Fuzzy rules (cont.) Most tools for working with fuzzy expert systems allow more than one conclusion per rule. A typical fuzzy expert system has more than one rule. Instead of assigning a single value to the output variable z, each rule assigns an entire fuzzy subset
Fuzzy rules (cont.) If a variable is set to a value by crisp rules, its value will change in steps as different rules fire. The only way to smooth those steps would be to have a large number of rules. However, only a small number of fuzzy rules is required to produce smooth changes in the outputs as the input values alter. The number of fuzzy rules required is dependent on the number of variables, the number of fuzzy sets, and the ways in which the variables are combined in the fuzzy rule conditions. The initial possibility values are assumed to be zero if these are the first rules to fire If several rules affect the same fuzzy set of the same variable, they are equivalent to a single rule whose conditions are joined by the disjunction OR
The Inference Process With the definition of the rules and membership functions in hand, we now need to know how to apply this knowledge to specific values of the input variables to compute the values of the output variables. This process is referred to as Inference Process In a fuzzy expert system, the inference process is a combination of four subprocesses: fuzzification, inference, composition, and defuzzification. The defuzzification subprocess is optional.
Fuzzification In the fuzzification subprocess, the membership functions defined on the input variables are applied to their actual values, to determine the degree of truth for each rule premise. The degree of truth for a rule's premise is sometimes referred to as its alpha. If a rule's premise has a nonzero degree of truth (if the rule applies at all…) then the rule is said to fire.
Inference In the inference subprocess, the truth value for the premise of each rule is computed, and applied to the conclusion part of each rule. This results in one fuzzy subset to be assigned to each output variable for each rule. Exists two inference methods: MIN and PRODUCT In MIN inferencing, the output membership function is clipped off at a height corresponding to the rule premise's computed degree of truth. This corresponds to the traditional interpretation of the fuzzy logic AND operation. In PRODUCT inferencing, the output membership function is scaled by the rule premise's computed degree of truth.
Composition In the composition subprocess, all of the fuzzy subsets assigned to each output variable are combined together to form a single fuzzy subset for each output variable. Exists two composition methods: MAX composition and SUM composition. In MAX composition, the combined output fuzzy subset is constructed by taking the pointwise maximum over all of the fuzzy subsets assigned to the output variable by the inference rule. In SUM composition the combined output fuzzy subset is constructed by taking the pointwise sum over all of the fuzzy subsets assigned to the output variable by the inference rule.
Composition (cont.) Note that this can result in truth values greater than one! For this reason, SUM composition is only used when it will be followed by a defuzzification method, such as the CENTROID method, that doesn't have a problem with this odd case.
Defuzzification Sometimes it is useful to just examine the fuzzy subsets that are the result of the composition process, but more often, this fuzzy value needs to be converted to a single number (a crisp value). This is what the defuzzification subprocess does. Defuzzification takes place in two stages: scaling the membership functions: adjust the fuzzy sets in accordance with the calculated possibilities: Larsen’s product operation rule: the membership functions are multiplied by their respective possibility values. The effect is to compress the fuzzy sets so that the peaks equal the calculated possibility values. An alternative approach in which the fuzzy sets are truncated
Defuzzification (cont.) finding the centroid. The most commonly used method of defuzzification is the centroid method, sometimes called the center of gravity, center of mass, or center of area method. If there are N membership functions with centroids ci and areas ai then the combined centroid C, i.e., the defuzzified value, is:
*Esta fórmula se emplea en el método de truncamiento
Defuzzification When the fuzzy sets are compressed using Larsen’s product operation rule, the values of ci are unchanged from the centroids of the uncompressed shapes, Ci, and ai is simply mi Ai where Ai is the area of the membership function prior to compression.
donde ai = mi * Ai
Deffuzzificación en los extremos En el caso de los extremos, se puede optar por 2 alternativas: Considerando el centroide del conjunto difuso involucrado o por la regla del espejo. Por la regla del espejo y el producto de Larsen, la obtención del centroide se simplifica a
Glossary Def 2.1 Intersection of Sets We call a new set generated from two given sets A and B intersection of A and B, if the new set contains exactly those elements that are contained in A and in B. Def 2.2 Unification of Sets We call a new set generated from two given sets A and B unification of A and B, if the new set contains all elements that are contained in A or in B or in both. Def 2.3 Negation of Sets We call a new set containing all elements which are in the universe of discourse but not in the set A the negation of A. Def 3.1 Linguistic Variable A linguistic variable is a quintuple (X,T(X),U,G,M,), where X is the name of the variable, T(X) is the term set, i.e. the set of names of linguistic values of X, U is the universe of discourse, G is the grammer to generate the names and M is a set of semantic rules for associating each X with its meaning.
References Fuzzy logic and fuzzy control http://www.flll.uni-linz.ac.at/aboutus/fuzzy A brief course in Fuzzy Logic and Fuzzy Control http://www.esru.strath.ac.uk/Reference/concepts/fuzzy/fuzzy.htm Hopgood, Adrian. Intelligent Systems for Engineers and Scientists. What is fuzzy logic? http://www.cs.cmu.edu/Groups/AI/html/faqs/ai/fuzzy/part1/faq.html
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