HEDGE ALGEBRAS WITH LIMITED NUMBER OF HEDGES AND APPLIED TO FUZZY CLASSIFICATION PROBLEMS

Abstract

SUMMARY

In this paper we introduce the Hedge Algebras with a limited number of hedges, called AX2. In the  AX2, we consider the  g-grade similarity fuzziness interval of a linguistic term  x, denote T g(x), which are constructed from two (g+k)-fuzziness intervals satisfy that  υ(x) is inside the interval (Definition 2.1,  k  =  l(x) is the length of  x). There is a system of  k-similarity fuzziness intervals, denote  S SS S(k), corresponding to a set of linguistic terms that their length is less than or equal to k, denote X(k) (Definition 2.2). Especially, we prove that the system is always exist and a partition of [0,1], it is constucted by a set of (k+2)-fuzziness intervals (Theorem 2.1), so the AX2
with its partition of  k-similarity fuzziness intervals can be used in any  real domain (Theorem 2.2).
Fuzzy rule-based systems are widely used for classification problems. There are two main goals in the design of fuzzy rule-based systems: one is the accuracy maximization and the other is the complexity minimization. Various approaches  have been proposed to deal with the problem [19, 22, 25, 23]. So in the section 3, we propose an extracting fuzzy rules algorithm
(RFRG) for classification problems base on the partitions  S SS S
(k) of domain of attributes. The generated rules, denote S0, of this algorithm content all attributes, i.e. their antecedents have full attributes of the problem, we call them a robust fuzzy rules-set. These rules can improve accuracy upto 100% by choosing particularly  k-similarity fuzziness intervals of attributes
(Corollary 2.2). However, this may increase the complexity of the fuzzy rules-set. To overcome this problem, we design an algorithm to optimize the fuzzy rules-set by using genetic algorithms and annealing simulation [1, 5, 7, 8, 26]. The solutions of this optimal problems are encoded in real encoding, which represents rule’s index and attribute’s index in  S0 to be selected, then the fitnessfunction is given as a weighting of three objectives: maximize  the  performance of rules-set, minimizethe number of rulesand minimize the average rule-length. In the section 4, we apply our method to the  glassproblem that posted in UCI machine learning repository. The results, in all patterns for training case, are better than [25] in comparision, the best accuracy of our method is 78.04% with 14 fuzzy rules while [Mansoori-07] is 78.5% with 95 fuzzy rules. In the 10-foldsexperiment, the best accuracy on testing patterns of our method is 64.67% with 15.9 average fuzzy rules, comparing with [19] is 62.97% with 28.32 average fuzzy rules. The comparision shows that the accuracy of our method is better than [19] and [25].