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III. FUZZY LOGIC – Lecture 3
OBJECTIVES1. To define the basic notions of fuzzy logic2. To introduce the logical operations and relations
on fuzzy sets3. To learn how to obtain results of fuzzy logical
operations4. To apply what we learn to GIS
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OUTLINEIII. FUZZY LOGIC
A. Introduction B. Inputs to fuzzy logic systems - fuzzification C. Fuzzy propositions D. Fuzzy hedges E. Computing the results of a fuzzy proposition given an input F. The resulting action
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A. Introduction (figure from Earl Cox)
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Introduction
Steps (Earl Cox based on previous slide):1. Input – vocabulary, fuzzification (creating fuzzy
sets)2. Fuzzy propositions – IF X is Y THEN Z (or Z is A) …
there are four types of propositions3. Hedges – very, extremely, somewhat, more, less4. Combination and evaluation – computation of the
results given the inputs5. Action - defuzzification
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Input – vocabulary, fuzzification (creating a fuzzy set) by using our previous methods of frequency, combination, experts/surveys (figure from Earl Cox)
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Input (figure from Klir&Yuan)
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Fuzzy Propositions – types 1 and 2
GENERAL FORMS1. Unconditional and unqualified proposition: Q is PExample: Temperature(Q) is high(P) 2. Unconditional and qualified proposition: proposition(Q is P) is RExample: That Coimbra and Catania are beautiful is
very true.
).( then )( 1 PQresultxx
)( then ,)}(),(min{ 1 vtruecaco resultxxx
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Fuzzy Proposition – type 1 and 2 (from Earl Cox)
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Fuzzy Propositions – type 1 and 2 (from Earl Cox)
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Fuzzy Propositions – type 3
3. Conditional and unqualifiedproposition: IF Q is P THEN R is SExample: If Robert is tall, then clothes are large. If car is slow, then gear is low.
)( then ,)(
)( then ,)(1
1
SR
PQ
lresultfinaresultx
resultxx
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Fuzzy Propositions – type 4
4. Conditional and qualifiedproposition: IF Q is P THEN R is S is T {proposition(IF Q is P THEN R is S )} is T
)()( then ,)(
)( then ,)(
1
1
1
T
SR
PQ
lresultfinaositionresultpropresultx
resultxx
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Fuzzy Hedges (from Earl Cox)
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Fuzzy Hedges (from Earl Cox)
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Illustrations of Fuzzy Propositions – Composition/Evaluation (from Klir&Yuan)
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Illustrations of Fuzzy Propositions – Composition/Evaluation (Earl Cox)
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Illustrations of Fuzzy Propositions – Composition/Evaluation (from Earl Cox)
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Illustrations of Fuzzy Propositions Decomposition – Defuzzification/Action (from Earl Cox)
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Defuzzification (from Earl Cox)
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Defuzzification (from Earl Cox)