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LINC, LINC-AN, and LIMD

Kai-Chun Fan presents 2010.10.21. LINC, LINC-AN, and LIMD. Reference. LINC / LINC-AN Identifying Linkage by Nonlinearity C heck Masaharu Munetomo & David E. Goldberg IlliGAL Report No. 98012 LINC-AN / LIMD Identifying Linkage Groups by Nonlinearity/Non-monotonicity Detection

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LINC, LINC-AN, and LIMD

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  1. Kai-Chun Fan presents 2010.10.21 LINC, LINC-AN, and LIMD

  2. Reference • LINC / LINC-AN • Identifying Linkage by Nonlinearity Check • MasaharuMunetomo & David E. Goldberg • IlliGAL Report No. 98012 • LINC-AN / LIMD • Identifying Linkage Groups by Nonlinearity/Non-monotonicity Detection • MasaharuMunetomo & David E. Goldberg • GECCO 1999

  3. Agenda • LINC • LINC-AN / LIMD • Population Sizing • Empirical Results • Conclusions

  4. Perturbation A string (chromosome) s s = s1s2s3…sl , where f(s) means the fitness of a string s

  5. Linearity & Nonlinearity Form in LINC • Linearity • Nonlinearity ( may exist linkage between loci i & j )

  6. Check the Whole Population • Checking nonlinearity in one string is not enough, because there may exist a linearity inside a BB in some contexts (for example, a trap function is linear along its deceptive attractor). • A trap function with order k = 3, s = s0s1s2 = 110 Δf01(s) = f (000) – f (110) = 0.9 = 0.45 + 0.45 = [ f (010) – f (110) ] + [ f (100) – f (110)] = Δf0(s) + Δf1(s) s = s0s1s2 = 111 Δf01(s) = f (001)– f (111)= -0.55 ≠ -1.0 + -1.0 = [ f (011)– f (111) ] + [ f (101)– f (111)] = Δf0(s) + Δf1(s)

  7. LINCLinkage Identification by Nonlinearity Check

  8. Problem for LINC f (s) = (# of 1’s in s)t, for some t ≠ 1 s = s0s1 = 00 Δf01(s) = f (11)– f (00) = 4 ≠ 1+ 1 = [ f (10)– f (00) ] + [ f (01)– f (00)] = Δf0(s) + Δf1(s)

  9. Agenda • LINC • LINC-AN / LIMD • Population Sizing • Empirical Results • Conclusions

  10. AN – Allowable Nonlinearity • Nonlinearity • Allowable nonlinearity

  11. AN – Allowable Nonlinearity (contd.) • Why allowable? • Problems that satisfies the above condition are considered GA-easy in the loci (i, j)because positive changes of Δfi (s),Δfj(s) will increase the number of strings through selection, and the combination of the changes will also improve their fitness values.

  12. LINC-ANLinkage Identification by Nonlinearity Check with Allowable Nonlinearity • Redefinition • fi (s) = f(s) + Δfi (s) • fj(s) = f(s) + Δfj(s) • fij(s) = f(s) + Δfij(s) • If the perturbations in si and sj cause monotone increase or decrease of fitness values along f(s) → fi (s) → fij(s) and f(s) → fj(s) → fij(s) for all strings (or almost all), the nonlinearity is considered allowable.

  13. LIMDLinkage Identification by Non-monotonicity Detection As the same definition in LINC-AN, rewrite the above conditions as follows:

  14. LINC-AN = LIMD • There exists linkage between loci i and j, if X X X X

  15. LINC-AN = LIMD (contd.) • For simplicity, the authors define the following predicates,

  16. LINC-AN = LIMD (contd.) ˄ ˅ ˄

  17. LINC-AN = LIMD (contd.)

  18. LINC-AN = LIMD (contd.)

  19. Agenda • LINC • LINC-AN / LIMD • Population Sizing • Empirical Results • Conclusions

  20. Population Sizing • Considering the worst case in which we have only one string which shows nonlinearity/non-monotonicity, the probability that we have the string in the population is: • If we fix a success probability r by solving P = r, we have: • When we set r = 1 - 2-k, at which a failure may occur in one of all the 2k combinations of order-k schemata, we have:

  21. Agenda • LINC • LINC-AN / LIMD • Population Sizing • Empirical Results • Conclusions

  22. Empirical Result (1) Problem length l= 10 x 5 = 50 # of strings (population size) = 100

  23. Empirical Result (1) (contd.)

  24. Empirical Result (2) • LINC: • All the loci are forced to be included in one linkage group. • LINC-AN / LIMD: • Same as the empirical result (1).

  25. Agenda • LINC • LINC-AN / LIMD • Population Sizing • Empirical Results • Conclusions

  26. Why D5?

  27. Conclusions • LINC, LINC-AN, and LIMD procedures are based on an idea that nonlinearity/non-monotonicity detection by order-2 simultaneous perturbations performed on O(2k) strings gives information on at most order-k linkage groups. • Since LINC-AN and LIMD further detect the non-monotonicity conditions, they can recognize GA-easiness more accurately than LINC and traditional nonlinearity-checking methods. • However, the cost for additional fitness evaluation is still a critical problem for detecting linkage by using perturbation.

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