Hedge Algebras with limited number of hedges and applied to fuzzy classification problems

Abstract

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 Tg(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_(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_(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  S0to be selected, then the fitnessfunction is given as a weighting of three objectives: maximize  the  performance of rulesset, minimizethe number of rulesand minimize the average rule-length.

In the  section 4, we apply our method to the glass problem 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-folds experiment, 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].