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.

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

Fuzzy set 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], [33])

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

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

Subsethood [21]

Embedded fuzzy sets ([32], [34], [35])

Fuzzy set ranking [25]

Fuzzy rule ranking and selection [31]

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.

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 argued, on the basis of Karl Popper's concept of Falsificationism ([19], [30]), 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. Much research is under way about CWW.

Applications

Type-2 fuzzy sets were applied in image processing, video processing and computer vision.