Type-2 fuzzy sets and systems

Although some researchers are beginning to explore higher than type-2 fuzzy sets, as of early 2009, this work is in its infancy.

The membership function of a general type-2 fuzzy set, Ã, is three-dimensional (Fig.

1), where the third dimension is the value of the membership function at each point on its two-dimensional domain that is called its "footprint of uncertainty"(FOU).

Work on type-2 fuzzy sets languished during the 1980s and early-to-mid 1990s, although a small number of articles were published about them.

This changed in the latter part of the 1990s as a result of Jerry Mendel and his student's works on type-2 fuzzy sets and systems.

[3] Since then, more and more researchers around the world are writing articles about type-2 fuzzy sets and systems.

Rules, that are either provided by subject experts or are extracted from numerical data, are expressed as a collection of IF-THEN statements, e.g., Fuzzy sets are associated with the terms that appear in the antecedents (IF-part) or consequents (THEN-part) of rules, and with the inputs to and the outputs of the FLS.

In most engineering applications of an FLS, a number (and not a fuzzy set) is needed as its final output, e.g., the consequent of the rule given above is "Rotate the valve a bit to the right."

Consequently, the fired-rule output fuzzy sets have to be converted into a number, and this is done in the Fig.

In a type-1 FLS, output processing, called "defuzzification", maps a type-1 fuzzy set into a number.

There are many ways for doing this, e.g., compute the union of the fired-rule output fuzzy sets (the result is another type-1 fuzzy set) and then compute the center of gravity of the membership function for that set; compute a weighted average of the centers of gravity of each of the fired rule consequent membership functions; etc.

Just as standard deviation is widely used in probability and statistics to provide a measure of unpredictable uncertainty about a mean value, the type-reduced set can provide a measure of uncertainty about the crisp output of an interval type-2 FLS.

Mendel[26][27] has argued, on the basis of Karl Popper's concept of "falsificationism",[28][25] that using a type-1 fuzzy set as a model for a word is scientifically incorrect.

An interval type-2 fuzzy set should be used as a (first-order uncertainty) model for a word.

Python library for type 1 and type 2 fuzzy sets is available at: https://github.com/carmelgafa/type2fuzzy Python library for interval type 2 fuzzy sets and systems is available at: https://github.com/Haghrah/PyIT2FLS An open source Matlab/Simulink Toolbox for Interval Type-2 Fuzzy Logic Systems is available at: http://web.itu.edu.tr/kumbasart/type2fuzzy.htm There are two IEEE Expert Now multi-media modules that can be accessed from the IEEE at: [1]