August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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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

August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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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

August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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A. Introduction (figure from Earl Cox)

August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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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

August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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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)

August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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Input (figure from Klir&Yuan)

August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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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 PQ resultxx

)( then ,)}(),(min{ 1 vtruecaco resultxxx

August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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Fuzzy Proposition – type 1 and 2 (from Earl Cox)

August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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Fuzzy Propositions – type 1 and 2 (from Earl Cox)

August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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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

lresultfinaresultxresultxx

August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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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

August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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Fuzzy Hedges (from Earl Cox)

August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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Fuzzy Hedges (from Earl Cox)

August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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Illustrations of Fuzzy Propositions – Composition/Evaluation (from Klir&Yuan)

August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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Illustrations of Fuzzy Propositions – Composition/Evaluation (Earl Cox)

August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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Illustrations of Fuzzy Propositions – Composition/Evaluation (from Earl Cox)

August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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Illustrations of Fuzzy Propositions Decomposition – Defuzzification/Action (from Earl Cox)

August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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Defuzzification (from Earl Cox)

August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 2003

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Defuzzification (from Earl Cox)