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Uncertainty in an FLS

The following sources of uncertainty can occur for the FLS in Figure 1:

  • Uncertainty about the meanings of the words that are used in the rules

  • Uncertainty about the consequent that is used in a rule

  • Uncertainty about the measurements that activate the FLS

  • Uncertainty about the data that is used to tune the parameters of an FLS

We have already seen that an FLS consists of rules and that rules use words. Our thesis is that words mean different things to different people, so there is uncertainty associated with words, which means that FL must somehow use this uncertainty when it computes with words.

Type-1 FL handles uncertainties about the meanings of words by using precise membership functions that the user believes capture the uncertainty of the words. Once the type-1 membership functions have been chosen, all uncertainty about the words disappears because type-1 membership functions are totally precise. Type-2 FL, on the other hand, handles uncertainties about the meanings of words by modeling the uncertainties. This is accomplished by blurring the boundaries of type-1 membership functions into what we call a footprint of uncertainty. Although a type-2 membership function will also be totally precise, it includes the footprint of uncertainty that provides new degrees of freedom that let uncertainties be handled by a type-2 FLS in totally new ways.

The uncertainty about the meanings of the words that are used in rules seems to me to be in accord with fuzziness, which results from imprecise boundaries of fuzzy sets.

Consequents for rules either are obtained from experts, by means of knowledge mining (engineering), or are extracted directly from data. Because experts don't all agree, a survey of experts will usually lead to a histogram of possibilities for the consequent of a rule. This histogram represents the uncertainty about the consequent of a rule, and this kind of uncertainty is different from that associated with the meanings of the words used in the rules. A histogram of consequent possibilities can be handled by a type-2 FLS.

Uncertainty about the consequent used in a rule, as established by a histogram of possibilities, seems to be in accord with strife, which expresses conflicts among the various sets of alternatives.

Measurements are usually corrupted by noise; hence, they are uncertain. I do not propose to abandon traditional ideas about noisy measurements (as in, measurement = signal + noise). What I do abandon is the frequently made assumption of existing knowledge of a probability model (that is, a probability density function) for either the signal or the noise. Doing this gets around the major shortcoming of a probability-based model—namely, the assumed probability model, for which results will be good if the data agrees with the model but may not be so good if the data does not. Uncertain measurements can be handled very naturally within the framework of an FLS.

Uncertain measurements (such as randomness in the data) can be modeled as fuzzy sets (type-1 or type-2); hence, uncertainty about the measurements that activate the FLS seems to be in accord with nonspecificity when nonspecificity is associated with information-based imprecision.

Finally, an FLS contains many design parameters whose values must be set by the designer before the FLS is operational. There are many ways to do this, and all make use of a set of data, usually called the training set. This set consists of input-output pairs for the FLS, and, if these pairs are measured signals, then they are as uncertain as the measurements that excite the FLS. In this case—one that is quite common in practice but has not received much attention in the FLS literature—the FLS must be tuned using unreliable data, which is yet another form of uncertainty that can be handled by a type-2 FLS.

The uncertainty about the data that is used to tune the parameters of an FLS also seems to be in accord with nonspecificity when nonspecificity is associated with information-based imprecision.

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