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

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  • Jin-Taek Seong

    (Department of Convergence Software, Mokpo National University, Muan 58554, Korea)

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.

Suggested Citation

  • Jin-Taek Seong, 2022. "Theoretical Bounds on the Number of Tests in Noisy Threshold Group Testing Frameworks," Mathematics, MDPI, vol. 10(14), pages 1-14, July.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:14:p:2508-:d:866139
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    References listed on IDEAS

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    1. Jin-Taek Seong, 2020. "Theoretical Bounds on Performance in Threshold Group Testing Schemes," Mathematics, MDPI, vol. 8(4), pages 1-13, April.
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    Cited by:

    1. Jin-Taek Seong, 2023. "Bounds on Performance for Recovery of Corrupted Labels in Supervised Learning: A Finite Query-Testing Approach," Mathematics, MDPI, vol. 11(17), pages 1-16, August.
    2. Alexander Bochkov & Gurami Tsitsiashvili, 2022. "Preface to the Special Issue on Probability and Stochastic Processes with Applications to Communications, Systems and Networks," Mathematics, MDPI, vol. 10(24), pages 1-4, December.

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    1. Jin-Taek Seong, 2023. "Bounds on Performance for Recovery of Corrupted Labels in Supervised Learning: A Finite Query-Testing Approach," Mathematics, MDPI, vol. 11(17), pages 1-16, August.

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