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October 2018 Thresholds for detecting an anomalous path from noisy environments
Shirshendu Chatterjee, Ofer Zeitouni
Ann. Appl. Probab. 28(5): 2635-2663 (October 2018). DOI: 10.1214/17-AAP1356

Abstract

We consider the “searching for a trail in a maze” composite hypothesis testing problem, in which one attempts to detect an anomalous directed path in a lattice 2D box of side $n$ based on observations on the nodes of the box. Under the signal hypothesis, one observes independent Gaussian variables of unit variance at all nodes, with zero mean off the anomalous path and mean $\mu_{n}$ on it. Under the null hypothesis, one observes i.i.d. standard Gaussians on all nodes. Arias-Castro et al. [Ann. Statist. 36 (2008) 1726–1757] showed that if the unknown directed path under the signal hypothesis has known initial location, then detection is possible (in the minimax sense) if $\mu_{n}\gg1/\sqrt{\log n}$, while it is not possible if $\mu_{n}\ll1/\log n\sqrt{\log\log n}$. In this paper, we show that this result continues to hold even when the initial location of the unknown path is not known. As is the case with Arias-Castro et al. [Ann. Statist. 36 (2008) 1726–1757], the upper bound here also applies when the path is undirected. The improvement is achieved by replacing the linear detection statistic used in Arias-Castro et al. [Ann. Statist. 36 (2008) 1726–1757] with a polynomial statistic, which is obtained by employing a multiscale analysis on a quadratic statistic to bootstrap its performance. Our analysis is motivated by ideas developed in the context of the analysis of random polymers in Lacoin [Comm. Math. Phys. 294 (2010) 471–503].

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Shirshendu Chatterjee. Ofer Zeitouni. "Thresholds for detecting an anomalous path from noisy environments." Ann. Appl. Probab. 28 (5) 2635 - 2663, October 2018. https://doi.org/10.1214/17-AAP1356

Information

Received: 1 May 2017; Revised: 1 September 2017; Published: October 2018
First available in Project Euclid: 28 August 2018

zbMATH: 06974761
MathSciNet: MR3847969
Digital Object Identifier: 10.1214/17-AAP1356

Subjects:
Primary: 62C20 , 62G10
Secondary: 60K37 , 82B20

Keywords: Detecting a chain of nodes in a network , minimax detection , Random scenery

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.28 • No. 5 • October 2018
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