Fuzzy Sets - Hedges. Adriano Joaquim de Oliveira Cruz – NCE e IM, UFRJ [email protected]

Fuzzy Sets - HedgesFuzzy Sets - Hedges

.

Adriano Joaquim de Oliveira Cruz – NCE e IM, UFRJ

[email protected]

@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 2

SummarySummary Hedges

– Definition

– Characteristics

– Examples


Hedges - CharacteristicsHedges - 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


The hedge veryThe 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.


The hedge veryThe hedge very


The hedge somewhatThe 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.


The hedge somewhatThe hedge somewhat

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Medium HeightSomewhat Medium Height


Hedges very - somewhatHedges 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.


Applying hedgesApplying 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


The Commutability of hedgesThe Commutability of hedges

very alto(x) <=alto(x)

not very alto(x) = 1 - [alto(x)]2

very not alto(x) = [1 - alto(x)]2


The Commutability of hedgesThe Commutability of hedges

160 165 170 175 180 185 190 195 2000

0.2

0.4

0.6

0.8

1

1.2

tallvery tallnot very tallvery not tall


Commutability of hedgesCommutability 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


Around and CloseAround 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.


Around e CloseAround e Close

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

scalar = 50around scalar = 50close scalar = 50


BelowBelow 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.


BelowBelow 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)


Greater or EqualGreater or Equal

100 110 120 130 140 150 160 170 180 190 2000

0.2

0.4

0.6

0.8

1

1.2

Medium HeightGreater or Equal Medium Height


Below = Not Greater or EqualBelow = Not Greater or Equal

100 110 120 130 140 150 160 170 180 190 2000

0.2

0.4

0.6

0.8

1

1.2

Medium HeightBelow Medium HeightGrEq Medium Height


AboveAbove Above should be applied to functions

that decrease in the universe of discourse.

If age is above around 35. if height is above short.


AboveAbove 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)


Smaller or EqualSmaller or Equal

100 110 120 130 140 150 160 170 180 190 2000

0.2

0.4

0.6

0.8

1

1.2

Medium HeightSmaller or Equal Medium Height


AboveAbove

100 110 120 130 140 150 160 170 180 190 2000

0.2

0.4

0.6

0.8

1

1.2

Medium HeightAbove Medium HeightSMEQ Medium Height


Intensifying and diluting contrastIntensifying and diluting contrast

0

1

Maximum fuzziness

Height1.70 1.901.80


Intensifying - positivelyIntensifying - 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.


Intensifying - positivelyIntensifying - positively

5.0)())(1(21

5.0)()(2)(

2

2

xifx

xifxx

AA

AAApos


Intensifying - positivelyIntensifying - positively

80 100 120 140 160 180 200 2200

0.2

0.4

0.6

0.8

1

1.2

TallPositively Tall


Diluting - generallyDiluting - 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.


Diluting - generallyDiluting - generally

5.0)()5.0)((25.0

5.0)())(5.0(25.0)(

2

2

xifx

xifxx

AA

AAAgen


GenerallyGenerally

80 100 120 140 160 180 200 2200

0.2

0.4

0.6

0.8

1

1.2

TallGenerally Tall


In Between 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))


From A to B From A to B From A to B = GREQ(A) and SMEQ(B)

Documents

Fuzzy Sets - Hedges. Adriano Joaquim de Oliveira Cruz – NCE e IM, UFRJ [email protected]