fuzzy sets and fuzzification michael j. watts
TRANSCRIPT
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Fuzzy Sets and Fuzzification
Michael J. Watts
http://mike.watts.net.nz
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Lecture Outline
• Crisp sets• Fuzzy sets• Fuzzy membership functions• Fuzzification• Fuzzy logic
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Crisp Sets
• Everything is either true or false• No uncertainty is allowed• An item either is
o entirely within a set, oro entirely not in a set
• The Law of the Excluded Middleo X must be either in set A or in set not-Ao no middle ground is allowed
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Crisp Sets
• Opposite sets (A and not-A) must between them contain everything
• Venn diagrams
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Fuzzy Sets
• Items can belong to a fuzzy set to different degreeso degrees of membership
• Completely within a set is a membership degree of 1
• Completely outside a set is a membership degree of 0
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Fuzzy Sets
• Degrees of membership must sum to 1• An item can be both A and not-A to different
degreeso e.g. A to a degree of 0.8, not-A 0.2
• Degrees of membership are expressed with membership functions
• Range of values a variable can take is called the universe of discourse
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Membership Functions
• A membership function describes the degree of membership of a value in a fuzzy set
• Referred to as MFo Also
where x is the value being fuzzified
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Membership Functions
• There are many different types of MF• Which one to use depends on the problem
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Singleton MF
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Singleton MF
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Rectangular MF
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Rectangular MF
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Triangular MF
• A family of MF• Constantly tend towards zero and one• Three in the family
o Left-shoulderedo Triangularo Right-shouldered
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Triangular MF
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Triangular MF
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Gaussian MF
• A family of MF• Smoothly tend towards one and zero• Three in the family
o Zo Gausso S
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Gaussian MF
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Gaussian MF
• c is the centre of the MF• sigma is the width of the MF• exp is the exponential function
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Gaussian MF
• S function
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Gaussian MF
• L is the left hand ‘breakpoint’ of the MF• r is the right hand ‘breakpoint’ of the MF• c is the centre of the MF
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Gaussian MF
• Z function is symmetrical to S function
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Membership Functions
• MF can also be represented by a set of ordered pairs
• Pairs are crisp-fuzzy valueso A={(0,1.0),(1,1.0),(2,0.75),(3,0.5),(4,0.25),(5,0.0),
(6,0.0),(7,0.0),(8,0.0),(9,0.0),(10,0.0)}o B={(0,0.0),(1,0.2),(2,0.4),(3,0.6),(4,0.8),(5,1.0),
(6,0.8),(7,0.6),(8,0.4),(9,0.2),(10,0.0)}o C={(0,0.0),(1,0.0),(2,0.0),(3,0.0),(4,0.0),(5,0.0)
(6,0.25),(7,0.5),(8,0.75),(9,1.0),(10,1.0)}
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Membership Functions
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Fuzzification
• The process of determining the degree to which a value belongs in a fuzzy set
• The value returned by a fuzzy MF• Most variables in a fuzzy system have
multiple MF attached to them• Fuzzifying that variable involves passing the
crisp value through each MF attached to that value
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Fuzzy Logic
• Same operations and function as in crisp logic• Must deal with degrees of truth rather than
absolute truths• Fuzzy logic is a superset of crisp (Boolean)
logic
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Fuzzy Logic
• AND, OR, NOT• Crisp logical functions
o AND true is both parameters are trueo OR true if either parameter is trueo NOT reverses truth of argument
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Fuzzy Logic
• AND function - crisp version
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Fuzzy Logic
• AND function - fuzzy versiono take the minimum of the two arguments
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Fuzzy Logic
• OR function - crisp version
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Fuzzy Logic
• OR function - fuzzy versiono take the maximum of the two arguments
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Fuzzy Logic
• NOT function - crisp version
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Fuzzy Logic
• NOT function - fuzzy versiono subtract the truth value from one
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Fuzzy Logic
• Output of fuzzy logical functions are the same as crisp functionso just calculated differentlyo handle degrees of truth, rather than absolute
truths• The basis of fuzzy rule based systems
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Summary
• Fuzzy logic deals with uncertainty• Allows degrees of truth• Allows partial membership in sets• Fuzzy membership functions describe
degrees of membership in fuzzy sets• Many different types of MF exist
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Summary
• Fuzzification = determining degree of membershipo uses fuzzy MF to do so
• Fuzzy logic extends Boolean operators to handle partial truthso the basis of fuzzy rules