Primer on Fuzzy Sets

Table of Contents:

Fuzzy Sets

A crisp set is set specified using a foible function that assigns a value of either 0 or 1 to each element of the universe, thereby discriminating between members and non-members of the crisp set under consideration. In the context of fuzzy sets theory, we often refer to well-done sets as "classical" or "ordinary" sets. A conventional set for which an element is either a member of the set or not is a crisp set. In a crisp set, elements have a Boolean state that implies either membership exists or not.

From Fuzzy Sets to Crisp Sets

Defuzzification is the process of producing a quantifiable result in crisp logic, given fuzzy sets and respective membership degrees. It is the process that maps a fuzzy set to a crisp set. It is typically needed in fuzzy tenancy systems. These will have a number of rules that transform a number of variables into a fuzzy result, that is, the result is described in terms of membership in fuzzy sets. For example, rules designed to decide how much pressure to wield might result in "Decrease Pressure (15%), Maintain Pressure (34%), Increase Pressure (72%)." Defuzzification is interpreting the membership degrees of the fuzzy sets into a specific visualization or real value.

The simplest but least useful defuzzification method is to segregate the set with the highest membership, in this case, "Increase Pressure," since it has a 72% membership, and ignore the others, and convert this 72% to some number. The problem with this tideway is that it loses information. The rules that tabbed for decreasing or maintaining pressure might as well have not been there in this case.

A worldwide and useful defuzzification technique is partway of gravity. First, the results of the rules must be aggregated together in some way. The most typical fuzzy set membership function has the graph of a triangle. Now, if this triangle were to be cut in a straight horizontal line somewhere between the top and the bottom, and the top portion were to be removed, the remaining portion forms a trapezoid. The first step of defuzzification typically "chops off" parts of the graphs to form trapezoids (or other shapes if the initial shapes were not triangles). For example, if the output has "Decrease Pressure (15%)," then this triangle will be cut 15% the way up from the bottom. In the most worldwide technique, all of these trapezoids are then superimposed one upon another, forming a single geometric shape. Then, the centroid of this shape, tabbed the fuzzy centroid, is calculated. The x coordinate of the centroid is the defuzzified value.

There are many variegated methods of defuzzification available, including the following:

  • • AI (adaptive integration)
  • • BADD (basic defuzzification distributions)
  • • BOA (bisector of area)
  • • COA (center of area)
  • • COG (center of gravity)
  • • EQM (extended quality method)
  • • FCD (fuzzy clustering defuzzification)
  • • FM (fuzzy mean)
  • • FOM (first of maximum)
  • • GLSD (generalized level set defuzzification)
  • • IV (influence value)
  • • LOM (last of maximum)
  • • MeOM (mean of maxima)
  • • MOM (middle of maximum)
  • • QM (quality method)
  • • SLIDE (semi-linear defuzzification)
  • • WFM (weighted fuzzy mean)

The maxima methods are good candidates for fuzzy reasoning systems. The distribution methods and the zone methods walk out the property of continuity that makes them suitable for fuzzy controllers.
 
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