# zbMATH — the first resource for mathematics

Sequential multi-sensor change-point detection. (English) Zbl 1267.62084
Summary: We develop a mixture procedure to monitor parallel streams of data for a change-point that affects only a subset of them, without assuming a spatial structure relating the data streams to one another. Observations are assumed initially to be independent standard normal random variables. After a change-point the observations in a subset of the streams of data have nonzero mean values. The subset and the post-change means are unknown. The procedure we study uses stream specific generalized likelihood ratio statistics, which are combined to form an overall detection statistic in a mixture model that hypothesizes an assumed fraction $$p_{0}$$ of affected data streams.
An analytic expression is obtained for the average run length (ARL) when there is no change and is shown by simulations to be very accurate. Similarly, an approximation for the expected detection delay (EDD) after a change-point is also obtained. Numerical examples are given to compare the suggested procedure to other procedures for unstructured problems and to a case where the problem is assumed to have a well-defined geometric structure. Finally we discuss sensitivity of the procedure to the assumed value of $$p_{0}$$ and suggest a generalization.

##### MSC:
 62L10 Sequential statistical analysis 65C60 Computational problems in statistics (MSC2010)
##### Keywords:
change-point detection; multi-sensor
Full Text:
##### References:
 [1] Aldous, D. (1989). Probability Approximations Via the Poisson Clumping Heuristic. Applied Mathematical Sciences 77 . Springer, New York. · Zbl 0679.60013 [2] Chen, M., Gonzalez, S., Vasilakos, A., Cao, H. and Leung, V. C. M. (2010). Body area networks: A survey. Mobile Netw. Appl. 16 171-193. [3] Lai, T. L. (1995). Sequential changepoint detection in quality control and dynamical systems. J. Roy. Statist. Soc. Ser. B 57 613-658. · Zbl 0832.62072 [4] Lévy-Leduc, C. and Roueff, F. (2009). Detection and localization of change-points in high-dimensional network traffic data. Ann. Appl. Stat. 3 637-662. · Zbl 1166.62094 [5] Lorden, G. (1971). Procedures for reacting to a change in distribution. Ann. Math. Statist. 42 1897-1908. · Zbl 0255.62067 [6] Mei, Y. (2010). Efficient scalable schemes for monitoring a large number of data streams. Biometrika 97 419-433. · Zbl 1406.62088 [7] Page, E. S. (1954). Continuous inspection schemes. Biometrika 41 100-115. · Zbl 0056.38002 [8] Page, E. S. (1955). A test for a change in a parameter occurring at an unknown point. Biometrika 42 523-527. · Zbl 0067.11602 [9] Petrov, A., Rozovskii, B. L. and Tartakovsky, A. G. (2003). Efficient Nonlinear Filtering Methods for Detection of Dim Targets by Passive Systems , Vol. IV . Artech House, Boston, MA. [10] Pollak, M. and Siegmund, D. (1975). Approximations to the expected sample size of certain sequential tests. Ann. Statist. 3 1267-1282. · Zbl 0347.62063 [11] Rabinowitz, D. (1994). Detecting clusters in disease incidence. In Change-point Problems ( South Hadley , MA , 1992). Institute of Mathematical Statistics Lecture Notes-Monograph Series 23 255-275. IMS, Hayward, CA. · Zbl 1158.60352 [12] Shafie, K., Sigal, B., Siegmund, D. and Worsley, K. J. (2003). Rotation space random fields with an application to fMRI data. Ann. Statist. 31 1732-1771. · Zbl 1043.92019 [13] Siegmund, D. (1985). Sequential Analysis : Tests and Confidence Intervals . Springer, New York. · Zbl 0573.62071 [14] Siegmund, D. and Venkatraman, E. S. (1995). Using the generalized likelihood ratio statistic for sequential detection of a change-point. Ann. Statist. 23 255-271. · Zbl 0821.62044 [15] Siegmund, D. and Yakir, B. (2007). The Statistics of Gene Mapping . Springer, New York. · Zbl 1280.62012 [16] Siegmund, D. and Yakir, B. (2008). Detecting the emergence of a signal in a noisy image. Stat. Interface 1 3-12. · Zbl 1230.62105 [17] Siegmund, D., Yakir, B. and Zhang, N. R. (2011). Detecting simultaneous variant intervals in aligned sequences. Ann. Appl. Stat. 5 645-668. · Zbl 1223.62166 [18] Širjaev, A. N. (1963). Optimal methods in quickest detection problems. Theory Probab. Appl. 8 22-46. · Zbl 0213.43804 [19] Tartakovsky, A. G. and Veeravalli, V. V. (2008). Asymptotically optimal quickest change detection in distributed sensor systems. Sequential Anal. 27 441-475. · Zbl 1247.93014 [20] Xie, Y. (2011). Statistical signal detection with multi-sensor and sparsity. Ph.D. thesis, Stanford Univ.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.