Fuzzy classification: Difference between revisions

Content deleted Content added

Inline

Latest revision as of 12:08, 12 June 2024

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 ${\textstyle \{\cdot |\cdot \}}$ . A class

$C=\{i|\Pi (i)\}$

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:

$\{\cdot |\cdot \}:V\times PF\rightarrow P(U)$

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 ${\textstyle V:\rightarrow R}$ 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".^[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.

$\mu :PF\times U\rightarrow T$

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.

$\mu C(i):=\tau (\Pi (i))$

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

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

[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.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

@@ Line 1: / Line 1: @@
+'''Fuzzy classification''' is the process of grouping elements into [[fuzzy set]]s<ref>Zadeh, L. A. (1965). Fuzzy sets. Information and Control (8), pp. 338–353.</ref> whose [[membership function]]s are defined by the [[truth value]] of a fuzzy [[propositional function]].<ref>Zimmermann, H.-J. (2000). ''Practical Applications of Fuzzy Technologies''. Springer.</ref><ref>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.</ref><ref>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).</ref> A fuzzy propositional function is analogous to<ref name="Russel 1919">Russel, B. (1919). ''Introduction to Mathematical Philosophy''. London: George Allen & Unwin, Ltd., S. 155</ref> an [[Expression (mathematics)|expression]] containing one or more variables, such that when values are assigned to these variables, the expression becomes a fuzzy [[Proposition (logic)|proposition]].<ref name="Zadeh 1975">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.</ref>
-{{context}}
+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]] <math display="inline">\mu_{\tilde{C}} : \tilde{PF} \times U \to \tilde{T}</math> that indicates the degree to which an individual <math display="inline">i\in U</math> is a member of the fuzzy class <math display="inline">\tilde{C}</math>, given its fuzzy classification [[Predicate (mathematical logic)|predicate]] <math display="inline">\tilde{\Pi}_{\tilde{C}} \in \tilde{PF}</math>. Here, <math display="inline">\tilde{T}</math> is the set of fuzzy [[truth value]]s, i.e., the [[unit interval]] <math display="inline">[0,1]</math>. The fuzzy classification predicate <math display="inline">\tilde{\Pi} _{\tilde{C}}(i)</math> corresponds to the fuzzy restriction "<math display="inline">i</math> is a member of <math display="inline">\tilde{C}</math>".<ref name="Zadeh 1975" />
-== Classification ==
+==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 (Glubrecht, Oberschelp, & Todt, 1983) is a logical system which supports set construction using logical predicates with the class operator { .| .}. A ''class''
+A [[class logic]]<ref>Glubrecht, J.-M., Oberschelp, A., & Todt, G. (1983). Klassenlogik. Mannheim/Wien/Zürich: Wissenschaftsverlag.</ref> is a logical system which supports set construction using logical predicates with the class operator <math display="inline">\{\cdot|\cdot\}</math>. A ''class''
-C = { i | Π(i) }
+<math>C = \{ i | \Pi(i) \}</math>
 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:
+<math>\{\cdot|\cdot\} :V\times PF \rightarrow P(U)</math>
+Here is an explanation of the logical elements that constitute this definition:
-{ .| .} ∶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 variable <math display="inline">V: \rightarrow R</math> 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” (Russel, 1919, S. 155).
+* 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".<ref name="Russel 1919" />
 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
+<math>\mu :PF \times U \rightarrow T </math>
 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.
+<math> \mu C(i):= \tau (\Pi(i)) </math>
+In classical logic the truth values are certain. Therefore a classification is crisp, since the truth values are either exactly true or exactly false.
-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.
-== Fuzzy Classification ==
-Fuzzy classification is the process of grouping elements into a fuzzy set (Zadeh 1965) whose membership function is defined by the truth value of a fuzzy propositional function. It has been discussed for example by (Zimmermann H.-J. , 2000),  (Meier, Schindler, & Werro, 2008) or (Del Amo, Montero, & Cutello, 1999).
-A fuzzy class ~C = { i | ~Π(i) } is defined as a fuzzy set ~C of individuals i satisfying a fuzzy classification predicate ~Π which is a fuzzy propositional function. The domain of the fuzzy class operator ~{ .| .}  is the set of variables V and the set of fuzzy propositional functions ~PF, and the range is the fuzzy powerset (the set of fuzzy subsets) of this universe, ~P(U):
-~{ .| .}∶V × ~PF ⟶ ~P(U)
-A fuzzy propositional function is, analogous to (Russel, 1919, S. 155), an expression containing one or more variables, such that, when values are assigned to these variables, the expression becomes a fuzzy proposition in the sense of (Zadeh 1975).
-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 that indicates whether an individual is a member of a class, given its fuzzy classification predicate ~Π.
-μ∶~PF × U ⟶ ~T
-Here, ~T is the set of fuzzy truth values (the interval between zero an one). The fuzzy classification predicate ~Π corresponds to a fuzzy restriction "i is R" (Zadeh, Calculus of fuzzy restrictions, 1975) of U, where R is a fuzzy set defined by a truth function. The degree of membership of an individual i in the fuzzy class ~C is defined by the truth value of the corresponding fuzzy predicate.
-μ~C(i):= τ(~Π(i))
-== References ==
-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).
-Glubrecht, J.-M., Oberschelp, A., & Todt, G. (1983). Klassenlogik. Mannheim/Wien/Zürich: Wissenschaftsverlag.
-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.
-Russel, B. (1919). Introduction to Mathematical Philosophy. London: George Allen & Unwin, Ltd.
-Zadeh, L. A. (1965). Fuzzy sets. Information and Control (8), pp. 338–353.
+==See also==
-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.
+* [[Fuzzy logic]]
+==References==
-Zimmermann, H.-J. (2000). Practical Applications of Fuzzy Technologies. Springer.
+<references/>
+{{DEFAULTSORT:Fuzzy Classification}}
 [[Category:Fuzzy logic]]