Application of Wald Function to OR and AND Fuzzy Operations in No- Data Problems
DOI:
https://doi.org/10.6000/2371-1647.2016.02.09Keywords:
Subjective qualifier, fuzzy event, membership function, subjective probability distribution, OR-conjunction, AND-conjunctionAbstract
Subjective qualifiers of Wald's theory of decision functions are the fuzzy events of the subsequent fuzzy set theory. Wald's notion of subjective qualifiers involves applying integral transforms to convert states of nature into fuzzy events. Probabilities of fuzzy events and arithmetic formulas for fuzzy utility function values are readily derived from Wald's integral transforms. We have applied Zadeh's extension principle to Wald's integral transforms and demonstrated that fuzzy mathematics is effective when applied to multiple subjective probability distributions conjoined by OR and AND operations. In this paper, we focus on no-data problems and construct a fuzzy Bayes' theorem for cases in which a membership function and multiple subjective probability distributions conjoined by OR or AND operations are given. In addition, we devise a formulation for the corresponding decision making problem.References
Wald A. Statistical Decision Functions, Chelsea Publishing Company 1950.
Zadeh LA. Fuzzy Sets. Information and Control 1965; 8: 338- 353. http://dx.doi.org/10.1016/S0019-9958(65)90241-X
Zadeh LA. Probability Measures of Fuzzy Events. J Mathematical Analysis and Applications 1968; 22: 421-427. http://dx.doi.org/10.1016/0022-247X(68)90078-4
Tanaka H, Okuda T, Asai K. Fuzzy Information and Decision in Statistical Model. Advances in Fuzzy Sets Theory and Applications 1979; 303-320.
Uemura Y. Decision Making on Fuzzy Events. Japanese Journal of Fuzzy Sets and Systems 1991; 3: 123-130. (In Japanese)
Uemura Y, Sakawa M. Simple Decision Making based on the Subjective Possibility Distribution. Japanese Journal of Fuzzy Sets and Systems 1993; 5: 528-536. (In Japanese).
Uemura Y. A Normal Possibility Decision Rule. Control and Cybernetics 1996; 24: 103-111.
Uemura Y. Application of Normal Possibility Theory to Silence. Control and Cybernetics 2001; 30: 465-472.
Hori H, Tsubaki H. Fuzzy Decision Making Rule under nodata problem. Applied Mathematics (to appear) 2016.
H. Hori and Tsubaki H. Case Study of Fuzzy Events, Journal of Fuzzy Mathematics (to appear) 2016.
Hori H, Tsubaki H. Identification of Type 2 Fuzzy Sets and its application into Fuzzy Decision Problem. Journal of Fuzzy Mathematics (to appear) 2016.
Savage LJ. The Foundations of Statistics, Dover Publications, Inc., 1972.
Tanaka H, Ishibutchi M. Evidence Theory based on Normal Possibility Distribution. Japanese Journal of System Control and Information 1992; 5: 235-243. (In Japanese) http://dx.doi.org/10.5687/iscie.5.235
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