The original fuzzy logic (FL), founded by Lotfi Zadeh, has been around for more than 35 years, and yet it is unable to handle uncertainties. By "handle," I mean "to model and minimize the effect of." That the original FL, type-1 FL, cannot do this sounds paradoxical because the word fuzzy has the connotation of uncertainty. The expanded FL, type-2 FL, is able to handle uncertainties because it can model them and minimize their effects. And, if all uncertainties disappear, type-2 FL reduces to type-1 FL, in much the same way that, if randomness disappears, probability reduces to determinism.
Although many applications were found for type-1 FL, it is its application to rule-based systems that has most significantly demonstrated its importance as a powerful design methodology.
A rule-based fuzzy logic system (FLS) is shown in Figure 1. Its fuzzifier, inference mechanism (which is associated with rules, the heart of an FLS), and output processor involve operations on fuzzy sets that are characterized by membership functions. (For more information on the FLS in this figure, see my first article in this series, "Uncertainty in Fuzzy Logic Systems.") An FLS that is described completely in terms of type-1 fuzzy sets is called a type-1 FLS, whereas an FLS that is described using at least one type-2 fuzzy set is called a type-2 FLS. The output processor for a type-1 FLS is a defuzzifier; it transforms a type-1 fuzzy set into a number, a type-0 fuzzy set. The output processor for a type-2 FLS has two components to it. First, type-2 fuzzy sets are transformed into type-1 fuzzy sets by means of type reduction. Then the type-reduced set is transformed into a number by means of defuzzifcation.
Figure 1. Fuzzy logic system.
Type-1 FLSs cannot directly handle rule uncertainties because they use type-1 fuzzy sets that are certain. Type-2 FLSs, on the other hand, are very useful in circumstances in which it is difficult to determine an exact membership function for a fuzzy set; hence, they can be used to handle rule uncertainties and even measurement uncertainties.
Type-2 FLSs move the world of FLSs into a fundamentally new and important direction. What is this new direction, and why is it important? To make the answers to these questions as clear as possible, let us briefly digress to review some things that are, no doubt, familiar.
Probability theory is used to model random uncertainty, and within that theory we begin with a probability density function (pdf) that embodies total information about random uncertainties. In most practical real-world applications, it is impossible to know or determine the pdf, so we fall back on the fact that a pdf is completely characterized by all of its moments (if they exist). If the pdf is Gaussian, then, as is well known, two momentsthe mean and the variancesuffice to completely specify it. For most pdfs, an infinite number of moments are required. Of course, in practice it is not possible to determine an infinite number of moments; 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 moments, the mean and the variance. In some cases, we even use higher-than-second-order moments.
To use just the first-order 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. For example, that is why designs based on minimizing a mean-squared error are so popular.
Should we expect any less of an FLS for rule uncertainties or any other types of uncertainties? To date, we may view the output of a type-1 FLSthe defuzzified outputas analogous to the mean of a pdf. (I do not want to get stuck in the quagmire about the equivalence between subjective probability and type-1 fuzzy sets; our "analogy" between the defuzzified output of an FLS and the mean of a pdf is meant to be just that and nothing more.) Just as variance provides a measure of dispersion about the mean and is almost always used to capture more about probabilistic uncertainty in practical statistical designs, an FLS also needs some measure of dispersionthe new directionto capture more about its uncertainties than just a single number. Type-2 FL provides this measure of dispersion and seems to be as fundamental to the design of systems that include linguistic or numerical uncertainties that translate into rule or input uncertainties as variance is to the mean.
Just as random uncertainties flow through a system and their effects can be evaluated using the mean and the variance, linguistic and random uncertainties flow through a type-2 FLS, and their effects can be evaluated using the defuzzified output and the type-reduced output of that system. Just as the variance provides a measure of dispersion about the mean and is often used in confidence intervals, the type-reduced output can be interpreted as providing a measure of dispersion about the defuzzified output. It can be thought of as (or related to) a linguistic confidence interval. Just as the variance increases as random uncertainty increases, the type-reduced set also increases as linguistic or random uncertainties increase. So, a type-2 FLS is analogous to a probabilistic system through first and second moments, whereas a type-1 FLS is analogous to a probabilistic system only through the first moment.
A type-2 FLS has more design degrees of freedom than does a type-1 FLS because its type-2 fuzzy sets are described by more parameters than are type-1 fuzzy sets. This is analogous to a probability density function being described by more parameters (for example, a Gaussian probability density function is described by its mean and standard deviation) than its deterministic counterpart (for example, a degenerate Gaussian probability density function is one whose standard deviation is 0 and is characterized just by its mean). This suggests that a type-2 FLS has the potential to outperform a type-1 FLS because of its larger number of design degrees of freedom. To date, there is no mathematical proof that this will always be the case; however, in every application to which I have applied type-2 FLSs, I have always observed that better performance is obtained using a type-2 FLS than is obtained using a type-1 FLS.
In summary, we need type-2 FLSs to directly model uncertainties and minimize their effects, all within the framework of rule-based FLSs