 MathSciNetzbMATHCrossRefGoogle Scholar
  40. Tartakovsky AG (2009a) Asymptotic optimality in Bayesian changepoint detection problems under global false alarm probability constraint. Theory Probab Appl 53:443–466. doi: 10.1137/S0040585X97983754 MathSciNetzbMATHCrossRefGoogle Scholar
  41. Tartakovsky AG (2009b) Discussion on “Optimal sequential surveillance for finance, public health, and other areas” by Marianne Frisén. Seq Anal 28(3):365–371. doi: 10.1080/07474940903041704 MathSciNetzbMATHCrossRefGoogle Scholar
  42. Tartakovsky AG, Moustakides GV (2010) State-of-the-art in Bayesian changepoint detection. Seq Anal 29(2):125–145. doi: 10.1080/07474941003740997 MathSciNetzbMATHCrossRefGoogle Scholar
  43. Tartakovsky AG, Polunchenko AS (2010) Minimax optimality of the Shiryaev-Roberts procedure. In: Proceedings of the 5th international workshop on applied probability, Universidad Carlos III of Madrid, SpainGoogle Scholar
  44. Tartakovsky AG, Veeravalli VV (2005) General asymptotic Bayesian theory of quickest change detection. Theory Probab Appl 49(3):458–497. doi: 10.1137/S0040585X97981202 MathSciNetCrossRefGoogle Scholar
  45. Tartakovsky AG, Pollak M, Polunchenko AS (2008) Asymptotic exponentiality of first exit times for recurrent Markov processes and applications to changepoint detection. In: Proceedings of the 2008 International workshop on applied probability, Compiégne, FranceGoogle Scholar
  46. Tartakovsky AG, Polunchenko AS, Moustakides GV (2009) Design and comparison of Shiryaev–Roberts- and CUSUM-type change-point detection procedures. In: Proceedings of the 2nd International workshop in sequential methodologies, University of Technology of Troyes, Troyes, FranceGoogle Scholar
  47. Tartakovsky AG, Pollak M, Polunchenko AS (2011) Third-order asymptotic optimality of the generalized Shiryaev-Roberts changepoint detection procedures. Theoiya Veroyatnostej i ee Primeneniya 56(3) (in press)Google Scholar
  48. Wald A (1947) Sequential analysis. J. Wiley & Sons, Inc., New YorkzbMATHGoogle Scholar
  49. Woodroofe M (1982) Nonlinear renewal theory in sequential analysis. Society for Industrial and Applied Mathematics, Philadelphia, PAGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Aleksey S. Polunchenko
    • 1
  • Alexander G. Tartakovsky
    • 1
  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

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