Description

Imprecision and Uncertainty

Fuzziness should not be confused with other forms of imprecision and uncertainty. There are several types of imprecision and uncertainty and fuzziness is just one aspect of it. Imprecision and uncertainty may be in the aspects of measurement, probability, or descriptions.

Imprecision in measurement is associated with a lack of precise knowledge. We sometimes have measurements that are inaccurate, inexact, or of low confidence.

Imprecision as a form of probability is associated with an uncertainty about the future occurrence of events or phenomena. It concerns the likelihood of non-deterministic events ( stochastic uncertainty). An example is a statement "It might rain tomorrow" which exhibits a degree of randomness.

Imprecision in description is the type of imprecision addressed by fuzzy logic. It is the ambiguity, vagueness, qualitativeness, or subjectivity in natural language (linguistic, lexical, or semantic uncertainty). It is the ambiguity found in the definition of a concept or the meaning of terms such as "tall building" or "low scores". It is also the ambiguity in human thinking, that is, perceptions and interpretations. Examples of statements that are fuzzy in nature are "Hemoglobin count is very low." and "Teddy is rather heavy compared to Ike."

The nature of fuzziness and randomness are therefore quite different. They are different aspects of imprecision and uncertainty. The former conveys subjective human thinking, feelings, or language, and the latter indicates an objective statistic in the natural sciences.

From the modeling point of view, fuzzy models and statistical models also possess philosophically different kinds of information: fuzzy memberships represent similarities of objects to imprecisely defined properties, while probabilities convey information about relative frequencies. Thus, fuzziness deals with deterministic plausability and not nondeterministic probability.

Linguistic Variables, Linguistic Values, Linguistic Terms

Just as numerical variables take numerical values, in fuzzy logic, linguistic variables take on linguistic values which are words (linguistic terms) with associated degrees of membership in the set. Thus, instead of a variable height assuming a numerical value of 1.75 meters, it is treated as a linguistic variable that may assume, for example, linguistic values of tall with a degree of membership of 0.92, "very short" with a degree of 0.06, or "very tall" with a degree of 0.7. This concept was introduced by Zadeh to provide a means of approximate characterization of phenomena that are too complex or too ill-defined to be amenable to description in conventional quantitative terms.

Linguistic variables take on values defined in its term set - its set of linguistic terms. Linguistic terms are subjective categories for the linguistic variable. For example, for linguistic variable age, the term set T(age) may be defined as follows:

T(age) = { "young", "not young", "not so young", "very young", ..., "middle aged", "not middle aged", ..., "old", "not old", "very old", "more or less old", "quite old", ..., "not very young and not very old", ... }

Each linguistic term is associated with a fuzzy set, each of which has a defined membership function (MF). Formally, a fuzzy set A in U is expressed as a set of ordered pairs

A = { (x, m_A(x)) | x in U }

It should be reiterated that membership functions are subjective measures for linguistic terms and are not probability functions.

There are many types of membership functions. Some of the more common ones are triangular MFs (such as the functions in the figure above), trapezoidal MFs, gaussian MFs, and generalized bell MFs. The figure below shows the definitions and graphs of these MFs.