Interpolation in Hierarchical Fuzzy Rule Bases Leila Muresan Dept. of Autovehicles and Dept. of Telecom. & Telematics, Technical University of Budapest, Sztoczek u. 6, 1111 Budapest, Hungary
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Abstract-- A major issue in the field of fuzzy applications is the complexity of the algorithms used. In order to obtain efficient methods, it is necessary to reduce complexity without losing the easy interpretability of the components. One of the possibilities to achieve complexity reduction is to combine fuzzy rule interpolation with the use of hierarchical structured fuzzy rule bases, as proposed by Sugeno. As an interpolation method the KH interpolation is used, but other techniques are also suggested. The difficulty of applying this method is that it is often impossible to determine a partition of any subspace of the original state space so that in all elements of the partition the number of variables can be locally reduced. Instead of this, a sparse fuzzy partition is searched for and so the local reduction of dimensions will be usually possible. In this case however, interpolation in the sparse partition itself, i.e. interpolation in the meta-rule level is necessary. This paper describes a method how such a multi-level interpolation is possible. Index terms-- Hierarchical rule bases, fuzzy interpolation, rule base size reduction
I. INTRODUCTION The classical approaches of fuzzy control deal with dense rule bases where the universe of discourse is fully covered by the antecedent fuzzy sets of the rule base in each dimension, thus for every input there is at least one activated rule. The main problem is the high computational complexity of these traditional approaches. If a fuzzy model contains k variables and maximum T linguistic (or other fuzzy) terms in each dimension, the O(T k ) . This order of the number of necessary rules is expression can be decreased either by decreasing T, or k, or both. The first method leads to sparse rule bases and rule interpolation, that was first introduced by Kóczy and Hirota (see e.g. [9,10]). The second one, more effective, aims to reduce the dimension of the sub-rule bases (k’s) by using meta-levels or hierarchical fuzzy rule bases. The combination of the two was first attempted by the authors in [11,12]. In this paper, this method will be further developed. II. BASIC CONCEPTS OF FUZZY INTERPOLATION AND HIERARCHICAL FUZZY RULE BASES The rule bases containing gaps require completely different techniques of reasoning from the traditional ones. The idea of first interpolation technique, proposed in [9,10],
considers the representation of a fuzzy set as the union of its α-cuts. The KH method calculates the conclusion by its α-cuts. Theoretically all α-cuts should be considered, but for practical reasons only a finite set is taken into consideration during the computation (as it will be seen in the last section, Practical issues). The KH rule interpolation algorithm requires the following conditions to be fulfilled: the fuzzy sets in both premises and consequences have to be convex and normal, for brevity, CNF sets, with bounded support, having also continuous membership functions. Further, there should exist a partial ordering among the CNF sets of each variable: A p B, A, B ∈ F ( X ) if
∀α ∈ [0,1] : inf{ Aα } ≤ sup{Bα } ∧ sup{ Aα } ≤ sup{Bα }. If two fuzzy sets are comparable, we can define a fuzzy distance, such that for all α∈[0,1], (in practice, only for each significant α), it is composed of the pairwise distances between the two extrema of these fuzzy sets: these pairs are named the ”lower” and the “upper fuzzy distance” of the two α-cuts. Using the concept of fuzzy distance, the Fundamental Equation of Rule Interpolation can be written as:
d(A*, A1) : d(A*, A2 ) = d(B*, B1) : d(B* , B2 )
(1) where A1 → B1 and A2 → B2 are the rules, A* is the observation, B* is the computed conclusion and The above mentioned distance is not the only possible choice for d, see [16,17]. A1 p A * p A2
∧
B1 p B 2 .
The conclusion generated by the solution of the Fundamental Equation is given below: This family of interpolation techniques has various advantageous properties, mainly simplicity, convenient
inf{B1α } inf{B2α } + * d α L ( A1α , Aα ) d α L ( A2α , Aα* ) * inf{Bα } = 1 1 + * d α L ( A1α , Aα ) d α L ( A2α , Aα* )
(2)
practical applicability, stability (see [1]), however, the direct applicability of this result is limited as the points defined by these equations often describe an abnormal membership function for B*, that needs further
transformation for obtaining a regular fuzzy set. Even then, the conclusion needs further transformation, since it is not normal. This fact is very important from the point of view of this paper, as the typical shape of membership functions in sparse fuzzy partitions never satisfies the conditions that have been determined in [14,5,7] Furthermore, the method does not conserve piecewise linearity of the rules and the observation. In order to avoid this difficulty, some alternative or modified interpolation algorithms are proposed, which solve the problem of abnormal solution. A part of these algorithms rely on the same idea of applying the Fundamental Equation of rule interpolation for some kind of metric, or in some cases non-metric dissimilarity degree. However, the way of measuring this distance or dissimilarity is varying. In [13], the distance is measured only for α=1 and so the position and length of the core in each of the k input dimensions determine the position and the core of conclusion in the output space Y. The remaining information is contained in the shape of the flanks of the fuzzy sets describing the rules, (the parts where 0
