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In summary, the FLAGID method pursues the following steps:
1. It uses the DDA/RecBF algorithm to get a first set of membership function from
the dataset.
2. It recombines the membership functions found by DDA/RecBF and obtains a new
set of membership functions.
3. Finally, it applies a genetic algorithm to find a small set of fuzzy rules that uses the
membership functions obtained in step 2 and provide high classification accuracy.
These steps will be explained deeply in the following three subsections.
2.3.1 Constructing a First Set of Membership Functions by Applying RecBFs
The DDA/RecBF clustering algortihm 13 constructs a set of membership functions
from a dataset that provides a set of linguistic labels that can be later used to concisely
express relationships and dependencies in the dataset. The algorithm creates hyper-
rectangles (called Rectangular Basis Functions or RecBFs ) belonging to every class,
from the input dataset and a defined fuzzy output variable. From these RecBFs, a set
of membership functions will be extracted for every variable.
Each RecBF is defined by a support-region and a core-region . In terms of mem-
bership function, a trapezoidal membership function is composed of four points
(a,b,c,d): the interval [a,d] defines the support-region and the [b,c] one, the
core-region. An example is shown in Fig. 2.2.
a b c d
Core Region
Support Region
Fig. 2.2. Example of a 2-dimensional Fuzzy Point and its membership functions defined by its
two regions. The a,b,c & d points are different for each dimension.
DDA/RecBF algorithm is based on three procedures, which are executed for every
training pattern: covered is executed when the support-region of a RecBF covers a
pattern, commit creates a new pattern if the previous condition is false and, finally, the
procedure shrink solves possible conflicts of the pattern with respect to RecBFs of a
different class. Fig. 2.3 shows the behaviour of the DDA/RecBF algorithm for one
epoch.
Fig. 2.4 shows an example of the execution of the DDA/RecBF algorithm. It is
shown how the patterns are passed to the DDA/RecBF algorithm and how the
different RecBFs are created.
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