Abstract
The “fuzzy dot” (or fuzzy relation) representation of fuzzy rules in fuzzy rule based systems, in case of classical fuzzy reasoning methods (e.g. the Zadeh-Mamdani- Larsen Compositional Rule of Inference (CRI) (Zadeh, 1973) (Mamdani, 1975) (Larsen, 1980) or the Takagi - Sugeno fuzzy inference (Sugeno, 1985) (Takagi & Sugeno, 1985)), are assuming the completeness of the fuzzy rule base. If there are some rules missing i.e. the rule base is “sparse”, observations may exist which hit no rule in the rule base and therefore no conclusion can be obtained. One way of handling the “fuzzy dot” knowledge representation in case of sparse fuzzy rule bases is the application of the Fuzzy Rule Interpolation (FRI) methods, where the derivable rules are deliberately missing. Since FRI methods can provide reasonable (interpolated) conclusions even if none of the existing rules fires under the current observation. From the beginning of 1990s numerous FRI methods have been proposed. The main goal of this article is to give a brief but comprehensive introduction to the existing FRI methods.
TopIntroduction
The “fuzzy dot” (or fuzzy relation) representation of fuzzy rules in fuzzy rule based systems, in case of classical fuzzy reasoning methods (e.g. the Zadeh-Mamdani-Larsen Compositional Rule of Inference (CRI) (Zadeh, 1973) (Mamdani, 1975) (Larsen, 1980) or the Takagi - Sugeno fuzzy inference (Sugeno, 1985) (Takagi & Sugeno, 1985)), are assuming the completeness of the fuzzy rule base. If there are some rules missing i.e. the rule base is “sparse”, observations may exist which hit no rule in the rule base and therefore no conclusion can be obtained. One way of handling the “fuzzy dot” knowledge representation in case of sparse fuzzy rule bases is the application of the Fuzzy Rule Interpolation (FRI) methods, where the derivable rules are deliberately missing. Since FRI methods can provide reasonable (interpolated) conclusions even if none of the existing rules fires under the current observation. From the beginning of 1990s numerous FRI methods have been proposed. The main goal of this article is to give a brief but comprehensive introduction to the existing FRI methods.
Key Terms in this Chapter
a-Cut of a Fuzzy Set: Is a crisp set, which holds the elements of a fuzzy set (on the same universe of discourse) whose membership grade is grater than, or equal to a. (In case of “strong” a -cut it must be grater than a.)
Sparse Fuzzy Rule Base: A fuzzy rule base is sparse, if an observation may exist, which hits no rule antecedent. (The rule base is not complete.)
Fuzzy Rule Interpolation: A way for fuzzy inference by interpolation of the existing fuzzy rules based on various distance and similarity measures of fuzzy sets. A suitable method for handling sparse fuzzy rule bases, since FRI methods can provide reasonable (interpolated/extrapolated) conclusions even if none of the existing rules fires under the current observation.
Convex and Normal Fuzzy (CNF) Set: A fuzzy set defined on a universe of discourse holds total ordering, which has a height (maximal membership value) equal to one (i.e. normal fuzzy set), and having membership grade of any elements between two arbitrary elements grater than, or equal to the smaller membership grade of the two arbitrary boundary elements (i.e. convex fuzzy set).
e-Covering Fuzzy Partition: The fuzzy partition (a set of linguistic terms (fuzzy sets)) e-covers the universe of discourse, if for all the elements in the universe of discourse a linguistic term exists, which have a membership value grater or equal to e.
Vague Environment (VE): The idea of a VE is based on the similarity (or in this case the indistinguishability) of the considered elements. In VE the fuzzy membership function is indicating level of similarity of x to a specific element a that is a representative or prototypical element of the fuzzy set, or, equivalently, as the degree to which is indistinguishable from (Klawonn, 1994). Therefore the a-cuts of the fuzzy set are the sets which contain the elements that are -indistinguishable from a. Two values in a VE are e-distinguishable if their distance is greater than e. The distances in a VE are weighted distances. The weighting factor or function is called scaling function (factor) (Klawonn, 1994). If the VE of a fuzzy partition (the scaling function or at least the approximate scaling function (Kovács, 1996), (Kovács & Kóczy, 1997b)) exists, the member sets of the fuzzy partition can be characterized by points in that VE.
“Fuzzy Dot” Representation of Fuzzy Rules: The most common understanding of the If-Then fuzzy rules. The fuzzy rules are represented as a fuzzy relation of the rule antecedent and the rule consequent linguistic terms. In case of the Zadeh - Mamdani - Larsen compositional rule of inference (Zadeh, 1973) (Mamdani, 1975) (Larsen, 1980) the fuzzy rule relations are calculated as the fuzzy cylindric closures (t-norm of the cylindric extensions) (Klir & Folger, 1988) of the antecedent and the rule consequent linguistic terms.
Fuzzy Compositional Rule of Inference (CRI): The most common fuzzy inference method. The fuzzy conclusion is calculated as the fuzzy composition (Klir & Folger, 1988) of the fuzzy observation and the fuzzy rule base relation (see “Fuzzy dot” representation of fuzzy rules). In case of the Zadeh - Mamdani - Larsen max-min compositional rule of inference (Zadeh, 1973) (Mamdani, 1975) (Larsen, 1980) the applied fuzzy composition is the max-min composition of fuzzy relations (“max” stands for the applied s-norm and “min” for the applied t-norm fuzzy operations).
Complete (or Dense) Fuzzy Rule Base: A fuzzy rule base is complete, or dense if all the input universes are e-covered by rule antecedents, where e>0. In case of Complete Fuzzy Rule Base, for all the possible multidimensional observations, a rule antecedent must exist, which has a nonzero activation degree. Note, that completeness of the fuzzy rule base is not equivalent with covering fuzzy partitions on each antecedent universe (required but not sufficient in multidimensional case). Usually the number of the rules of a complete rule base is O(MI), where M is the average number of the linguistic terms in the fuzzy partitions and I is the number of the input universe.