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Frequently Asked Questions About Type-2 Fuzzy Logic and Fuzzy Sets

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Fuzzy logic systems expert Jerry Mendel answers some frequently asked questions about type-2 fuzzy sets.
From the author of Question: I thought that fuzzy logic is all about uncertainty. Now you tell us that it isn't. What's going on here?

Answer: That the original fuzzy logic (FL), type-1 FL, cannot handle (that is, model and minimize the effects of) uncertainties sounds paradoxical because the word fuzzy has the connotation of uncertainty. Type-1 FL handles uncertainties by using precise membership functions (MFs) that the user believes capture the uncertainties. When the type-1 MFs have been chosen, all uncertainty disappears because type-1 MFs are totally precise. Type-2 FL, on the other hand, handles uncertainties about the meanings of words by modeling the uncertainties using type-2 MFs.

Question: I have heard about FL but have never heard it referred to as type-1 FL. Why do we now need to refer to it as type-1 FL?

Answer: Before the work on type-2 FL, it was never necessary to refer to FL as type-1. Now, to distinguish between the two kinds of FL, we need to use "type-1" or "type-2."

Question: What exactly is type-2 FL?

Answer: First, let's recall that FL is all about IF-THEN rules (as in, IF the sky is blue and the temperature is between 60 and 75° Fahrenheit, THEN it is a lovely day). The IF and THEN parts of a rule are called its antecedent and consequent, and they are modeled as fuzzy sets. Rules are described by the MFs of these fuzzy sets. In type-1 FL, the antecedents and consequents are all described by the MFs of type-1 fuzzy sets. In type-2 FL, some or all of the antecedents and consequents are described by the MFs of type-2 fuzzy sets.

Question: Do the rules change as we go from type-1 FL to type-2 FL?

Answer: Good news, the rules do not change. Paraphrasing Gertrude Stein, "A rule is a rule is a rule…." What does change is the way in which we model a rule's antecedent and consequent fuzzy sets. In type-1 FL, they are all modeled as type-1 fuzzy sets, whereas in type-2 FL, some or all are modeled as type-2 fuzzy sets.

Question: What's the difference between a fuzzy set, a type-1 fuzzy set, and a type-2 fuzzy set?

Answer: The term "fuzzy set" is general and includes type-1 and type-2 fuzzy sets (and even higher-type fuzzy sets). All fuzzy sets are characterized by MFs. A type-1 fuzzy set is characterized by a two-dimensional MF, whereas a type-2 fuzzy set is characterized by a three-dimensional MF.

As an example, suppose that the variable of interest is eye contact, which we denote as x. Let's put eye contact on a scale of values 0–10. One of the terms that might characterize the amount of perceived eye contact (for example, during flirtation) is "some eye contact." Suppose that we surveyed 100 men and women and asked them to locate the ends of an interval for "some eye contact" on the scale of 0–10. Surely, we will not get the same results from all of them because words mean different things to different people.

One approach to using the 100 sets of two endpoints is to average the endpoint data and use the average values for the interval associated with "some eye contact." We could then construct a triangular (or other shape) MF whose base endpoints (on the x-axis) are at the two average values and whose apex is midway between the two endpoints. This type-1 triangle MF can be displayed in two dimensions and can be expressed mathematically as follows:

{(x, MF(x))| x an element of X}

Unfortunately, this MF has completely ignored the uncertainties associated with the two endpoints.

A second approach is to make use of the average values and the standard deviations for the two endpoints. By doing this, we are blurring the location of the two endpoints along the x-axis. Now locate triangles so that their base endpoints can be anywhere in the intervals along the x-axis associated with the blurred average endpoints. Doing this leads to a continuum of triangular MFs sitting on the x-axis—for example, picture a whole bunch of triangles all having the same apex point but different base points, as in Figure 1.

Figure 1 Triangular MFs when base endpoints (l and r) have uncertainty intervals associated with them.

For the purposes of this discussion, suppose that there are exactly N such triangles. Then at each value of x, there can be up to N MF values: MF1(x), MF2(x), …, MFN(x). Let's assign a weight to each of the possible MF values, say wx1, wx2, …, wxN (see Figure 1). We can think of these weights as the possibilities associated with each triangle at this value of x. The resulting type-2 MF can be expressed as follows:

{(x, {( MFi(x), wxi)| i = 1, …, N}| x an element of X}

Another way to write this is:

{(x, MF(x, w)| x an element of X and w an element of Jx}

MF(x, w) is a type-2 MF. It is three-dimensional because MF(x, w) depends on two variables, x and w.

Question: How are type-2 fuzzy sets visualized?

Answer: We just indicated that type-2 fuzzy sets are three-dimensional, so they can be visualized as three-dimensional plots. Unfortunately, it is not as easy to sketch such plots as it is to sketch the two-dimensional plots of a type-1 MF. Another way to visualize type-2 fuzzy sets is to plot their so-called footprint of uncertainty (FOU). The type-2 MF, MF(x, w), sits atop a two-dimensional x-w plane. It sits only on the permissible (sometimes called "admissible") values of x and w. This means that x is defined over a range of values (its domain)—say, X. In addition, w is defined over its range of values (its domain)—say, W.

An example of an FOU is shown in Figure 2. It is an FOU for a Gaussian MF whose standard deviation is known with perfect certainty but whose mean, m, is uncertain and varies anywhere in the interval from m1 to m2, which can be expressed as m_[m1, m2]. The uniform shading over the entire FOU means that, for this example, we are assuming uniform weighting (possibilities). Because of the uniform weighting, this type-2 fuzzy set is called an interval type-2 fuzzy set.

Figure 2 FOU for Gaussian (primary) membership function with uncertain mean.

Question: Is there new terminology for type-2 fuzzy sets?

Answer: Yes, there is. The fact that we must now distinguish between type-1 and type-2 MFs is one example of the new terminology. A lot of the new terminology is due to the three-dimensional nature of a type-2 MF. Another term that we have already explained is the FOU. Some other new terms are: primary membership, primary MF, secondary grade, secondary MF, secondary set, upper and lower MFs, principal MF, embedded type-1 fuzzy set, and embedded type-2 fuzzy set. All of these terms can be defined mathematically and let us communicate effectively about type-2 fuzzy sets.

Question: What can we do with type-2 fuzzy sets?

Answer: Whatever we can do with type-1 fuzzy sets we can also do with type-2 fuzzy sets. It is how we do it that is different. For type-1 fuzzy sets, we perform set theoretic operations, such as union, intersection, and complement. We can do the same for type-2 fuzzy sets. Procedures for how to do this have been worked out and are especially simple for interval type-2 fuzzy sets.

Question: Where are type-2 fuzzy sets being used?

Answer: They are being used in type-2 fuzzy logic systems, and they let us model uncertainties totally within the framework of fuzzy logic. See the fourth and fifth articles in this collection—titled "Frequently Asked Questions About Rule-Based Type-2 Fuzzy Logic Systems" and "Applications for Rule-Based Type-2 Fuzzy Logic Systems"—for a description about these FLSs.

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