Defuzzification

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Defuzzification is the process of producing a quantifiable result in fuzzy logic. Typically, a fuzzy system 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 apply might result in "Decrease Pressure (15%), Maintain Pressure (34%), Increase Pressure (72%)". Defuzzification would transform this result into a single number indicating the change in pressure. The simplest but least useful defuzzification method is to choose 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 approach is that it loses information. The rules that called for decreasing or maintaining pressure might as well have not been there in this case.

A useful defuzzification technique must first add the results of the rules 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 common technique, all of these trapezoids are then superimposed one upon another, forming a single geometric shape. Then, the centroid of this shape, called the fuzzy centroid, is calculated. The x coordinate of the centroid is the defuzzified value.

[edit] Methods

There are many different methods of defuzzification available, including the following^[1]:

RCOM (random choice of maximum)
FOM (first of maximum)
LOM (last of maximum)
MOM (middle of maximum)
COG (center of gravity)
MeOM (mean of maxima)
BADD (basic defuzzification distributions)
GLSD (generalized level set defuzzification)
ICOG (indexed center of gravity)
SLIDE (semi-linear defuzzification)
FM (fuzzy mean)
WFM (weighted fuzzy mean)
QM (quality method)
EQM (extended quality method)
COA (center of area)
ECOA (extended center of area)
CDD (constraint decision defuzzification)
FCD (fuzzy clustering defuzzification)

The maxima methods are good candidates for fuzzy reasoning systems. The distribution methods and the area methods exhibit the property of continuity that makes them suitable for fuzzy controllers^[1] .

[edit] Notes

^ ^a ^b Defuzzification: criteria and classification, from the journal Fuzzy Sets and Systems, Van Leekwijck and Kerre, Vol. 108 (1999), pp. 159-178 [1]