Chapter
1. Introduction
Fuzzy logic systems are, as is well known, comprised of rules.
Quite often, the knowledge that is used to construct these rules
is uncertain. Such uncertainty leads to rules whose antecedents
or consequents are uncertain, which translates into uncertain
antecedent or consequent membership functions. Type1 Fuzzy
Logic Systems (FLS), whose membership functions are type1
fuzzy sets, are unable to directly handle such rule uncertainties.
We introduce a new class of fuzzy logic systemstype2
fuzzy logic systemsin which the antecedent or consequent
membership functions are type2 fuzzy sets. Such sets are
fuzzy sets whose membership grades themselves are type1
fuzzy sets; they are very useful in circumstances where it is
difficult to determine an exact membership function for a fuzzy
set; hence, they are useful for incorporating uncertainties.
There is no prior work that we have been able to find on type2
FLSs; hence, this work moves the world of fuzzy logic in a fundamentally
new and important direction. What is this important direction
and why is it important? To make the answers as clear as possible,
let us briefly digress to review some things that are, no doubt,
familiar to the reader.
Probability theory is used to model random uncertainty, and
within that theory we begin with a probability density function
(pdf), which embodies total information about random uncertainties.
In most practical realworld applications it is impossible
to know or determine the pdf; so, we fall back on using the fact
that a pdf is completely characterized by all of its moments.
If the pdf is Gaussian, then, as is well known, two momentsthe
mean and variancesuffice to completely specify this pdf. For
most pdfs, an infinite number of moments are required. Of course,
it is not possible, in practice, to determine an infinite number
of moments; so, instead, we compute as many moments as we believe
are necessary to extract as much information as possible from
the data. At the very least, we use two momentsthe mean and
variance; and, in some case, we even use higherthan secondorder
moments.
To just use the firstorder moments would not be very
useful, because random uncertainty requires an understanding
of dispersion about the mean, and this information is provided
by the variance. So, our accepted probabilistic modeling of random
uncertainty focuses to a large extent on methods that use at
least the first two moments of a pdf. This is, for example,
why designs based on minimizing meansquared errors are so
popular.
Should we expect any less of a FLS for rule uncertainties?
Todate, we may view the output of a type1 FLS as analogous
to the mean of a pdf. We may view computing the defuzzified output
of a type1 FLS as analogous to computing the mean of a pdf.
Just as variance provides a measure of dispersion about the mean,
and is almost always used to capture more about probabilistic
uncertainty in practical statisticalbased designs, FLSs
also need some measure of dispersion to capture more about rule
uncertainties than just a single number. Type2 FL provides
this measure of dispersion, and seems to be as fundamental to
the design of systems that include linguistic and/ or numerical
uncertainties, that translate into rule uncertainties, as variance
is to the mean.
Let us now familiarize ourselves with the concept of a type2
fuzzy set. (More Information)
Chapter
5. Fuzzy Logic
Systems
The tenets of.fuzzy logic
do not change from type1 to type2 fuzzy sets, and,
in general, will not change for any typen. A highertype
number just indicates a higher degree of fuzziness. Since
a higher type changes the nature of the membership functions,
the operations that depend on the membership functions change;
however, the basic principles of fuzzy logic are independent
of the nature of the membership functions and hence do not change.
Rules of inference like Generalized Modus Ponens or Generalized
Modus Tollens continue to apply. (More
Information)
Chapter
6. Examples of Type2 Fuzzy Logic
Systems
In this chapter,.we describe
two examples of type2 FLSs that will give the reader an
idea of the circumstances under which a type2 FLS can be
used and the kind of output that one can expect from a type2
FLS. Applications for type2 FLSs are by no means limited
just to the situations described here and will continue to be
a topic of future research. (More
Information)
In Section 6.1 we consider the problem of designing a FLS
from rules collected by surveying multiple experts. We show how
linguistic uncertainty about membership functions of the FLS,
as well as rule uncertainty from the multiple experts, each of
whom may give different answers to the same question, can be
handled in a type2 framework. In Section 6.2 we demonstrate
how information associated with numerical uncertainty in the
training data for a type1 FLS can be interpreted in a type2
framework, so as to obtain bounds on the type1 FLS output.
We do this for the problem of forecasting the MackeyGlass
chaotic time series.
