The Annals of Applied Probability

Thresholds for detecting an anomalous path from noisy environments

Shirshendu Chatterjee and Ofer Zeitouni

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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].

Article information

Source
Ann. Appl. Probab., Volume 28, Number 5 (2018), 2635-2663.

Dates
Received: May 2017
Revised: September 2017
First available in Project Euclid: 28 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1535443230

Digital Object Identifier
doi:10.1214/17-AAP1356

Mathematical Reviews number (MathSciNet)
MR3847969

Zentralblatt MATH identifier
06974761

Subjects
Primary: 62C20: Minimax procedures 62G10: Hypothesis testing
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 60K37: Processes in random environments

Keywords
Detecting a chain of nodes in a network minimax detection random scenery

Citation

Chatterjee, Shirshendu; Zeitouni, Ofer. Thresholds for detecting an anomalous path from noisy environments. Ann. Appl. Probab. 28 (2018), no. 5, 2635--2663. doi:10.1214/17-AAP1356. https://projecteuclid.org/euclid.aoap/1535443230


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References

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