Transforming probability distributions into membership functions of fuzzy classes: A hypothesis test approach

Abstract In fuzzy Decision Support Systems, methods are strongly required for eliciting knowledge in the form of interpretable fuzzy sets from numerical data. In medical settings, statistical data are often available, or can be obtained from rough data, typically in the form of probability distributions. Moreover, since physicians are used to think and work according to a statistical interpretation of medical knowledge, the definition of fuzzy sets starting from statistical data is thought to be able to significantly reduce the existing lack of familiarity of physicians with fuzzy set theory, with respect to the classical statistical methods. Some methods based on different assumptions transform probability distributions into fuzzy sets. However, no theoretical approach was proposed up to now, for extracting fuzzy knowledge according to a fuzzy class interpretation, which can be used for inference purposes in fuzzy rule based systems. In this paper, a method for transforming probability distributions into fuzzy sets is shown, which generalizes some existing approaches and gives them a justification. It is based on the application of statistical test of hypothesis, and the resulting fuzzy sets are interpretable as fuzzy classes. The method enables the construction of normal fuzzy sets, which can be adapted to have pseudo-triangular or pseudo-trapezoidal shape, both coherently with the corresponding probability distributions, by tuning the method parameters. The properties of this method are illustrated by applying it to simulated probability distributions and its experimental comparison with existing methods is shown. Moreover, an application is performed on a real case study involving the detection of Multiple Sclerosis lesions.

[1]  Ronald Christensen,et al.  Testing Fisher, Neyman, Pearson, and Bayes , 2005 .

[2]  I. Turksen,et al.  Measurement of Membership Functions: Theoretical and Empirical Work , 2000 .

[3]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

[4]  I. B. Turksen,et al.  Measurement of Membership Functions , 1986 .

[5]  J. Reid.  Experimental Design and Data Analysis for Biologists , 2003 .

[6]  Didier Dubois,et al.  Practical representations of incomplete probabilistic knowledge , 2006, Comput. Stat. Data Anal..

[7]  Philippe Smets,et al.  Belief functions on real numbers , 2005, Int. J. Approx. Reason..

[8]  George J. Klir,et al.  A MATHEMATICAL ANALYSIS OF INFORMATION-PRESERVING TRANSFORMATIONS BETWEEN PROBABILISTIC AND POSSIBILISTIC FORMULATIONS OF UNCERTAINTY , 1992 .

[9]  PradeHenri,et al.  Gradualness, uncertainty and bipolarity , 2012 .

[10]  Diego A. Alvarez,et al.  On the calculation of the bounds of probability of events using infinite random sets , 2006, Int. J. Approx. Reason..

[11]  D. Dubois,et al.  Fuzzy sets and statistical data , 1986 .

[12]  Giuseppe De Pietro,et al.  Data Driven Generation of Fuzzy Systems: An Application to Breast Cancer Detection , 2010, CIBB.

[13]  Klaus-Peter Adlassnig,et al.  Definition of Fuzzy Sets Representing Medical Concepts and Acquisition of Fuzzy Relationships Between Them by Semi-Automatic Procedures , 1999 .

[14]  Didier Dubois,et al.  Modeling uncertain and vague knowledge in possibility and evidence theories , 2013, UAI.

[15]  Didier Dubois,et al.  Probability-Possibility Transformations, Triangular Fuzzy Sets, and Probabilistic Inequalities , 2004, Reliab. Comput..

[16]  Didier Dubois,et al.  Gradualness, uncertainty and bipolarity: Making sense of fuzzy sets , 2012, Fuzzy Sets Syst..

[17]  Arthur P. Dempster,et al.  Upper and Lower Probabilities Induced by a Multivalued Mapping , 1967, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[18]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[19]  Didier Dubois,et al.  Unifying practical uncertainty representations - I: Generalized p-boxes , 2008, Int. J. Approx. Reason..

[20]  H. Carter.  Fuzzy Sets and Systems — Theory and Applications , 1982 .

[21]  D. Dubois,et al.  On Possibility/Probability Transformations , 1993 .

[22]  DesterckeS.,et al.  Unifying practical uncertainty representations -- I , 2008 .

[23]  Inés Couso,et al.  A random set characterization of possibility measures , 2004, Inf. Sci..

[24]  Vladik Kreinovich,et al.  Entropy conserving probability transforms and the entailment principle , 2007, Fuzzy Sets Syst..

[25]  George J. Klir,et al.  A principle of uncertainty and information invariance , 1990 .

[26]  Giuseppe De Pietro,et al.  Transformation of probability distribution into fuzzy set interpretable with likelihood view , 2011, 2011 11th International Conference on Hybrid Intelligent Systems (HIS).

[27]  Monique Frize,et al.  Does Bayes have it? Decision support systems in diagnostic medicine , 2009 .

[28]  K. Yamada.  Probability-possibility transformation based on evidence theory , 2001, Proceedings Joint 9th IFSA World Congress and 20th NAFIPS International Conference (Cat. No. 01TH8569).

[29]  Mohammad R. Akbarzadeh-Totonchi,et al.  A hierarchical fuzzy rule-based approach to aphasia diagnosis , 2007, J. Biomed. Informatics.

[30]  L. Zadeh.  Fuzzy sets as a basis for a theory of possibility , 1999 .

[31]  Henri Prade,et al.  Fuzzy sets and probability: misunderstandings, bridges and gaps , 1993, [Proceedings 1993] Second IEEE International Conference on Fuzzy Systems.

[32]  Mathieu Serrurier,et al.  Maximum-likelihood principle for possibility distributions viewed as families of probabilities , 2011, 2011 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE 2011).

[33]  Didier Dubois,et al.  A semantics for possibility theory based on likelihoods , 1995, Proceedings of 1995 IEEE International Conference on Fuzzy Systems..

[34]  R. Geetha Ramani,et al.  Mining of classification patterns in clinical data through data mining algorithms , 2012, ICACCI '12.

[35]  Didier Dubois,et al.  A definition of subjective possibility , 2008, Int. J. Approx. Reason..