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Fuzzy ES
What is fuzzy logic?
- Words (and the thoughts they describe) are often vague and imprecise: How hot is āhotā? How tall is ātallā?
- Even more concrete concepts are not necessarily black and white. What makes a chair a chair?
- Fuzzy logic attempts to capture the idea of degrees of membership in a class. Contrast with Boolean logic, in which membership is absolute.
- Class question: What makes a chair a chair?
www.moosemaple.com/ www.johnston.k12.ia.us
www.greenwoodworking.com
www.backstreet.demon.co.uk (^) www.petdiscounters.com
Which of these is the most āchair-likeā? What features make a chair a chair?
Classical (ācrispā) sets
If X is the universe of discourse (also called the finite reference super set) , with elements x, then the characteristic function of crisp set A is:
f (^) A ( x ): X ā 0 , 1 Where:
ļ£³
ļ£²
ļ£±
0
1 f (^) A ( x ) x A
x A
ā
ā
Fuzzy sets
Ī¼ A ( x ): X ā[ 0 , 1 ]
If X is the universe of discourse , with elements x, then the characteristic function of fuzzy set A is:
Where: If x is totally in A If x is totally not in A 0 ( ) 1 If x is partially in A
x
x
x
A
A
A
Ī¼
Ī¼
Ī¼
Defining fuzzy sets with fit vectors A can be defined as:
Where linear fit functions join the points given in the set definition. So, for example: Tall men = (0/180, 1/190) Short men=(1/160, 0/170) Average men=(0/165,1/175,0/185)
A ={Ī¼A ( x 1 )/ x 1 }...{ Ī¼ A ( xn )/ xn }
Graphical representation of set definition
short average tall
0
Degree of membership 1
Height
160165 170 180 185 190
What if key features donāt have a
scale?
- Determine list of binary features (either present or absent)
- Determine weight (importance) of each feature
- Sum over weights, and normalize
- This gives your membership function! Class exercise: develop a membership function for āchairā.
Hedges
Hedges are fuzzy set qualifiers, and modify the shape of the set. Can be:
- Truth values: e.g. quite true, mostly false
- Probabilities: e.g. likely, unlikely
- Quantifiers: e.g. most, several, few
- Possibilities: e.g. almost impossible, quite possible
Interpretations of hedges
Hedges are given mathematical interpretations, for example, the operation of concentration (i.e. making the set smaller):
2 4
3
2
( ) [ ( )] [ ( )]
( ) [ ( )]
( ) [ ( )]
x x x
x x
x x
A
very A
veryvery A
A
extremely A
A
very A
Ī¼ Ī¼ Ī¼
Ī¼ Ī¼
Ī¼ Ī¼
= =
=
=
Interpretation of hedges (2)
Also, operations of dilation (i.e. make set bigger):
( x ) (^) A ( x )
moreorless Ī¼ (^) A = Ī¼ And operations of intensification (exaggerates the extremes of the set):
2
1 2 [ 1 ( )]
2 [ ( )]
x
x x A
indeed A A Ī¼
Ī¼ Ī¼
- 5 ( ) 1
x
x A
A Ī¼
Ī¼
Fuzzy Operations (1)
The complement of a set is its opposite, given by:
Ī¼ (^) Ā¬ A ( x ) = 1 ā Ī¼ A ( x )
Monotonic selection
Allows estimation of the variable in the consequent. However, it only works for monotonic membership functions ā not common!
0
1
height weight
Tall men Heavy men
180 90
Antecedents/consequents with multiple parts
- Antecedents: Use unification (OR) or intersection (AND) operations to calculate a membership value for the whole antecedent.
- Consequents: Each consequent is affected equally by the membership in the antecedent class(es).
Fuzzy inference: Mamdani-style
(see Negnevitsky)
- Fuzzify the input variables
- Evaluate the rules
- Aggregate the rule outputs
- Defuzzify the output
Fuzzification
- Using the crisp inputs from the user, turn them into fuzzy memberships for all the relevant (i.e. right Universe of Discourse) classes.
Rule evaluation
- Apply the fuzzified inputs to all the relevant rules, using union and intersection operations to handle complex antecedents.
Aggregate the results
Build a membership function for each output UoD, by aggregating all the relevant classes.
Defuzzify
Use a center of mass formula to calculate the crisp output value (integration approximated as summation):
=
b x a A
b
x a
A
x
x x
COG
Ī¼
Ī¼
Class exercise
Write 3-5 fuzzy rules that determine heart attack risk, using:
- Three āuniverses of discourseā (UoD): diet, exercise, and risk
- 2 or 3 fuzzy classes per UoD, and their membership functions (represent graphically)
- Show fuzzy inference for one set of sample data
One solution: Rules
- diet is poor AND exercise is low risk is high
- diet is good AND exercise is high risk is low
- diet is good OR exercise is high risk is average Your rules and membership functions may vary.
Membership function: Diet
0
0.
0.
0.
0.
1
1000 2000 3000 4000 Calories ingested per day
Membership
Poor Good
Membership function: Exercise
0
0.
0.
0.
0.
1
1000 2000 3000 4000 Calories used per day
Membership
High Low
Membership function: Risk
0
0.
0.
0.
0.
1
0 25 75 100 Likelihood of heart disease
Membership
Low Medium High
Defuzzification
- Use the COG formula to determine the center the aggregate function.
ā
ā
=
= = b x a A
b
xa
A
x
xx COG ( )
( )
Ī¼
Ī¼
Defuzzification(2)
- To simplify the calculation, we will sample every 12.5 units (better to do more, or to integrate properly):
COG =^12.^5 *.^2 +(^25 +^37.^5 +. 250 +) 3 **..^44 ++^62. 5 .+^53 *.^5. 6 +(^75 +^87.^5 +^100 ).^6 = 70
Result
- So, the likelihood of heart disease for our patient is 70%.
- Please keep in mind that I made up these membership functions and rules without any medical knowledge ā they are just for the exercise!!
