Change detection with an unknown sensor subset: More information is not always better

Marco Guerriero, Dayu Huang, Jayakrishnan Unnikrishnan, Michael Lexa, Satish Iyengar, Frederick W. Wheeler

Abstract: We study the problem of detecting changes in the environment based on observations taken by multiple sensors under the setting in which the change affects an unknown subset of the sensors. We explore different approaches to this problem and relate different stopping rules for change detection with multiple sensors in a single framework. We introduce four new different stopping rules: T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">MAP</sub> , T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SOFT-MAP</sub> , T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ML</sub> and T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Order</sub> . While the first three rules rely on the information contained in the posterior probability of the sensor being affected by the change and on the Maximum Likelihood (ML) estimator of the sensor being affected, respectively, the last one is based on the order statistic of the local likelihood ratios at the sensors. We show that: i) T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ML</sub> , which is based on the "scan statistic", is equivalent to the Bayesian stopping rule T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> (p <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> = 1) that was recently introduced by Xie and Siegmund; ii) T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Order</sub> , is the counterpart of T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ML</sub> when the cardinality of the affected sensors is known. Surprisingly, the additional information about the cardinality does not always lead to better detection performance. A derivation of an upper bound for the false alarm rate for T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Order</sub> , is given and a comparative numerical analysis of the different stopping rules is provided in order to relate their performances.