Theoretical Bounds on the Number of Tests in Noisy Threshold Group Testing Frameworks

Jin-Taek Seong
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Jin-Taek Seong: Department of Convergence Software, Mokpo National University, Muan 58554, Korea

Mathematics, 2022, vol. 10, issue 14, 1-14

Abstract: We consider a variant of group testing (GT) models called noisy threshold group testing (NTGT), in which when there is more than one defective sample in a pool, its test result is positive. We deal with a variant model of GT where, as in the diagnosis of COVID-19 infection, if the virus concentration does not reach a threshold, not only do false positives and false negatives occur, but also unexpected measurement noise can reverse a correct result over the threshold to become incorrect. We aim to determine how many tests are needed to reconstruct a small set of defective samples in this kind of NTGT problem. To this end, we find the necessary and sufficient conditions for the number of tests required in order to reconstruct all defective samples. First, Fano’s inequality was used to derive a lower bound on the number of tests needed to meet the necessary condition. Second, an upper bound was found using a MAP decoding method that leads to giving the sufficient condition for reconstructing defective samples in the NTGT problem. As a result, we show that the necessary and sufficient conditions for the successful reconstruction of defective samples in NTGT coincide with each other. In addition, we show a trade-off between the defective rate of the samples and the density of the group matrix which is then used to construct an optimal NTGT framework.

Keywords: noisy threshold group testing; defective samples; number of tests; bounds; COVID-19 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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