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Fuzzy hypotheses testing in the framework of fuzzy logic. (English) Zbl 1050.68137
Summary: Testing hypotheses about the probability distribution underlying available empirical data is one of the fundamental data-analytic tasks in any application domain. Basically, it consists in checking the null hypothesis that the probability distribution, a priori assumed to belong to a certain set of distributions, actually belongs to some of its narrow subsets, which must be precisely delimited in advance. However, sometimes there are not enough clues for such a precise delimitation, especially if the purpose of the data analysis is explorative, a situation encountered increasingly often, due to the growing amount of routinely collected data and the increasing importance of data mining. That is why generalizations of statistical hypotheses testing to vague hypotheses have been investigated for more than a decade, so far based on the most straightforward approach – to replace the set defining the null hypothesis by a fuzzy set. In this paper, a principally different approach is presented, motivated by the observational logic and its success in automated knowledge discovery. Its key idea is to view statistical testing of a fuzzy hypothesis as the application of an appropriate generalized quantifier of a fuzzy predicate calculus to predicates describing the data. The theoretical principles of the approach are explained and its first implementations are briefly sketched.

MSC:
68T37 Reasoning under uncertainty in the context of artificial intelligence
03B52 Fuzzy logic; logic of vagueness
62-07 Data analysis (statistics) (MSC2010)
Software:
GUHA
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[1] Alexander, J.A.; Mozer, M.C., Template-based procedures for neural network interpretation, Neural networks, 12, 479-498, (1999)
[2] Arnold, B.F., Statistical tests optimally meeting certain fuzzy requirements on the power function and on the sample size, Fuzzy sets and systems, 75, 365-372, (1995) · Zbl 0851.62025
[3] Arnold, B.F., An approach to fuzzy hypotheses testing, Metrika, 42, 119-126, (1996) · Zbl 0862.62019
[4] Arnold, B.F., Testing fuzzy hypotheses with crisp data, Fuzzy sets and systems, 94, 232-333, (1998) · Zbl 0940.62015
[5] Casals, M.R., Bayesian testing of fuzzy parametric hypotheses from fuzzy information, Rech. oper./oper. res., 27, 189-199, (1993) · Zbl 0773.62001
[6] De Raedt, L., Interactive theory revision: an inductive logic programming approach, (1992), Academic Press London
[7] Delgado, M.; Verdegay, J.L.; Vila, M.A., Testing fuzzy hypotheses. a Bayesian approach, (), 307-316
[8] Džeroski, S., Inductive logic programming and knowledge discovery in databases, (), 117-152
[9] Grabisch, M.; Nguyen, H.T.; Walker, E.A., Fundamentals of uncertainty calculi with applications to fuzzy inference, (1995), Kluwer Academic Publishers Dordrecht
[10] Grzegorzewski, P.; Hryniewicz, O., Testing statistical hypotheses in fuzzy environment, Mathware soft comput., 4, 203-217, (1997) · Zbl 0893.68139
[11] Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Academic Publishers Dordrecht · Zbl 0937.03030
[12] Hájek, P.; Havránek, T., On generating of inductive hypotheses, Internat. J. man – mach. stud., 9, 415-438, (1977) · Zbl 0372.68026
[13] Hájek, P.; Havránek, T., Mechanizing hypothesis formation, (1978), Springer Berlin · Zbl 0371.02002
[14] Hájek, P.; Sochorová, A.; Zvárová, J., Guha for personal computers, Comput. statist. data anal., 19, 149-153, (1995) · Zbl 0875.62013
[15] Harmancová, D.; Holeňa, M.; Sochorová, A., Overview of the guha method for automating knowledge discovery in statistical data sets, (), 65-77
[16] Havránek, T., Statistical quantifiers in observational Calculian application in guha methods, Theory decision, 6, 213-230, (1975) · Zbl 0313.68070
[17] Holeňa, M., Exploratory data processing using a fuzzy generalization of the guha approach, (), 213-229
[18] Holeňa, M., Fuzzy hypotheses for guha implications, Fuzzy sets and systems, 98, 101-125, (1998)
[19] M. Holeňa, A fuzzy logic framework for testing vague hypotheses with empirical data in: Proc. Fourth Internat. ICSC Symp. on Soft Computing and Intelligent Systems for Industry, ICSC Academic Press, Sliedrecht, 2001, pp. 401-407.
[20] Holeňa, M., A fuzzy logic generalization of a data mining approach, Neural network world, 11, 595-610, (2001)
[21] M. Holeňa, Mining rules from empirical data with an ecological application, Tech. Report, Brandenburg University of Technology, Cottbus, 2002, ISBN 3-934934-07-2, 62pp.
[22] Holeňa, M.; Sochorová, A.; Zvárová, J., Increasing the diversity of medical data mining through distributed object technology, (), 442-447
[23] Klösgen, W., Explora: a multipattern and multistrategy discovery assistant, (), 249-272
[24] Mareš, M.; Mesiar, R., Calculation over verbal quantities, (), 409-427 · Zbl 0953.68133
[25] Muggleton, S., Inductive logic programming, (1992), Academic Press London · Zbl 0838.68093
[26] Narayanan, A., Revisable knowledge discovery in databases, Internat. J. intell. systems, 11, 75-96, (1996)
[27] Rauch, J., Logical calculi for knowledge discovery in databases, (), 47-57
[28] Römer, C.; Kandel, A., Statistical tests for fuzzy data, Fuzzy sets and systems, 72, 1-26, (1995) · Zbl 0843.62003
[29] Saade, J.J.; Schwarzlander, H., Fuzzy hypotheses testing with hybrid data, Fuzzy sets and systems, 35, 213-217, (1990) · Zbl 0713.62010
[30] Shen, W.M.; Ong, K.L.; Mitbander, B.; Zamiolo, C., Metaqueries for data mining, (), 375-398
[31] Watanabe, N.; Imaizumi, T., A fuzzy statistical test of fuzzy hypotheses, Fuzzy sets and systems, 53, 167-178, (1993) · Zbl 0795.62025
[32] Zembowicz, R.; Żytkov, J., From contingency tables to various forms of knowledge in databases, (), 329-352
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