A new class of fuzzy logic systems
(FLS)  type2 fuzzy logic systems  is introduced, one that
makes use of type2 fuzzy sets for representing linguistic and/or
numerical uncertainties.
What are type2 fuzzy sets? They are fuzzy sets having fuzzy
membership functions, i.e., the membership grade of each element
of such a set is an ordinary (type1) fuzzy set. Type2 sets
are useful in circumstances where it is difficult to define the
exact membership function for a fuzzy set, as in computing with
words, when words mean different things to different people.
The report includes the very basic operations of union, intersection
and complement of type2 sets, and develops results that are
needed to implement a type2 FLS, including: set theoretic and
algebraic operations for type2 sets, properties of membership
grades of type2 sets, and type2 relations and compositions.
A new operation called typereduction is introduced.
It is an extended version of type1 defuzzification. Typereduction
as well as defuzzification are examined in great detail. Results
are provided that greatly simplify the implementation of interval
and Gaussian type2 FLS's. Whenever actual results are difficult
to implement or generalize, practical approximations are provided.
The report demonstrates the use of a type2 FLS for two applications:
managing rules collected by means of a survey, and timeseries
prediction.
In the survey application, the report shows how linguistic
uncertainty about membership functions, as well as rule uncertainty
from multiple experts (each of whom may give different answers
to the same questions), can be handled in the type2 framework.
Type2 FLSs lets us combine expert opinions in a rational way.
In the timeseries application, the report shows how information
about noise in the training data (i.e., unreliable training data)
can be incorporated into a type2 FLS to obtain bounds on the
predictions as well as better predictions. The bounds are linguistic
confidence intervals.
Contents
1. Introduction
2. Operations on Type2 Sets
3. Properties of Membership Grades
4. Relations and Compositions
5. Fuzzy
Logic Systems
6. Examples
of Type2 Fuzzy Logic
Systems
7. Conclusions
... Four Appendixes with detailed
...derivations and proofs
... Index
