linc, linc-an, and limd
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LINC, LINC-AN, and LIMD
Kai-Chun Fan presents2010.10.21
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
Agenda
LINC LINC-AN / LIMD Population Sizing Empirical Results Conclusions
Perturbation
A string (chromosome) ss = s1s2s3…sl
, where f(s) means the fitness of a string s
Linearity & Nonlinearity Form in LINC
Linearity
Nonlinearity ( may exist linkage between loci i & j )
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)
LINCLinkage Identification by Nonlinearity Check
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)
Agenda
LINC LINC-AN / LIMD Population Sizing Empirical Results Conclusions
AN – Allowable Nonlinearity
Nonlinearity
Allowable nonlinearity
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.
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.
LIMDLinkage Identification by Non-monotonicity Detection
As the same definition in LINC-AN, rewrite the above conditions as follows:
LINC-AN = LIMD
There exists linkage between loci i and j, if
X X
X X
LINC-AN = LIMD (contd.)
For simplicity, the authors define the following predicates,
LINC-AN = LIMD (contd.)
˅˄
˄
LINC-AN = LIMD (contd.)
LINC-AN = LIMD (contd.)
Agenda
LINC LINC-AN / LIMD Population Sizing Empirical Results Conclusions
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:
Agenda
LINC LINC-AN / LIMD Population Sizing Empirical Results Conclusions
Empirical Result (1)
Problem length l = 10 x 5 = 50# of strings (population size) = 100
Empirical Result (1) (contd.)
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).
Agenda
LINC LINC-AN / LIMD Population Sizing Empirical Results Conclusions
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
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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