Towards Making Fuzzy Techniques More Adequate for Combining Knowledge of Several Experts
Hoang Phuong Nguyen*, and Vladik Kreinovich**
*Division Informatics, Math-Informatics Faculty, Thang Long University
Nghiem Xuan Yem Road, Hoang Mai District, Hanoi, Vietnam
**Department of Computer Science, University of Texas at El Paso
500 West University Avenue, El Paso, Texas 79968, USA
In medical and other applications, expert often use rules with several conditions, each of which involve a quantity within the domain of expertise of a different expert. In such situations, to estimate the degree of confidence that all these conditions are satisfied, we need to combine opinions of several experts – i.e., in fuzzy techniques, combine membership functions corresponding to different experts. In each area of expertise, different experts may have somewhat different membership functions describing the same natural-language (“fuzzy”) term like small. It is desirable to present the user with all possible conclusions corresponding to all these membership functions. In general, even if, for each area of expertise, we have only a 1-parametric family characterizing different membership function, then for rules with 3 conditions, we already have a difficult-to-interpret 3-parametric family of possible consequences. It is thus desirable to limit ourselves to the cases when the resulting family is still manageable – e.g., is 1-parametric. In this paper, we provide a full description of all such families. Interestingly, it turns out that such families are possible only if we allow non-normalized membership functions, i.e., functions for which the maximum may be smaller than 1. We argue that this is a way to go, since normalization loses some information that we receive from the experts.
-  R. Bělohlávek, J. W. Dauben, and G. J. Klir, “Fuzzy Logic and Mathematics: A Historical Perspective,” Oxford University Press, 2017.
-  G. J. Klir and B. Yuan, “Fuzzy Sets and Fuzzy Logic: Theory and Applications,” Prentice Hall, 1995.
-  J. M. Mendel, “Uncertain Rule-Based Fuzzy Systems: Introduction and New Directions,” 2nd Edition, Springer, 2017.
-  H. T. Nguyen, C. L. Walker, and E. A. Walker, “A First Course in Fuzzy Logic,” 4th Edition, CRC Press, 2019.
-  V. Novák, I. Perfilieva, and J. Močkoř, “Mathematical Principles of Fuzzy Logic,” Kluwer Academic Publishers, 1999.
-  L. A. Zadeh, “Fuzzy sets,” Information and Control, Vol.8, Issue 3, pp. 338-353, 1965.
-  H. T. Nguyen, V. Kreinovich, and P. Wojciechowski, “Strict Archimedean t-norms and t-conorms as universal approximators,” Int. J. of Approximate Reasoning, Vol.18, Issues 3-4, pp. 239-249, 1998.