Pattern Classification Based on New Interpretation of MFI

Abstract

Our aim is to design a pattern classifier using fuzzy relational calculus (FRC) which was initially proposed by Pedrycz (Pattern Recogn 23 (1/2):121–146, 1990). In the course of doing this we introduce a new interpretation of multidimensional fuzzy implication (MFI) (Sugeno and Takagi Fuzzy Sets Syst 9:313–325, 1983); (Tsukamoto in Advance in Fuzzy Set Theory and Applications North-Holland, Amsterdam, pp. 137–149, 1979) (see Eq. (A.4) of Appendix-A) to represent our knowledge about the training data set. The new interpretation is basically a set of one-dimensional fuzzy implications. The consequences of all one-dimensional fuzzy implications are finally collected through one intersection operator ‘∩’. Subsequently, we consider the notion of a fuzzy pattern vector, which is formed by the cylindrical extension of the antecedent part of each one-dimensional fuzzy implication. Thus, we get a set of fuzzy pattern vectors (see Fig. A.2 and A.3 of Appendix-A) for the new interpretation of MFI and represent the population of training patterns in the pattern space. We also consider the new approach to the computation of the derivative of the fuzzy max and min functions, as stated earlier, using the concept of a generalized function (see Appendix-B and Sect. 2.3). During the construction of the classifier, based on FRC, we use fuzzy linguistic statements (or fuzzy membership function to represent the linguistic statement) to represent the ranges of features (e.g. feature F _i is small/medium/big, etc. ∀i) for a population of patterns. Note that the construction of the classifier essential depends on the estimate of a fuzzy relation ℜi between the antecedent part and consequent part of each one-dimensional fuzzy implication. Thus, a set of fuzzy relations is formed from the new interpretation of MFI. This set of fuzzy relations is termed as a core of the classifier. Once the classifier is constructed the non-fuzzy features of a pattern can be classified. At the time of classification of the test patterns, we use the concept of fuzzy singleton to fuzzy the non-fuzzy feature values of the test patterns. The performance of the proposed scheme is tested on synthetic data. Finally, we use the proposed scheme for the vowel classification problem of Indian languages.