Intelligent Tuning of Fuzzy Controllers by Learning and Optimization - Fuzzy Modeling and Control: Theory and Applications - page 133

input-output space. The mountain-clustering methods provide a systemic approach
for identifying the most important rules from the input-output data (Yager and Filev
1994 ).
Clustering is the partitioning of data into subsets or groups based on similarities
between the data. Among the clustering algorithms, we can find fuzzy clustering
methods. A normal clustering algorithm could be used to form a fuzzy controller,
but a fuzzy clustering algorithm provides an extra potentiality. One of the advantages
of fuzzy clustering algorithms is that the fuzziness between data remains intact (a
datum can belong to a several sets) when the membership functions have to be create
and, therefore, also the fuzzy rules.
Here, we will introduce different methods to perform fuzzy clustering where
we seek to use fuzzy sets to define soft boundaries to separate data into groups.
Among soft-computing techniques, fuzzy clustering is one of the most intensively
used strategies for process modeling and monitoring. Certainly, there are a great
amount of fuzzy clustering algorithms developed in last years. However, the classical
algorithms such as Fuzzy c-means, Gustafson-Kessel (G-K), Gath-Geva (G-G), and
fuzzy k-nearest neighbor are the most commonly applied methods. Nevertheless,
there are currently a lot of works with a clear tend to improve and complete classical
methods.
In this chapter, we will do a briefly revision of two classical fuzzy clustering
algorithms (FCM and FK-NN), a recent research that improves a classical fuzzy
clustering algorithm (Generalized Fuzzy C-Means with Improved Fuzzy Partitions)
and how a non-fuzzy clustering algorithm could also be useful to help us for adjusting
the structure of a fuzzy controller (Evolving Clustering Method).
Compared to other techniques, Fuzzy k-Nearest Neighbors (F-kNN) approach
is simple, easily interpretable and can achieve an acceptable accuracy rate (Keller
et al. 1985 ). The fuzzy version of k-NN averages the value of the points closest
to the query point, on the assumption that points close to each other have similar
values. However, standard k-nearest neighbor methods place equal weights on all
the selected neighbors, regardless of their distances from the query data. Learning
in fuzzy k-NN is simple, in the sense that, at creating fuzzy controllers, it is only
necessary to store the known data.
Among classical methods, Fuzzy c-means (FCM) clustering algorithm have some
features that make it very attractive for information extraction in modeling and con-
troling (Bezdek et al. 1984 ). FCM and its variants realize the clustering task for a
data set by minimizing an objective-function subject to some constraints, e.g., the
summation of all the membership degrees of every data point to all clusters must be
one. Due to its good performance, simplicity and broad knowledge, many authors
have focused on improvement it, particularly considering the fuzziness index or with
the use of different mathematical distances.
Among others, the Generalized Fuzzy c-Means with improved Fuzzy Partitions
algorithm aims at improving the slow convergence of FCM, to help us in the right
selection of the reward coefficient , and to expansion the fuzziness coefficient m

(without limitation). The main advantage with regard to algorithm with Improved

λ