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February 2011 Detection of an anomalous cluster in a network
Ery Arias-Castro, Emmanuel J. Candès, Arnaud Durand
Ann. Statist. 39(1): 278-304 (February 2011). DOI: 10.1214/10-AOS839

Abstract

We consider the problem of detecting whether or not, in a given sensor network, there is a cluster of sensors which exhibit an “unusual behavior.” Formally, suppose we are given a set of nodes and attach a random variable to each node. We observe a realization of this process and want to decide between the following two hypotheses: under the null, the variables are i.i.d. standard normal; under the alternative, there is a cluster of variables that are i.i.d. normal with positive mean and unit variance, while the rest are i.i.d. standard normal. We also address surveillance settings where each sensor in the network collects information over time. The resulting model is similar, now with a time series attached to each node. We again observe the process over time and want to decide between the null, where all the variables are i.i.d. standard normal, and the alternative, where there is an emerging cluster of i.i.d. normal variables with positive mean and unit variance. The growth models used to represent the emerging cluster are quite general and, in particular, include cellular automata used in modeling epidemics. In both settings, we consider classes of clusters that are quite general, for which we obtain a lower bound on their respective minimax detection rate and show that some form of scan statistic, by far the most popular method in practice, achieves that same rate to within a logarithmic factor. Our results are not limited to the normal location model, but generalize to any one-parameter exponential family when the anomalous clusters are large enough.

Citation

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Ery Arias-Castro. Emmanuel J. Candès. Arnaud Durand. "Detection of an anomalous cluster in a network." Ann. Statist. 39 (1) 278 - 304, February 2011. https://doi.org/10.1214/10-AOS839

Information

Published: February 2011
First available in Project Euclid: 3 December 2010

zbMATH: 1209.62097
MathSciNet: MR2797847
Digital Object Identifier: 10.1214/10-AOS839

Subjects:
Primary: 62C20, 62G10
Secondary: 82B20

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 1 • February 2011
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