Explain four methods for deffuzzification in detail
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Defuzzification. Defuzzification is the process of producing a quantifiable result in Crisp logic, given fuzzy sets and corresponding membership degrees. ... Defuzzification is interpreting the membership degrees of the fuzzy sets into a specific decision or real
A brief explanation of each defuzzification strategy are shown as follows:
Centroid of area zCOA:
zCOA=∫zμA(z)zdz∫zμA(z)dzzCOA=∫zμA(z)zdz∫zμA(z)dz'
where μA(z) is the aggregationed output MF. This is the most widely adopted defuzzification strategy, which is reminiscent of the calculation of expected values of probability distributions.
Bisector of area zBOA: zBOA satisfies
∫zBOAαμA(z)dz=∫βzBOAμA(z)dz,∫αzBOAμA(z)dz=∫zBOAβμA(z)dz,
where α=min{z|z ∈ Z} and β=max{z|z ∈ Z}. That is, the vertical line z=zBOA partitions the region between z=α, z=β, y=0 and y=μA(z) into two regions with the same area.
Mean of maximum zMOM: zMOM is the average of the maximizing z at which the MF reach a maximum μ*. In symbols
zMOM=∫ZZ′dz∫Z′dzzMOM=∫Z′Zdz∫Z′dz
where Z' = {z | μA(z)=μ * }. In particular, if μA(z) has a single maximum at z=z *, then zMOM=z *. Moreover, if μA(z) reaches its maximum whenever z ∈ [zleft, zright] (This is the case in Figure 3), then zMOM = (zleft + zright)/2. The mean of maximum is the defuzzification strategy employed in Mamdani's fuzzy logic controllers.
Smallest of maximum zSOM: zSOM is the minimum (in terms of magnitude) of the maximizing z.
Largest of maximum zLOM: zLOM is the maximum (in terms of magnitude) of the maximizing z. Because of their obvious bias, zSOM and zLOM are not used as often as the other three defuzzification methods.
The calculation needed to carry out any of these five defuzzification operations is time-consuming unless special hardware support is available. Furthermore, these defuzzification operations are not easily subject to rigorous mathematical analysis, so most of the studies are based on experimental results. This leads to the propositions of other types of fuzzy inference systems that do not need defuzzification at all; two of them are introduced:
1.Sugeno Fuzzy Models (Also known as Takagi-Sugeno or TSK Fuzzy Model);
2.Tsukamoto Fuzzy Models ;
A brief explanation of each defuzzification strategy are shown as follows:
Centroid of area zCOA:
zCOA=∫zμA(z)zdz∫zμA(z)dzzCOA=∫zμA(z)zdz∫zμA(z)dz'
where μA(z) is the aggregationed output MF. This is the most widely adopted defuzzification strategy, which is reminiscent of the calculation of expected values of probability distributions.
Bisector of area zBOA: zBOA satisfies
∫zBOAαμA(z)dz=∫βzBOAμA(z)dz,∫αzBOAμA(z)dz=∫zBOAβμA(z)dz,
where α=min{z|z ∈ Z} and β=max{z|z ∈ Z}. That is, the vertical line z=zBOA partitions the region between z=α, z=β, y=0 and y=μA(z) into two regions with the same area.
Mean of maximum zMOM: zMOM is the average of the maximizing z at which the MF reach a maximum μ*. In symbols
zMOM=∫ZZ′dz∫Z′dzzMOM=∫Z′Zdz∫Z′dz
where Z' = {z | μA(z)=μ * }. In particular, if μA(z) has a single maximum at z=z *, then zMOM=z *. Moreover, if μA(z) reaches its maximum whenever z ∈ [zleft, zright] (This is the case in Figure 3), then zMOM = (zleft + zright)/2. The mean of maximum is the defuzzification strategy employed in Mamdani's fuzzy logic controllers.
Smallest of maximum zSOM: zSOM is the minimum (in terms of magnitude) of the maximizing z.
Largest of maximum zLOM: zLOM is the maximum (in terms of magnitude) of the maximizing z. Because of their obvious bias, zSOM and zLOM are not used as often as the other three defuzzification methods.
The calculation needed to carry out any of these five defuzzification operations is time-consuming unless special hardware support is available. Furthermore, these defuzzification operations are not easily subject to rigorous mathematical analysis, so most of the studies are based on experimental results. This leads to the propositions of other types of fuzzy inference systems that do not need defuzzification at all; two of them are introduced:
1.Sugeno Fuzzy Models (Also known as Takagi-Sugeno or TSK Fuzzy Model);
2.Tsukamoto Fuzzy Models ;
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Defuzzification is the process of producing a quantifiable result in Crisp logic, given fuzzy sets and corresponding membership degrees. ... A common and useful defuzzification technique is center of gravity. First, the results of the rules must be added together in some way.
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