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

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.