Neuro-Fuzzy Control Adriano Joaquim de Oliveira Cruz NCE/UFRJ [email protected].
Fuzzy Sets - Hedges. Adriano Joaquim de Oliveira Cruz – NCE e IM, UFRJ [email protected].
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Transcript of 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
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 2
SummarySummary Hedges
– Definition
– Characteristics
– Examples
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 3
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
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 4
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.
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 5
The hedge veryThe hedge very
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 6
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.
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 7
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
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 8
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.
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 9
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
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 10
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
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 11
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
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 12
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
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 13
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.
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 14
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
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 15
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.
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 16
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)
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 17
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
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 18
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
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 19
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.
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 20
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)
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 21
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
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 22
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
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 23
Intensifying and diluting contrastIntensifying and diluting contrast
0
1
Maximum fuzziness
Height1.70 1.901.80
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 24
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.
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 25
Intensifying - positivelyIntensifying - positively
5.0)())(1(21
5.0)()(2)(
2
2
xifx
xifxx
AA
AAApos
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 26
Intensifying - positivelyIntensifying - positively
80 100 120 140 160 180 200 2200
0.2
0.4
0.6
0.8
1
1.2
TallPositively Tall
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 27
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.
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 28
Diluting - generallyDiluting - generally
5.0)()5.0)((25.0
5.0)())(5.0(25.0)(
2
2
xifx
xifxx
AA
AAAgen
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 29
GenerallyGenerally
80 100 120 140 160 180 200 2200
0.2
0.4
0.6
0.8
1
1.2
TallGenerally Tall
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 30
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))
@2001 Adriano Cruz NCE e IM - UFRJ Fuzzy Sets Hedges 31
From A to B From A to B From A to B = GREQ(A) and SMEQ(B)