The Annals of Applied Probability

Robustness of the N-CUSUM stopping rule in a Wiener disorder problem

Hongzhong Zhang, Neofytos Rodosthenous, and Olympia Hadjiliadis

Full-text: Open access


We study a Wiener disorder problem of detecting the minimum of $N$ change-points in $N$ observation channels coupled by correlated noises. It is assumed that the observations in each dimension can have different strengths and that the change-points may differ from channel to channel. The objective is the quickest detection of the minimum of the $N$ change-points. We adopt a min–max approach and consider an extended Lorden’s criterion, which is minimized subject to a constraint on the mean time to the first false alarm. It is seen that, under partial information of the post-change drifts and a general nonsingular stochastic correlation structure in the noises, the minimum of $N$ cumulative sums (CUSUM) stopping rules is asymptotically optimal as the mean time to the first false alarm increases without bound. We further discuss applications of this result with emphasis on its implications to the efficiency of the decentralized versus the centralized systems of observations which arise in engineering.

Article information

Ann. Appl. Probab., Volume 25, Number 6 (2015), 3405-3433.

Received: October 2013
Revised: October 2014
First available in Project Euclid: 1 October 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62L10: Sequential analysis 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 62L15: Optimal stopping [See also 60G40, 91A60] 62C20: Minimax procedures 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

CUSUM correlated noise quickest detection Wiener disorder problem


Zhang, Hongzhong; Rodosthenous, Neofytos; Hadjiliadis, Olympia. Robustness of the N -CUSUM stopping rule in a Wiener disorder problem. Ann. Appl. Probab. 25 (2015), no. 6, 3405--3433. doi:10.1214/14-AAP1078.

Export citation


  • [1] Basseville, M., Abdelghani, M. and Benveniste, A. (2000). Subspace-based fault detection algorithms for vibration monitoring. Automatica J. IFAC 36 101–109.
  • [2] Basseville, M., Benveniste, A., Goursat, M. and Mevel, L. (2007). Subspace-based algorithms for structural identification, damage detection, and sensor data fusion. Journal of Applied Signal Processing, Special Issue on Advances in Subspace-Based Techniques for Signal Processing and Communications 2007 200–213.
  • [3] Basseville, M., Mevel, L. and Goursat, M. (2004). Statistical model-based damage detection and localization, subspace-based residuals and damage-to-noise sensitivity ratios. J. Sound Vib. 275 769–794.
  • [4] Basseville, M. and Nikiforov, I. V. (1993). Detection of Abrupt Changes: Theory and Application. Prentice Hall, Englewood Cliffs, NJ.
  • [5] Bayraktar, E., Dayanik, S. and Karatzas, I. (2006). Adaptive Poisson disorder problem. Ann. Appl. Probab. 16 1190–1261.
  • [6] Bayraktar, E. and Poor, H. V. (2007). Quickest detection of a minimum of two Poisson disorder times. SIAM J. Control Optim. 46 308–331 (electronic).
  • [7] Beibel, M. (1996). A note on Ritov’s Bayes approach to the minimax property of the CUSUM procedure. Ann. Statist. 24 1804–1812.
  • [8] Beibel, M. (1997). Sequential change-point detection in continuous time when the post-change drift is unknown. Bernoulli 3 457–478.
  • [9] Beibel, M. and Lerche, H. R. (2003). Sequential Bayes detection of trend changes. In Foundations of Statistical Inference (Shoresh, 2000) (Y. Haitovsky, H. R. Lerche and Y. Ritov, eds.) 117–130. Physica, Heidelberg.
  • [10] Cuchiara, R., Piccardi, M. and Mello, P. (2000). Image analysis and rule-based reasoning for a traffic monitoring system. IEEE Intelligent Transportation Systems Society 1 119–130.
  • [11] Dayanik, S., Poor, H. V. and Sezer, S. O. (2008). Multisource Bayesian sequential change detection. Ann. Appl. Probab. 18 552–590.
  • [12] Ewins, D. J. (2000). Modal Testing: Theory, Practice and Applications, 2nd ed. Research Studies Press, Letchworth, Hertfordshire, UK.
  • [13] Fellouris, G. and Moustakides, G. V. (2011). Decentralized sequential hypothesis testing using asynchronous communication. IEEE Trans. Inform. Theory 57 534–548.
  • [14] Gapeev, P. V. (2005). The disorder problem for compound Poisson processes with exponential jumps. Ann. Appl. Probab. 15 487–499.
  • [15] Hadjiliadis, O., Hernandez del-Valle, G. and Stamos, I. (2009). A comparison of 2-CUSUM stopping rules for quickest detection of two-sided alternatives in a Brownian motion model. Sequential Anal. 28 92–114.
  • [16] Hadjiliadis, O. and Moustakides, V. (2005). Optimal and asymptotically optimal CUSUM rules for change point detection in the Brownian motion model with multiple alternatives. Teor. Veroyatn. Primen. 50 131–144.
  • [17] Hadjiliadis, O., Schäfer, T. and Poor, H. V. (2009). Quickest detection in coupled systems. In Proceedings of the 48th IEEE Conference on Decision and Control 4723–4728. IEEE, Shanghai, China.
  • [18] Hadjiliadis, O., Zhang, H. and Poor, H. V. (2009). One shot schemes for decentralized quickest change detection. IEEE Trans. Inform. Theory 55 3346–3359.
  • [19] Heylen, W., Lammens, S. and Sas, P. (1995). Modal analysis theory and testing. Technical report, Leuven, Belgium.
  • [20] Ivanoff, B. G. and Merzbach, E. (2010). Optimal detection of a change-set in a spatial Poisson process. Ann. Appl. Probab. 20 640–659.
  • [21] Juang, J. N. (1994). Applied System Identification. Prentice Hall, Englewood Cliffs, NJ.
  • [22] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [23] Moustakides, G. V. (1986). Optimal stopping times for detecting changes in distributions. Ann. Statist. 14 1379–1387.
  • [24] Moustakides, G. V. (2004). Optimality of the CUSUM procedure in continuous time. Ann. Statist. 32 302–315.
  • [25] Moustakides, G. V. (2006). Decentralized CUSUM change detection. In Proceedings of the 9th International Conference on Information Fusion (ICIF) 1–6. IEEE, Florence, Italy.
  • [26] Moustakides, G. V. (2008). Sequential change detection revisited. Ann. Statist. 36 787–807.
  • [27] Øksendal, B. (2002). Stochastic Differential Equations. Springer, New York.
  • [28] Peeters, B. and Roeck, G. D. (1999). Reference-based stochastic subspace identification for output-only modal analysis. Mech. Syst. Signal Process. 13 855–877.
  • [29] Polunchenko, A. S. and Tartakovsky, A. G. (2010). On optimality of the Shiryaev–Roberts procedure for detecting a change in distribution. Ann. Statist. 38 3445–3457.
  • [30] Poor, H. V. and Hadjiliadis, O. (2008). Quickest Detection. Cambridge Univ. Press, Cambridge, UK.
  • [31] Raghavan, V. and Veeravalli, V. V. (2008). Quickest detection of a change process across a sensor array. In Proceedings of the 11th International Conference on Information Fusion 1–8. IEEE, Cologne, Germany.
  • [32] Sezer, S. O. (2010). On the Wiener disorder problem. Ann. Appl. Probab. 20 1537–1566.
  • [33] Shiryaev, A. N. (1996). Minimax optimality of the method of cumulative sums (CUSUM) in the continuous time case. Uspekhi Mat. Nauk 51 173–174.
  • [34] Shreve, S. E. (2004). Stochastic Calculus for Finance 2. Springer, New York.
  • [35] Tartakovsky, A. G. and Kim, H. (2006). Performance of certain decentralized distributed change detection procedures. In Proceedings of the 9th International Conference on Information Fusion (ICIF) 1–8. IEEE, Florence, Italy.
  • [36] Tartakovsky, A. G. and Veeravalli, V. V. (2004). Change-point detection in multichannel and distributed systems. In Applied Sequential Methodologies (N. Mukhopadhay, S. Datta and S. Chattopadhay, eds.). Statist. Textbooks Monogr. 173 339–370. Dekker, New York.
  • [37] Tartakovsky, A. G. and Veeravalli, V. V. (2008). Asymptotically optimal quickest change detection in distributed sensor systems. Sequential Anal. 27 441–475.
  • [38] Zhang, H. and Hadjiliadis, O. (2012). Quickest detection in a system with correlated noise. In Proceedings of the 51st IEEE Conference on Decision and Control 4757–4763. IEEE, Maui, HI.
  • [39] Zhang, H., Hadjiliadis, O., Schäfer, T. and Poor, H. V. (2014). Quickest detection in coupled systems. SIAM J. Control Optim. 52 1567–1596.