fuzzy set

• Fuzzy Logic


  A way of mapping an input space to an output space using fuzzy logic
• Fuzzy Sets


  • Height
  • Boundary Region
  • Crisp Sets
• Relations


  Connections between elements
• Composition of Relations


  Combining multiple relations
• Linguistic Hedges


  Adjectives (nouns) or adverbs (verbs) like very, low, slight, more or less, fairly, slightly, almost, barely, mostly, roughly, approximately etc. that modify fundamental atomic terms
• Concentrations


  Linguistic hedges like very, very very, plus that concentrate the elements of a fuzzy set by reducing the degree of membership of all elements that are only "partly" in the set
• Dilations


  Linguistic hedges like slightly, minus that stretch or dilate a fuzzy set by increasing the membership of elements that are "partly" in the set
• Intensification


  An operation on linguistic fuzzy sets that increases the degree of membership of those elements in the set with original membership values greater than 0.5, and decreases the degree of membership of those elements in the set with original membership values less than 0.5
• Fuzzy Inference System (FIS)


  A way of mapping an input space to an output space using fuzzy logic, formalizing the reasoning process of human language by means of fuzzy logic and building fuzzy IF-THEN rules
• Structure of FIS


  • Fuzzification module, Knowledge base, Inference engine, Defuzzification module
• Fuzzification


  Transforming crisp numbers into fuzzy sets by applying a fuzzification function
• Membership Function


  Defines a fuzzy set A on the universe of discourse X as a mapping from X to the interval [0,1], quantifying the grade of membership of each element in X to the fuzzy set A
• Membership Functions


  • Triangular function, Trapezoidal function, Gaussian function
• If-Then Rules


  Fuzzy if-then rules in the form "if x is A then y is B" where A and B are linguistic values defined by fuzzy sets
• Antecedent/Premise


  The "if x is A" part of a fuzzy if-then rule
• Consequent/Conclusion


  The "then y is B" part of a fuzzy if-then rule
• Example If-Then Rule


  • If service is good then tip is average
• Interpreting If-Then Rules


  1. Evaluate the antecedent (fuzzify input and apply fuzzy operators)

  2. Apply the result to the consequent (implication)
• Mamdani FIS


  The most commonly seen fuzzy inference method, proposed in 1975 by Ebrahim Mamdani as an attempt to control a steam engine and boiler combination
• Tipping Problem


  • Non-fuzzy approach, Fuzzy approach
• Overview of FIS


  Fuzzification

  2. Apply fuzzy operator

  3. Apply implication method

  4. Aggregate all outputs

  5. Defuzzify
• Defuzzification to Scalars


  Max membership principle, Weighted average method, Centroid method, Mean max membership
• The weighted average method is the most frequently used in fuzzy applications since it is one of the more computationally efficient methods
• The centroid method returns the center of area under the curve
• The mean max membership method is similar to max membership but it is used when max membership is not unique