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Introduction
This article is about type-2 fuzzy sets and systems. Such sets and systems stand on the shoulders of (type-1) fuzzy sets and systems and provide for expanded and richer fuzzy sets and systems.

A fuzzy set Fuzzy_sets is comprised of elements that reside in it to a degree of belonging that can be any number from 0 to 1. It is this property that distinguishes it from a crisp set, for which a member is either in the set or is not.

Example: When a car is evaluated as being domestic or foreign by the manufacturer’s headquarter location, then it is either domestic or foreign, and there is nothing fuzzy about this situation. On the other hand, when a car is evaluated as being domestic or foreign by the percentage of its parts made in the US or abroad, then it may simultaneously belong in the sets domestic car and foreign car but to degrees equal to the percentage of its parts made in the US and abroad. ■

The elements of the fuzzy set, along with their degrees of belonging, are paired into a membership function of the fuzzy set. This function, whose values are single numbers from 0 to 1, that are all given the same (equal) weighting (e.g., 1, to indicate absolute certainty about the membership function value), completely characterizes the fuzzy set. In the most common kind of fuzzy set—now called a type-1 fuzzy set—one is absolutely certain about the degree of membership (belonging) of the elements in the fuzzy set, so there is therefore nothing very “fuzzy” about such a set.

From the very beginning of fuzzy sets, criticism was made about the fact that the membership function of a type-1 fuzzy set has no uncertainty associated with it, something that seems to contradict the word fuzzy, since that word has the connotation of lots of uncertainty. So, what does one do when there is uncertainty about the value of the membership function?

The answer to this question was already provided in 1976 by the inventor of fuzzy sets, Prof. Lotfi A. Zadeh [27], when he proposed more sophisticated kinds of fuzzy sets, the first of which he called a type-2 fuzzy set. A type-2 fuzzy set lets us incorporate uncertainty about the membership function into fuzzy set theory, and is a way to address the above criticism of type-1 fuzzy sets head-on. And, if there is no uncertainty, then a type-2 fuzzy set reduces to a type-1 fuzzy set, which is analogous to probability reducing to determinism when unpredictability vanishes.

General Type-2 Fuzzy Sets
In order to symbolically distinguish between a type-1 fuzzy set and a type-2 fuzzy set, a tilde symbol is put over the symbol for the fuzzy set; so, A denotes a type-1 fuzzy set, whereas Ã denotes the comparable type-2 fuzzy set. In the following discussions it is assumed that the reader is already familiar with type-1 fuzzy sets and systems.

The distinguishing feature of Ã versus A is the membership function values of Ã are blurred, i.e. they are no longer a single number from 0 to 1, but are instead a continuous range of values between 0 and 1, say [a, b] (some people call this a blurring of the membership function value). Now things get even more interesting, because one can either assign the same weighting or a variable weighting to the interval of membership function values [a, b]. When the former is done, the resulting type-2 fuzzy set is called either an interval type-2 fuzzy set or an interval valued fuzzy set (although different names may be used, they are the same fuzzy set). When the latter is done, the resulting type-2 fuzzy set is called a general type 2 fuzzy set (to distinguish it from the special interval type-2 fuzzy set).

Prof. Zadeh didn’t stop with type-2 fuzzy sets, because in that 1976 paper [27] he also generalized all of this to type-n fuzzy sets. The present article focuses only on type-2 fuzzy sets because they are the next step in the logical progression from type-1 to type-n fuzzy sets, where n = 1, 2, …. 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).

Figure 1. The membership function of a general type-2 fuzzy set is three-dimensional. A cross-section of one slice of the third dimension is shown. This cross-section, as well as all others, sits on the FOU. Only the boundary of the cross-section is used to describe the membership function of a general type-2 fuzzy set. It is shown filled-in for artistic purposes.

For an interval type-2 fuzzy set that third-dimension value is the same (e.g., 1) everywhere, which means that no new information is contained in the third dimension of an interval type-2 fuzzy set. So, for such a set, the third dimension is ignored, and only the FOU is used to describe it. It is for this reason that an interval type-2 fuzzy set is sometimes called a first-order uncertainty fuzzy set model, whereas a general type-2 fuzzy set (with its useful third-dimension) is sometimes referred to as a second-order uncertainty fuzzy set model.

The FOU represents the blurring of a type-1 membership function, and is completely described by its two bounding functions (Fig. 2), a lower membership function (LMF) and an upper membership function (UMF), both of which are type-1 fuzzy sets! Consequently, it is possible to use type-1 fuzzy set mathematics to characterize and work with interval type-2 fuzzy sets. This means that engineers and scientists who already know type-1 fuzzy sets will not have to invest a lot of time learning about general type-2 fuzzy set mathematics in order to understand and use interval type-2 fuzzy sets.

Figure 2. FOU for an interval type-2 fuzzy set. Many other shapes are possible for the FOU.

Work on type-2 fuzzy sets languished during the 1980’s and early-to-mid 1990’s, although a small number of articles were published about them. People were still trying to figure out what to do with type-1 fuzzy sets, so even though Zadeh proposed type-2 fuzzy sets in 1976, the time was not right for researchers to drop what they were doing with type-1 fuzzy sets to focus on type-2 fuzzy sets. This changed in the latter part of the 1990’s as a result of Prof. Jerry Mendel and his student’s works on type-2 fuzzy sets and systems (e.g., [12]). Since then, more and more researchers around the world are writing articles about type-2 fuzzy sets and systems.

Interval Type-2 Fuzzy Sets
Interval type-2 fuzzy sets have received the most attention because the mathematics that is needed for such sets—primarily interval arithmetic—is much simpler than the mathematics that is needed for general type-2 fuzzy sets. So, the literature about interval type-2 fuzzy sets is large, whereas the literature about general type-2 fuzzy sets is much smaller. Both kinds of fuzzy sets are being actively researched by an ever-growing number of researchers around the world.

Formulas for the following have already been worked out for interval type-2 fuzzy sets:


 * Set theoretic operations: union, intersection and complement ([6], [12])


 * Centroid (a very widely-used operation by practitioners of such sets, and also an important uncertainty measure for them) ([7], [12])


 * Other uncertainty measures [fuzziness, cardinality, variance and skewness [22] and uncertainty bounds [26]


 * Similarity ([1], [24], [25])


 * Subsethood [21]


 * Ranking [25]


 * Type-reduction methods ([7], [12])


 * Firing intervals for an interval type-2 fuzzy logic system ([3], [8], [12])


 * Fuzzy weighted average [9]


 * Linguistic weighted average [23]


 * Synthesizing an FOU from data that are collected from a group of subject [10]

Interval Type-2 Fuzzy Logic Systems
Type-2 fuzzy sets are finding very wide applicability in rule-based fuzzy logic systems (FLSs) because they let uncertainties be modeled by them whereas such uncertainties cannot be modeled by type-1 fuzzy sets. A block diagram of a type-2 FLS is depicted in Fig. 3. This kind of FLS is used in fuzzy logic control, fuzzy logic signal processing, rule-based classification, etc., and is sometimes referred to as a function approximation application of fuzzy sets, because the FLS is designed to minimize an error function.

Figure 3. Type-2 FLS.

The following discussions, about the four components in the Fig. 3 rule-based FLS, are given for an interval type-2 FLS, because to-date they are the most popular kind of type-2 FLS; however, most of the discussions are also applicable for a general type-2 FLS.

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.,


 * IF temperature is moderate and pressure is high, then rotate the valve a bit to the right.

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. Membership functions are used to describe these fuzzy sets, and in a type-1 FLS they are all type-1 fuzzy sets, whereas in an interval type-2 FLS at least one membership function is an interval type-2 fuzzy set.

An interval type-2 FLS lets any one or all of the following kinds of uncertainties be quantified:


 * 1) Words that are used in antecedents and consequents of rules—because words can mean different things to different people.
 * 2) Uncertain consequents—because when rules are obtained from a group of experts, consequents will often be different for the same rule, i.e. the experts will not necessarily be in agreement.
 * 3) Membership function parameters—because when those parameters are optimized using uncertain (noisy) training data, the parameters become uncertain.
 * 4) Noisy measurements—because very often it is such measurements that activate the FLS.

In Fig. 3, measured (crisp) inputs are first transformed into fuzzy sets in the Fuzzifier block because it is fuzzy sets and not numbers that activate the rules which are described in terms of fuzzy sets and not numbers. Three kinds of fuzzifiers are possible in an interval type-2 FLS. When measurements are:
 * Perfect, they are modeled as a crisp set;
 * Noisy, but the noise is stationary, they are modeled as a type-1 fuzzy set; and,
 * Noisy, but the noise is non-stationary, they are modeled as an interval type-2 fuzzy set (this latter kind of fuzzification cannot be done in a type-1 FLS).

In Fig. 3, after measurements are fuzzified, the resulting input fuzzy sets are mapped into fuzzy output sets by the Inference block. This is accomplished by first quantifying each rule using fuzzy set theory, and by then using the mathematics of fuzzy sets to establish the output of each rule, with the help of an inference mechanism. If there are M rules then the fuzzy input sets to the Inference block will activate only a subset of those rules, where the subset contains at least one rule and usually way fewer than M rules. Inference is done one rule at a time. So, at the output of the Inference block, there will be one or more fired-rule fuzzy output sets.

In most engineering applications of a 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.” No automatic valve will know what this means because “a bit to the right” is a linguistic expression, and a valve must be turned by numerical values, i.e. by a certain number of degrees. Consequently, the fired-rule output fuzzy sets have to be converted into a number, and this is done in the Fig. 3 Output Processing block.

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 center of gravities of each of the fired rule consequent membership functions; etc.

Things are somewhat more complicated for an interval type-2 FLS, because to go from an interval type-2 fuzzy set to a number (usually) requires two steps (Fig. 3). The first step, called type-reduction, is where an interval type-2 fuzzy set is reduced to an interval-valued type-1 fuzzy set. There are as many type-reduction methods as there are type-1 defuzzification methods. An algorithm developed by Karnik and Mendel ([7], [12]) now known as the KM Algorithm is used for type-reduction. Although this algorithm is iterative, it is very fast.

The second step of Output Processing, which occurs after type-reduction, is still called defuzzification. Because a type-reduced set of an interval type-2 fuzzy set is always a finite interval of numbers, the defuzzified value is just the average of the two end-points of this interval.

It is clear from Fig. 3 that there can be two outputs to an interval type-2 FLS—crisp numerical values and the type-reduced set. The latter provides a measure of the uncertainties that have flowed through the interval type-2 FLS, due to the (possibly) uncertain input measurements that have activated rules whose antecedents or consequents or both are uncertain. 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 provided a measure of uncertainty about the crisp output of an interval type-2 FLS.

Computing with words
Another application for fuzzy sets has also been inspired by Prof. Zadeh ([28]–[30])—Computing With Words. Different acronyms have been used for “computing with words,” e.g., CW and CWW. According to Zadeh:
 * CWW is a methodology in which the objects of computation are words and propositions drawn from a natural language. [It is] inspired by the remarkable human capability to perform a wide variety of physical and mental tasks without any measurements and any computations.

Of course, he did not mean that computers would actually compute using words—single words or phrases—rather than numbers. He meant that computers would be activated by words, which would be converted into a mathematical representation using fuzzy sets and that these fuzzy sets would be mapped by a CWW engine into some other fuzzy set after which the latter would be converted back into a word. A natural question to ask is: Which kind of fuzzy set—type-1 or type-2—should be used as a model for a word? Mendel ([13], 16]) has proved (using Karl Popper’s Falsificationism ([19], 30]) that to use 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. Much research is under way about CWW.

Conclusions
Works on type-2 fuzzy sets and systems has and continues to move in theoretical, computational and applied directions.

To conclude this article, some guidance is now provided to a reader who wants to learn more about such things and wants to start using type-2 fuzzy sets.

What to Read
 * For the reader who is new to interval type-2 fuzzy sets and systems and wants to learn more about them, without getting into lots of details, the easiest way to do this is to read Mendel’s 2007 magazine article [14].


 * For the reader who is new to interval type-2 fuzzy sets and systems and wants to learn more about them, with lots of details, but does not want to first learn about general type-2 fuzzy sets and systems, the easiest way to do this is to read the journal article by Mendel, John and Liu [18].


 * For the reader who is new to type-2 fuzzy sets and systems and wants to learn more about them, wants all of the details, wants a top-down presentation—from general type-2 to interval type-2—, and wants to see how they compare with type-1 fuzzy sets and systems, the easiest way to do this is to read Mendel’s 2001 textbook [12].


 * For the reader who wants to learn about a very powerful and useful representation for general type-2 fuzzy sets, in terms of simpler type-2 fuzzy sets that are called embedded type-2 fuzzy sets, read the paper by Mendel and John [17].


 * For the reader who may already be familiar with type-2 fuzzy sets and systems and who wants to know what has happened since the 2001 publication of Mendel’s book, see Mendel’s 2007 journal article [15] and also the 2008 book by Castillo and Melin [2].


 * The February 2007 issue of the IEEE Computational Intelligence Magazine is a special issue that is about type-2 fuzzy sets and systems. This issue contains articles (see the reference list below) about:
 * The history of type-2 fuzzy logic, by Bob John and Simon Coupland [5]
 * Type-2 fuzzy logic controllers, by Hani Hagras [4]
 * Fuzzy clustering using type-2 fuzzy sets, by Frank Rhee [20]
 * Hardware implementation for a type-2 fuzzy system, by Miguel Melgarejo [11]

Where else to get information
 * The type-2 website http://www.type2fuzzylogic.org is loaded with references and is being continually updated with new ones.


 * There are two IEEE Expert Now multi-media modules that can be accessed from the IEEE at: http://www.ieee.org/web/education/Expert_Now_IEEE/Catalog/AI.html
 * “Introduction to Type-2 Fuzzy Sets and Systems” by Jerry Mendel, sponsored by the IEEE Computational Intelligence Society
 * “Type-2 Fuzzy Logic Controllers: Towards a New Approach for Handling Uncertainties in Real World Environments” by Hani Hagras, sponsored by the IEEE Computational Intelligence Society

Software

Freeware Matlab M-file implementations, which cover general and interval type-2 fuzzy sets and systems, as well as type-1 fuzzy systems, are available at: http://sipi.usc.edu/~mendel/software.