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Fuzzy Sets - Hedges. Adriano Joaquim de Oliveira Cruz – NCE e IM, UFRJ adriano@nce.ufrj.br

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 2 Summary Hedges –Definition –Characteristics –Examples

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 3 Hedges - Characteristics Hedges behave like adverbs and adjectives, they modify the meaning of nouns (very tall, near 35). Hedges change the shape of membership functions. Hedges are heuristic. The definition of the hedge functions are arbitrary

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 4 The hedge very Zadeh defined the hedge very as the square of the membership function. Very: very A (x)=[ A (x)] 2 Very intensifies the membership function. very A (x)<= A (x) Points representing absolute inclusion (1.0) or exclusion (0.0) do not change.

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 5 The hedge very

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 6 The hedge somewhat Zadeh defined the hedge somewhat as the square root of the membership function. Very: somewhat A (x)=[ A (x)] 1/2 Very dilutes the membership function. somewhat A (x)>= A (x) Points representing absolute inclusion (1.0) or exclusion (0.0) do not change.

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 7 The hedge somewhat

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 8 Hedges very - somewhat Very intensifies the membership function. Somewhat has the opposite effect. The powers (2, 1/2) are arbitrary choices The power 3 is sometimes used as the hedge extremely A number in the range 2 to 3 is used as the hedge slightly.

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 9 Applying hedges Hedges can be applied in different orders. Not very high = not (very high) very not high = very (not high) very not high <> not very high

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 10 The Commutability of hedges very alto (x) <= alto (x) not very alto (x) = 1 - [ alto (x)] 2 very not alto (x) = [1 - alto (x)] 2

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 11 The Commutability of hedges

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 12 Commutability of hedges Very and somewhat are the only hedges that are commutative. Somewhat very alto = very somewhat alto This is against the rules of language

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 13 Around and Close Around and close are hedges used to approximate scalars. If age is around 50. If age is around middle age. If age is close to 50. Is age is close to middle age.

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 14 Around e Close

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 15 Below Below should be applied to functions that increase in the universe of discourse. Below is not the same as not! If age is below around 35. if height is below medium.

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 16 Below Let A = A (x) Below A = not GREQ (A) GREQ(A)= A (x) for x < x *.= 1 for x >= x * x * = min(x | A (x) = 1) (leftmost value of X with membership = 1)

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 17 Greater or Equal

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 18 Below = Not Greater or Equal

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 19 Above Above should be applied to functions that decrease in the universe of discourse. If age is above around 35. if height is above short.

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 20 Above A = A (x) Above A = not SMEQ (A) SMEQ(A)= 1 for x < x * Above is not the same as not!.= A (x) for x >= x * x * = min(x | A (x) = 1) (leftmost value of X with membership = 1)

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 21 Smaller or Equal

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 22 Above

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 23 Intensifying and diluting contrast 0 1 Maximum fuzziness Height1.701.901.80

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 24 Intensifying - positively Positively increases the values of the membership function when (x)>=0.5 and diminishes all the values when (x)<0.5 It approximates the values to 0 and 1, therefore reducing the fuzziness.

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 25 Intensifying - positively

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 26 Intensifying - positively

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 27 Diluting - generally Generally diminishes the values of the membership function when (x) >= 0.5 and increases all the values when x)<0.5 It moves the values away from 0 and 1, therefore increasing the fuzziness.

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 28 Diluting - generally

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 29 Generally

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 30 In Between In between A and B = Norm(above A and below B) Norm( (x)) = (x) / max( (x)) Norm (not SMEQ(A) and not GREQ(B))

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@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 31 From A to B From A to B = GREQ(A) and SMEQ(B)

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Learning from Examples Adriano Cruz ©2004 NCE/UFRJ e IM/UFRJ.

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