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Stochastic Threshold Group Testing

Stochastic Threshold Group Testing. Chun Lam Chan, Sheng Cai , Mayank Bakshi , Sidharth Jaggi The Chinese University of Hong Kong. Venkatesh Saligrama Boston University. q. q. Classical Group testing. Adaptive vs. Non-adaptive. What’s known. [CCJS11].

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Stochastic Threshold Group Testing

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  1. Stochastic Threshold Group Testing • Chun Lam Chan, ShengCai, • MayankBakshi, SidharthJaggi The Chinese University of Hong Kong • VenkateshSaligrama • Boston University

  2. q q Classical Group testing Adaptive vs. Non-adaptive What’s known [CCJS11] For Pr(error)< ε , Lower bound of number of tests: Chun Lam Chan; Pak HouChe; Jaggi, S.; Saligrama, V.; , "Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms,"  49th Annual Allerton Conference on Communication, Control, and Computing, pp.1832-1839, 28-30 Sept. 2011 [CCJS11]

  3. Classical vs. Stochastic Threshold Prob. of positive outcome Prob. of positive outcome # defective items # defective items Classical Group Testing Stochastic Threshold Group Testing

  4. Classical vs. Stochastic Threshold Prob. of positive outcome Prob. of positive outcome Classical Group Testing Stochastic Threshold Group Testing # defective items # defective items Fair Coin

  5. Linear vs. BernoulliGap Prob. of positive outcome Prob. of positive outcome Linear Gap Bernoulli Gap # defective items # defective items Biased Coin with more and more weight on positive Noiseless Non-adaptive Bernoulli Gap

  6. Our results: Non-adaptive algorithm with Bernoulli gap model Two-stage Adaptive algorithm Non-adaptive algorithm with linear gap model Faster implementation by [CJBJ13] Previous work: (NA) Non-adaptive algorithm (A) Adaptive algorithm [CJBJ13] M. Jahangoshahi, S. Cai, M. Bakshi, S. Jaggi, “GROTESQUE: Noisy Group Testing (Quick and Efficient),” submitted to the IEEE Transactions on Information Theory, Mar. 2013 * If not specified, the algorithm allows up to g misclassifications

  7. Transversal Design 1st family (partition) Groups of the same size [BBTK96] D. Balding, W. Bruno, D. Torney, and E. Knill, “A comparative survey of non-adaptive pooling designs,” in Genetic Mapping and DNA Sequencing, ser. The IMA Volumes in Mathematics and its Applications, T. Speed and M. Waterman, Eds. Springer New York, 1996, vol. 81, pp. 133–154.

  8. Transversal Design 2nd family (partition)

  9. Transversal Design 3rd family (partition)

  10. Transversal Design Last family (partition)

  11. Transversal Design 1st family (partition) positive Prob. of positive outcome # defective items Classical Group Testing

  12. Transversal Design 2nd family (partition) positive Prob. of positive outcome # defective items Classical Group Testing

  13. Transversal Design 3rd family (partition) positive Prob. of positive outcome # defective items Classical Group Testing

  14. Transversal Design Last family (partition) k positive Prob. of positive outcome # defective items Classical Group Testing

  15. Transversal Design Prob. of positive outcome 100% positive tests # defective items Classical Group Testing

  16. Transversal Design 1st family (partition) negative Prob. of positive outcome # defective items Classical Group Testing

  17. Transversal Design 2nd family (partition) negative Prob. of positive outcome # defective items Classical Group Testing

  18. Transversal Design 3nd family (partition) negative Prob. of positive outcome # defective items Classical Group Testing

  19. Transversal Design Last family (partition) positive Prob. of positive outcome # defective items Classical Group Testing

  20. Transversal Design Prob. of positive outcome 25% positive tests # defective items Classical Group Testing

  21. Transversal Design Prob. of positive outcome 100% positive tests # defective items Classical Group Testing defective (higher ratio)

  22. Transversal Design Prob. of positive outcome Statistical Difference!! 25% positive tests non-defective (lower ratio) # defective items Classical Group Testing

  23. Transversal Design for “Stochastic” Group Testing negative Fair Coin Prob. of positive outcome # defective items “Stochastic” Group Testing Classical Group Testing negative Fair Coin

  24. Transversal Design for “Stochastic” Group Testing Fair Coin Prob. of positive outcome 100% positive tests 50% positive tests on expectation Defective (higher ratio) # defective items “Stochastic” Group Testing

  25. Transversal Design for “Stochastic Group” Testing # of families Fair Coin Prob. of positive outcome Statistical Difference!! Statistical Difference 12.5% positive tests on expectation 25% positive tests Union bound non-defective (lower ratio) # defective items “Stochastic” Group Testing

  26. Stochastic Threshold Group Testing negative Prob. of positive outcome # defective items negative negative Stochastic Threshold Group Testing

  27. Stochastic Threshold Group Testing negative Prob. of positive outcome Critical Reference group (R): Exactly l defectives # defective items negative Fair Coin Stochastic Threshold Group Testing

  28. Stochastic Threshold Group Testing Critical Reference group (R): Exactly l defectives Prob. of positive outcome Prob. of positive outcome # defective items # defective items “Stochastic” Group Testing Stochastic Threshold Group Testing

  29. Existence of Critical Reference Group Prob. of picking l defectives …… Size of reference group

  30. Identification of Critical Reference Group 1st family (partition) 2nd family (partition) 3rd family (partition) Last family (partition)

  31. Identification of Critical Reference Group Exactly l defectives Less than l defectives More than l defectives

  32. Identification of Critical Reference Group 1st family (partition) 2nd family (partition) negative 3rd family (partition) Last family (partition) negative

  33. Identification of Critical Reference Group 1st family (partition) 2nd family (partition) positive 3rd family (partition) Last family (partition) positive

  34. Identification of Critical Reference Group 1st family (partition) 2nd family (partition) negative 3rd family (partition) Last family (partition) positive

  35. Identification of Critical Reference Group Exactly l defectives Critical! Less than l defectives 50% positive tests 25% positive tests Statistical Difference!! More than l defectives 75% positive tests

  36. Threshold Group Testing with Linear Gap Smoother Gap, smaller statistical difference Many candidate critical reference groups Prob. of positive outcome # defective items

  37. Threshold Group Testing with other Gap Models As long as there exists a statistical difference Prob. of positive outcome # defective items Other Gap

  38. Thank you謝謝

  39. Identification within Critical Reference Group is partitioned into and Reference groups are randomly picked from each complement set.

  40. Threshold Group Testing Prob. of positive outcome ? # defective items Threshold Group Testing [Dam06]

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