Detection by Block- and Band-Permuted Data

Abstract

Unlabeled detection is an important problem in large sensor networks faced with big-data applications. The focus of this paper is on decision problems in which the data vector received at the fusion center (FC), tasked to perform the final inference, undergoes an unknown permutation. Two representative permutation models are considered, whose structure is imposed by practical considerations. The first is the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> -block permutation model, where the observations are split into different blocks of size <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> , and independent data permutations affect each block. The second is the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$r$</tex-math></inline-formula> -banded permutation model, in which each sample of the vector received by the FC lies at most <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$r$</tex-math></inline-formula> positions ahead or behind its original location. To solve these challenging detection tasks, three detectors for both permutation models are proposed. The first is inspired by the uniformly most powerful invariant principle; the second relies on a distribution-averaged strategy; and the third is designed according the generalized likelihood ratio approach. The performance of these detectors is investigated by computer experiments, and their relative merits are discussed.