Evolving Neural and Fuzzy Systems - Computational Intelligence in Time Series Forecasting - page 306

x optimum search capabilities of evolutionary algorithms can help design and
optimally tune the parameters of fuzzy logic systems
x during the evolutionary processes the rule base of the fuzzy system could
be used to automatically tune the algorithm parameters in order to avoid its
premature convergence and other undesired behaviour of the search
process
x the fuzziness can be embedded into the algorithms for internal calculations
of fitness function, etc .
The work on the design of fuzzy logic systems using evolutionary computation was
effectively initiated in early 1990s and was made public by the reports of Thrift
(1991) and Karr (1991) on the use of genetic algorithms in synthesis of fuzzy logic
controllers. This was later extended to the synthesis of a model-reference adaptive
controller (Hwang and Thompson, 1994). In the early considerations of the
evolving procedures, triangular membership functions were preferred because their
encoding within the chromosome as finite-length bit strings was relatively simple.
This kind of membership functions is parameterized by the left and the right base
and by the distance from the previous centre point. For evolving purposes, the
same triangular form for all membership functions and the same number of
membership functions for each variable were taken.
Some researchers (Hwang and Thompson, 1994) encoded all the rules and the
fixed membership functions into the chromosome. Under this condition, the
evolving process, however, did not evolve an optimal fuzzy system, because the
shape of the membership functions is strongly related to the character of the rules.
As a consequence, both the rules and the membership functions have to be evolved
simultaneously. Homaifar and McCormick (1995) solved the problem of
simultaneous tuning of the membership functions and evolving the rule set by
encoding all the rules and the base length of each triangular membership function
into chromosomes.
Thrift (1991) pleaded for building the fuzzy rule base in tabular form by
assigning to each input variable a number of partition domains, say n , that are to be
specified in detail. This, however, was not applicable, because in this way a huge
number of detailed data are generated that cannot be stored in a transparent form.
The idea of Thrift, of representing the generated data in matrix form, was
acceptable only for small fuzzy systems, say for systems with two input variables
for which an n u matrix is to be built. However, for a system with a higher
number of input variables an

uuu dimensional matrix has to be built.
To avoid the super-dimensionality problem, Lee and Takagi (1993a)
recommended numerating the rules instead of tabulating them. They also encoded
the membership functions and the rule set into the chromosomes, but they took
another route to encoding the triangular membership function by restricting the
adjacent membership functions from fully overlapping and by some additional
restrictions. This considerably reduced the total number of membership functions
required. Further reduction is still possible by grouping the given rules into
relevant (needed) and non-relevant ones, and by encoding only the relevant rules.
This enables fuzzy systems of higher dimensionality to be evolved.
While considering the Takagi-Sugeno model, in which the consequent part is

nn ... n