Biomedical Engineering Reference
In-Depth Information
4.2 ExactSchemaTheoremforJumpingGeneTransposition
For. a. GA. incorporating. jumping. gene. (JG). operations,. the. exact. schema.
formulae.for.selection,.crossover,.and.mutation.can.be.directly.borrowed.
from. the. ones. derived. by. Stephens. and. Waelbroeck,. as. described. in.
Section 4.1.3..Therefore,.our.focus.is.to.derive.the.schema.evolution.equa-
tions. for. the. two. JG. transpositions,. that. is,. the. copy-and-paste. and. cut-
and-paste.operations.[16,20].
4.2.1 Notations and Functional Definitions
Before.describing.the.derivation.of.the.equations,.the.notations.and.functions.
are.given.
4.2.1.1 Notations
L
.
The.length.of.the.binary.string.(i.e.,.the.chromosome.length).
S
(
i
)
.
The.set.of.all.schemata.with.length.
i
,.and.for.simplicity,.
(
L
≡ .
)
S
S
V
(
i
)
.
.The.superset.of.all.sets.formed.by.integers.0.to.(
i
.−.1),.and.for.simplicity,.
(
L
≡ ..For.example,.if.
L
.=.3,.
(3)
V
)
V
V
= φ
{ , {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}}
.
Z
,
Z
even
.
.The.subsets.of.
S
,.
even
= ξ ξ = ,.for.some.
integer.
n
,.and.
zeros
(ξ).returns.the.number.of.zeros.in.schema.ξ∈
S
..It.is.also.
denoted.that.
Z
Z
= ξ
{
:
zeros
( )
ξ =
2
n
+
1}
,.
Z
{
:
zeros
( )
2 }
n
odd
odd
′
=
Z
.and.
Z
′
=
Z
.
odd
even
even
odd
. .
For.example,.if.ξ=*01**0,.since.
zeros
ξ =
( )
2
,.we.can.get.
ξ ∈
Z
even
.
o
(ξ).
The.order.of.schema.ξ.
L
d
ξ .
The.defining.length.of.schema.ξ.
( )
L
g
.
The.length.of.transposon,.where.1
L
≤
<
L
.
g
4.2.1.2 Functional Definitions
Deinition4.1
A. map.
f
L
. is. deined. as.
v f
L
( )
. returns. the.
locations. of. all. the. actual. bits. in. schema. ξ
∈
S
.. It. is. also. assumed. that. the.
location.begins.from.0.
f
L
:
S V
→
,. such. that. for.
v
∈
V
,.
=
ξ
Deinition4.2
( )
,.such.that.for.
i
S
i
( )
A.map.
f
T
.is.defined.as.
f
S V
S
ξ
∈
,
. ξ
=
f
T
(
ξ
,
v
)
.
:
×
→
T
2
2
1
is.formed.by.copying.the.bits.from.schema.
ξ
∈
S
.according.to.the.locations.
1
speciied.in.
v
∈
V
,.where.
i
=
size(v)
.
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