T Cell–Inspired Algorithms - Immunological Computation: Theory and Applications - page 87

1 1 n n

S 1 ,..., S m S 1 ,..., S m

…

Gene 1 Gene n

Figure 4.10 Structure of the chromosome representing the condition part of a
rule. Each gene represents an atomic condition x i ∊

T i and each bit s i j is “on” if and
only if the corresponding basic fuzzy set S j is part of the composite fuzzy set T j .

Each atomic condition, x i ∈ T i , corresponds to a gene in the chromosome that is

represented by a sequence ( s i 1 , …, s i m ) of bits, where m = | S | (the size of the set of

T i . h at is, the bit s j is “on” if and
only if the corresponding basic fuzzy set S j is part of the composite fuzzy set T j .
Figure 4.10 shows the structure of a chromosome which is n

linguistic values) and s j

= ⊆

1 if and only if S j

×

m bits long ( n is the
dimension of the space and m the number of basic fuzzy set). Hamming distance
was used as a distance measure. For example, if the s i bit (see Figure 4.10) in both
parent and child fuzzy rule detectors is set to 1, both individuals include the atomic
sentence x i

∊

s j , that is, they use the j th fuzzy set to cover some part of the i th
attribute. h en, the more bits the parent and the child have in common, the more
common area they cover.
h e fi tness of a rule R i is calculated by taking into account the following two
factors: the fuzzy true value produced when the condition part of a rule, Cond i , is
evaluated for each element x from the self-set:

∑

Cond () x

i

∈

xS f

selfC ov ering R ()

Self

h e fuzzy measure of the volume of the subspace represented by the rule:

n

∏

volume R () measure T i ( )

i 1

where “measure” ( T i ) corresponds to the area under the membership function of
the fuzzy set T i .
h e fi tness is defi ned as follows:

= ⋅ − + − ⋅

fi t n e s s ( R ) C (1 selfCovering(R)) (1 C) volume(R)

1, is a coe cient that determines the amount of penalization
that a rule suff ers if it covers normal samples. h e closer the coe cient to 1, the
higher the penalization value (values between 0.8 and 0.9 were used).
h e pseudocode in NS Algorithm 6 show the details of Negative Selection
with Fuzzy Detection Rules (NSFDR) implementation; the time complexity of the
algorithm is O ( num _ gen

≤ ≤

where C , 0 C

⋅ ⋅

pop _ size | Self ' |) .