linc, linc-an, and limd

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

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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 - PowerPoint PPT Presentation

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Page 1: LINC, LINC-AN, and LIMD

LINC, LINC-AN, and LIMD

Kai-Chun Fan presents2010.10.21

Page 2: LINC, LINC-AN, and LIMD

Reference

LINC / LINC-AN Identifying Linkage by Nonlinearity Check Masaharu Munetomo & David E. Goldberg IlliGAL Report No. 98012

LINC-AN / LIMD Identifying Linkage Groups by

Nonlinearity/Non-monotonicity Detection Masaharu Munetomo & David E. Goldberg GECCO 1999

Page 3: LINC, LINC-AN, and LIMD

Agenda

LINC LINC-AN / LIMD Population Sizing Empirical Results Conclusions

Page 4: LINC, LINC-AN, and LIMD

Perturbation

A string (chromosome) ss = s1s2s3…sl

, where f(s) means the fitness of a string s

Page 5: LINC, LINC-AN, and LIMD

Linearity & Nonlinearity Form in LINC

Linearity

Nonlinearity ( may exist linkage between loci i & j )

Page 6: LINC, LINC-AN, and LIMD

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

Δ f01(s) = f (000) – f (110) = 0.9 = 0.45 + 0.45 = [ f (010) – f (110) ] + [ f (100) – f (110) ] = Δ f0(s) + Δ f1(s)

s = s0s1s2 = 111

Δ f01(s) = f (001) – f (111) = -0.55 ≠ -1.0 + -1.0 = [ f (011) – f (111) ] + [ f (101) – f (111) ] = Δ f0(s) + Δ f1(s)

Page 7: LINC, LINC-AN, and LIMD

LINCLinkage Identification by Nonlinearity Check

Page 8: LINC, LINC-AN, and LIMD

Problem for LINC

f (s) = (# of 1’s in s)t , for some t ≠ 1

s = s0s1 = 00

Δ f01(s)

= f (11) – f (00)

= 4

≠ 1 + 1

= [ f (10) – f (00) ] + [ f (01) – f (00) ]

= Δ f0(s) + Δ f1(s)

Page 9: LINC, LINC-AN, and LIMD

Agenda

LINC LINC-AN / LIMD Population Sizing Empirical Results Conclusions

Page 10: LINC, LINC-AN, and LIMD

AN – Allowable Nonlinearity

Nonlinearity

Allowable nonlinearity

Page 11: LINC, LINC-AN, and LIMD

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 Δ fi (s), Δ fj (s) will increase the number of strings through selection, and the combination of the changes will also improve their fitness values.

Page 12: LINC, LINC-AN, and LIMD

LINC-ANLinkage Identification by Nonlinearity Check with Allowable Nonlinearity

Redefinition fi (s) = f (s) + Δ fi (s)

fj (s) = f (s) + Δ fj (s)

fij (s) = f (s) + Δ fij (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.

Page 13: LINC, LINC-AN, and LIMD

LIMDLinkage Identification by Non-monotonicity Detection

As the same definition in LINC-AN, rewrite the above conditions as follows:

Page 14: LINC, LINC-AN, and LIMD

LINC-AN = LIMD

There exists linkage between loci i and j, if

X X

X X

Page 15: LINC, LINC-AN, and LIMD

LINC-AN = LIMD (contd.)

For simplicity, the authors define the following predicates,

Page 16: LINC, LINC-AN, and LIMD

LINC-AN = LIMD (contd.)

˅˄

˄

Page 17: LINC, LINC-AN, and LIMD

LINC-AN = LIMD (contd.)

Page 18: LINC, LINC-AN, and LIMD

LINC-AN = LIMD (contd.)

Page 19: LINC, LINC-AN, and LIMD

Agenda

LINC LINC-AN / LIMD Population Sizing Empirical Results Conclusions

Page 20: LINC, LINC-AN, and LIMD

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:

Page 21: LINC, LINC-AN, and LIMD

Agenda

LINC LINC-AN / LIMD Population Sizing Empirical Results Conclusions

Page 22: LINC, LINC-AN, and LIMD

Empirical Result (1)

Problem length l = 10 x 5 = 50# of strings (population size) = 100

Page 23: LINC, LINC-AN, and LIMD

Empirical Result (1) (contd.)

Page 24: LINC, LINC-AN, and LIMD

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).

Page 25: LINC, LINC-AN, and LIMD

Agenda

LINC LINC-AN / LIMD Population Sizing Empirical Results Conclusions

Page 26: LINC, LINC-AN, and LIMD

Why D5 ?

LINC-AN / LIMD D5

Nonlinearity Form

Δ fij (s) = Δ fi (s) + Δ fj (s) Δ fi (s p) = f (s p) - f (si p)

Detect Nonlinear GA-easy

Non-monotonicity Entropy?

Additional Fitness

Evaluation

Every perturbationNo need (average schema

fitness)

Clustering Mechanis

mLinkage set Entropy

Page 27: LINC, LINC-AN, and LIMD

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.