Type-2 Fuzzy membership function design method through a piecewise-linear approach

Type-2 Fuzzy sets (T2FS) represent a large amount of uncertainty and complexity; however, these benefits are closely related to the definition of its footprint of uncertainty (FOU). The FOU can be so freely designed that difficulties are encountered when trying to compare any two of them, or conclude how a system is improved by varying FOU's characteristics. Consequently, a parametrization method is needed, which permits a systematic analysis and comparison in terms of design criteria. This paper presents a novel method which allows the description of the FOU based on the primary membership function (MF). The provided piecewise-linear output depends only on two design parameters and the underlying primary function. A complete review about the preference for linear functions is also given as well as continuity considerations. In addition, it can be applied to an arbitrary monotonic primary MF around its mean. Close-form equations are derived for normal and Laplace distributions, as well as a regression approach to deal with sampled data where no assumptions are made about symmetry. According to the literature, such an approximation has not been published.