Abstract
It is well accepted that in many real life situations information is not certain and precise but rather uncertain or imprecise. To describe uncertainty probability theory emerged in the 17th and 18th century. Bernoulli, Laplace and Pascal are considered to be the fathers of probability theory. Today probability can still be considered as the prevalent theory to describe uncertainty. However, in the year 1965 Zadeh seemed to have challenged probability theory by introducing fuzzy sets as a theory dealing with uncertainty (Zadeh, 1965). Since then it has been discussed whether probability and fuzzy set theory are complementary or rather competitive (Zadeh, 1995). Sometimes fuzzy sets theory is even considered as a subset of probability theory and therefore dispensable. Although the discussion on the relationship of probability and fuzziness seems to have lost the intensity of its early years it is still continuing today. However, fuzzy set theory has established itself as a central approach to tackle uncertainty. For a discussion on the relationship of probability and fuzziness the reader is referred to e.g. Dubois, Prade (1993), Ross et al. (2002) or Zadeh (1995). In the meantime further ideas how to deal with uncertainty have been suggested. For example, Pawlak introduced rough sets in the beginning of the eighties of the last century (Pawlak, 1982), a theory that has risen increasing attentions in the last years. For a comparison of probability, fuzzy sets and rough sets the reader is referred to Lin (2002). Presently research is conducted to develop a Generalized Theory of Uncertainty (GTU) as a framework for any kind of uncertainty whether it is based on probability, fuzziness besides others (Zadeh, 2005). Cornerstones in this theory are the concepts of information granularity (Zadeh, 1979) and generalized constraints (Zadeh, 1986). In this context the term Granular Computing was first suggested by Lin (1998a, 1998b), however it still lacks of a unique and well accepted definition. So, for example, Zadeh (2006a) colorfully calls granular computing “ballpark computing” or more precisely “a mode of computation in which the objects of computation are generalized constraints”.
TopIntroduction
It is well accepted that in many real life situations information is not certain and precise but rather uncertain or imprecise. To describe uncertainty probability theory emerged in the 17th and 18th century. Bernoulli, Laplace and Pascal are considered to be the fathers of probability theory. Today probability can still be considered as the prevalent theory to describe uncertainty.
However, in the year 1965 Zadeh seemed to have challenged probability theory by introducing fuzzy sets as a theory dealing with uncertainty (Zadeh, 1965). Since then it has been discussed whether probability and fuzzy set theory are complementary or rather competitive (Zadeh, 1995). Sometimes fuzzy sets theory is even considered as a subset of probability theory and therefore dispensable. Although the discussion on the relationship of probability and fuzziness seems to have lost the intensity of its early years it is still continuing today. However, fuzzy set theory has established itself as a central approach to tackle uncertainty. For a discussion on the relationship of probability and fuzziness the reader is referred to e.g. Dubois, Prade (1993), Ross et al. (2002) or Zadeh (1995).
In the meantime further ideas how to deal with uncertainty have been suggested. For example, Pawlak introduced rough sets in the beginning of the eighties of the last century (Pawlak, 1982), a theory that has risen increasing attentions in the last years. For a comparison of probability, fuzzy sets and rough sets the reader is referred to Lin (2002).
Presently research is conducted to develop a Generalized Theory of Uncertainty (GTU) as a framework for any kind of uncertainty whether it is based on probability, fuzziness besides others (Zadeh, 2005). Cornerstones in this theory are the concepts of information granularity (Zadeh, 1979) and generalized constraints (Zadeh, 1986).
In this context the term Granular Computing was first suggested by Lin (1998a, 1998b), however it still lacks of a unique and well accepted definition. So, for example, Zadeh (2006a) colorfully calls granular computing “ballpark computing” or more precisely “a mode of computation in which the objects of computation are generalized constraints”.
Key Terms in this Chapter
Linguistic Variable: A linguistic variable is a linguistic expression (one or more words) labeling an information granular. For example a membership function is labeled by the expressions like “hot temperature” or “rich customer”.
Granular Computing: The idea of granular computing goes back to Zadeh (1979). The basic idea of granular computing is that an object is describe by a bunch of values in possible dimensions like indistinguishability, similarity and proximity. If a granular is labeled by a linguistic expressing it is called a linguistic variable. Zahed (2006a) defines granular computing as “a mode of computation in which the objects of computation are generalized constraints”.
Rough Set Theory: Rough set theory was introduced by Pawlak in 1982. The central idea of rough sets is that some objects distinguishable while others are indiscernible from each other.
Fuzzy Set Theory: Fuzzy set theory was introduced by Zahed in 1965. The central idea of fuzzy set theory is that an object belongs to more than one sets simultaneously. the closeness of the object to a set is indicated by membership degrees.
Generalized Theory of Uncertainty (GTU): GTU is a framework that shall subsume any kind of uncertainty (Zadeh 2006a). The core idea is to formulate generalized constraints (like possibilistic, probabilistic, veristic etc.). The objective of GTU is not to replace existing theories like probability or fuzzy sets but to provide an umbrella that allows to formulate any kind of uncertainty in a unique way.
Soft Computing: In contrast to “hard computing” soft computing is collection of methods (fuzzy sets, rough sets neutral nets etc.) for dealing with ambiguous situations like imprecision, uncertainty, e.g. human expressions like “high profit at reasonable risks”. The objective of applying soft computing is to obtain robust solutions at reasonable costs.
Hybridization: Combination of methods like probabilistic, fuzzy, rough concepts, or neural nets, e.g. fuzzy-rough, rough-fuzzy or probabilistic-rough, or fuzzy-neural approaches.
Membership Function: A membership function shows the membership degrees of a variable to a certain set. For example, a temperature t=30° C belongs to the set “hot temperature“ with a membership degree ?HT(30°)=0.8. The membership functions are not objective but context and subject-dependent.