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Fuzzy classification

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

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 μ C ~ : P F ~ × U → T ~ {\textstyle \mu _{\tilde {C}}:{\tilde {PF}}\times U\to {\tilde {T}}} that indicates the degree to which an individual i ∈ U {\textstyle i\in U} is a member of the fuzzy class C ~ {\textstyle {\tilde {C}}} , given its fuzzy classification predicate Π ~ C ~ ∈ P F ~ {\textstyle {\tilde {\Pi }}_{\tilde {C}}\in {\tilde {PF}}} . Here, T ~ {\textstyle {\tilde {T}}} is the set of fuzzy truth values, i.e., the unit interval [ 0 , 1 ] {\textstyle } . The fuzzy classification predicate Π ~ C ~ ( i ) {\textstyle {\tilde {\Pi }}_{\tilde {C}}(i)} corresponds to the fuzzy restriction " i {\textstyle i} is a member of C ~ {\textstyle {\tilde {C}}} ".

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 is a logical system which supports set construction using logical predicates with the class operator { ⋅ | ⋅ } {\textstyle \{\cdot |\cdot \}} . A class

C = { i | Π ( i ) } {\displaystyle C=\{i|\Pi (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 × P F → P ( U ) {\displaystyle \{\cdot |\cdot \}:V\times PF\rightarrow 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 {\textstyle V:\rightarrow 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".

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 : P F × U → T {\displaystyle \mu :PF\times U\rightarrow 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.

μ C ( i ) := τ ( Π ( i ) ) {\displaystyle \mu C(i):=\tau (\Pi (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.

See also

References

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