Fuzzy classification

Fuzzy classification is the process of grouping elements into fuzzy sets[1] whose membership functions are defined by the truth value of a fuzzy propositional function.[2][3][4] A fuzzy propositional function is analogous to[5] an expression containing one or more variables, such that when values are assigned to these variables, the expression becomes a fuzzy proposition.[6]

Accordingly, fuzzy classification is the process of grouping individuals having the same characteristics into a fuzzy set. A fuzzy classification corresponds to a membership function ${\textstyle \mu _{\tilde {C}}:{\tilde {PF}}\times U\to {\tilde {T}}}$ that indicates the degree to which an individual ${\textstyle i\in U}$ is a member of the fuzzy class ${\textstyle {\tilde {C}}}$ , given its fuzzy classification predicate ${\textstyle {\tilde {\Pi }}_{\tilde {C}}\in {\tilde {PF}}}$ . Here, ${\textstyle {\tilde {T}}}$ is the set of fuzzy truth values, i.e., the unit interval ${\textstyle [0,1]}$ . The fuzzy classification predicate ${\textstyle {\tilde {\Pi }}_{\tilde {C}}(i)}$ corresponds to the fuzzy restriction " ${\textstyle i}$ is a member of ${\textstyle {\tilde {C}}}$ ".[6]

Classification

Intuitively, a class is a set that is defined by a certain property, and all objects having that property are elements of that class. The process of classification evaluates for a given set of objects whether they fulfill the classification property, and consequentially are a member of the corresponding class. However, this intuitive concept has some logical subtleties that need clarification.

A class logic[7] is a logical system which supports set construction using logical predicates with the class operator { .| .}. A class

C = { i | Π(i) }

is defined as a set C of individuals i satisfying a classification predicate Π which is a propositional function. The domain of the class operator { .| .} is the set of variables V and the set of propositional functions PF, and the range is the powerset of this universe P(U) that is, the set of possible subsets:

{ .| .} :V×PF⟶P(U)

Here is an explanation of the logical elements that constitute this definition:

An individual is a real object of reference.
A universe of discourse is the set of all possible individuals considered.
A variable V:⟶R is a function which maps into a predefined range R without any given function arguments: a zero-place function.
A propositional function is "an expression containing one or more undetermined constituents, such that, when values are assigned to these constituents, the expression becomes a proposition".[5]

In contrast, classification is the process of grouping individuals having the same characteristics into a set. A classification corresponds to a membership function μ that indicates whether an individual is a member of a class, given its classification predicate Π.

μ:PF × U ⟶ T

The membership function maps from the set of propositional functions PF and the universe of discourse U into the set of truth values T. The membership μ of individual i in Class C is defined by the truth value τ of the classification predicate Π.

μC(i):=τ(Π(i))

In classical logic the truth values are certain. Therefore a classification is crisp, since the truth values are either exactly true or exactly false.

References

Zadeh, L. A. (1965). Fuzzy sets. Information and Control (8), pp. 338–353.
Zimmermann, H.-J. (2000). Practical Applications of Fuzzy Technologies. Springer.
Meier, A., Schindler, G., & Werro, N. (2008). Fuzzy classification on relational databases. In M. Galindo (Hrsg.), Handbook of research on fuzzy information processing in databases (Bd. II, S. 586-614). Information Science Reference.
Del Amo, A., Montero, J., & Cutello, V. (1999). On the principles of fuzzy classification. Proc. 18th North American Fuzzy Information Processing Society Annual Conf, (S. 675 – 679).
Russel, B. (1919). Introduction to Mathematical Philosophy. London: George Allen & Unwin, Ltd., S. 155
Zadeh, L. A. (1975). Calculus of fuzzy restrictions. In L. A. Zadeh, K.-S. Fu, K. Tanaka, & M. Shimura (Hrsg.), Fuzzy sets and Their Applications to Cognitive and Decision Processes. New York: Academic Press.
Glubrecht, J.-M., Oberschelp, A., & Todt, G. (1983). Klassenlogik. Mannheim/Wien/Zürich: Wissenschaftsverlag.

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[1] Zadeh, L. A. (1965). Fuzzy sets. Information and Control (8), pp. 338–353.

[2] Zimmermann, H.-J. (2000). Practical Applications of Fuzzy Technologies. Springer.

[3] Meier, A., Schindler, G., & Werro, N. (2008). Fuzzy classification on relational databases. In M. Galindo (Hrsg.), Handbook of research on fuzzy information processing in databases (Bd. II, S. 586-614). Information Science Reference.

[4] Del Amo, A., Montero, J., & Cutello, V. (1999). On the principles of fuzzy classification. Proc. 18th North American Fuzzy Information Processing Society Annual Conf, (S. 675 – 679).

[Russel_1919-5] Russel, B. (1919). Introduction to Mathematical Philosophy. London: George Allen & Unwin, Ltd., S. 155

[Zadeh_1975-6] Zadeh, L. A. (1975). Calculus of fuzzy restrictions. In L. A. Zadeh, K.-S. Fu, K. Tanaka, & M. Shimura (Hrsg.), Fuzzy sets and Their Applications to Cognitive and Decision Processes. New York: Academic Press.

[7] Glubrecht, J.-M., Oberschelp, A., & Todt, G. (1983). Klassenlogik. Mannheim/Wien/Zürich: Wissenschaftsverlag.

Fuzzy classification

Classification

See also

References