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Fuzzy Rules
1965 paper: “Fuzzy Sets” (Lotfi Zadeh)Apply natural language terms to a formal
system of mathematical logichttp://www.cs.berkeley.edu/~zadeh
1973 paper outlined a new approach to capturing human knowledge and designing expert systems using fuzzy rules
Fuzzy Rules
A fuzzy rule is a conditional statementin the familiar form:
IF x is ATHEN y is B
x and y are linguistic variablesA and B are linguistic values determined by
fuzzy sets on the universe of discourses X and Y, respectively
Linguistic Variables
A linguistic variable is a fuzzy variablee.g. the fact “John is tall” implies linguistic
variable “John” takes the linguistic value “tall”
Use linguistic variables to form fuzzy rules:IF ‘project duration’ is longTHEN ‘risk’ is high
IF risk is very highTHEN ‘project funding’ is very low
Fuzzy Expert Systems
A fuzzy expert system is an expert system thatuses fuzzy rules, fuzzy logic, and fuzzy sets
Many rules in a fuzzy logic system will fire to some extentIf the antecedent is true to some degree of
membership, then the consequent is true to the same degree
Fuzzy Expert Systems
Two distinct fuzzy sets describing tall and heavy:
Tall men Heavy men
180
Degree ofMembership1.0
0.0
0.2
0.4
0.6
0.8
Height, cm
190 200 70 80 100160
Weight, kg
120
Degree ofMembership1.0
0.0
0.2
0.4
0.6
0.8
Tall men Heavy men
180
Degree ofMembership1.0
0.0
0.2
0.4
0.6
0.8
Height, cm
190 200 70 80 100160
Weight, kg
120
Degree ofMembership1.0
0.0
0.2
0.4
0.6
0.8
Fuzzy Expert Systems
IF height is tallTHENweight is heavy
Tall menHeavy men
180
Degree ofMembership1.0
0.0
0.2
0.4
0.6
0.8
Height, cm
190 200 70 80 100160
Weight, kg
120
Degree ofMembership1.0
0.0
0.2
0.4
0.6
0.8
Fuzzy Expert Systems
Other examples (multiple antecedents):e.g. IF ‘project duration’ is long
AND ‘project staffing’ is largeAND ‘project funding’ is
inadequateTHEN risk is high
e.g. IF service is excellentOR food is deliciousTHEN tip is generous
Fuzzy Expert Systems
Other examples (multiple consequents):e.g. IF temperature is hot
THEN ‘hot water’ is reduced;‘cold water’ is increased
Fuzzy Inference
Named after Ebrahim Mamdani, the Mamdani method for fuzzy inference is:1. Fuzzify the input variables2. Evaluate the rules3. Aggregate the rule outputs4. Defuzzify
Fuzzy Inference – Example
Rule 1:IF x is A3OR y is B1THEN z is C1
Rule 2:IF x is A2AND y is B2THEN z is C2
Rule 3:IF x is A1THEN z is C3
Rule 1:IF ‘project funding’ is adequateOR ‘project staffing’ is smallTHEN risk is low
Rule 2:IF ‘project funding’ is marginalAND ‘project staffing’ is largeTHEN risk is normal
Rule 3:IF ‘project funding’ is inadequateTHEN risk is high
x, y, and z are linguistic variablesA1, A2, and A3 are linguistic values on XB1 and B2 are linguistic values on YC1, C2, and C3 are linguistic values on Z
Fuzzy Inference – Example
1. Fuzzification
CrispInput
0.1
0.71
0y1
B1 B2
Y
CrispInput
0.20.5
1
0
A1 A2 A3
x1
x1 X(x =A1) = 0.5(x =A2) = 0.2
(y =B1) = 0.1(y =B2) = 0.7
project funding project staffing
inadequate
marginal large
small
Fuzzy Inference – Example
2. Rule 1 evaluation
A31
0 X
1
y10 Y
0.0
x1 0
0.1C1
1
C2
Z
1
0 X
0.2
0
0.2 C11
C2
Z
A2
x1
Rule 3: IF x isA1 (0.5)
A11
0 X 0
1
Zx1
THEN
C1 C2
1
y1
B2
0 Y
0.7
B10.1
C3
C3
C30.5 0.5
OR(max)
AND(min)
OR THENRule 1: IF x isA3 (0.0)
AND THENRule 2: IF x isA2 (0.2)
y isB1 (0.1) z isC1 (0.1)
y isB2 (0.7) z isC2 (0.2)
z isC3 (0.5)
project fundingproject staffing
adequate small
risk
low
Fuzzy Inference – Example
2. Rule 2 evaluation
A31
0 X
1
y10 Y
0.0
x1 0
0.1C1
1
C2
Z
1
0 X
0.2
0
0.2 C11
C2
Z
A2
x1
Rule 3: IF x isA1 (0.5)
A11
0 X 0
1
Zx1
THEN
C1 C2
1
y1
B2
0 Y
0.7
B10.1
C3
C3
C30.5 0.5
OR(max)
AND(min)
OR THENRule 1: IF x isA3 (0.0)
AND THENRule 2: IF x isA2 (0.2)
y isB1 (0.1) z isC1 (0.1)
y isB2 (0.7) z isC2 (0.2)
z isC3 (0.5)
project funding project staffing
marginal large
risk
normal
Fuzzy Inference – Example
2. Rule 3 evaluation
A31
0 X
1
y10 Y
0.0
x1 0
0.1C1
1
C2
Z
1
0 X
0.2
0
0.2 C11
C2
Z
A2
x1
Rule 3: IF x isA1 (0.5)
A11
0 X 0
1
Zx1
THEN
C1 C2
1
y1
B2
0 Y
0.7
B10.1
C3
C3
C30.5 0.5
OR(max)
AND(min)
OR THENRule 1: IF x isA3 (0.0)
AND THENRule 2: IF x isA2 (0.2)
y isB1 (0.1) z isC1 (0.1)
y isB2 (0.7) z isC2 (0.2)
z isC3 (0.5)
project funding
inadequate
risk
high
00.1
1C1
z isC1 (0.1)
C2
0
0.2
1
z isC2 (0.2)
0
0.5
1
z isC3 (0.5)
ZZZ
0.2
Z0
C30.5
0.1
Fuzzy Inference – Example
3. Aggregation of the rule outputs
risk
highnormallow
00.1
1C1
z isC1 (0.1)
C2
0
0.2
1
z isC2 (0.2)
0
0.5
1
z isC3 (0.5)
ZZZ
0.2
Z0
C30.5
0.1
Fuzzy Inference – Example
4. Defuzzificatione.g. use the centroid method in
which a vertical lineslices the aggregate setinto two equal halves
How can we calculate this?
Fuzzy Inference – Example
4. DefuzzificationCalculate the centre of gravity (cog):
b
aA
b
aA
COG
x x dx
x dx
Fuzzy Inference – Example
4. DefuzzificationUse a reasonable sampling of points
4.675.05.05.05.02.02.02.02.01.01.01.0
5.0)100908070(2.0)60504030(1.0)20100(
COG
1.0
0.0
0.2
0.4
0.6
0.8
0 20 30 40 5010 70 80 90 10060
Z
DegreeofMembership
67.4
4.675.05.05.05.02.02.02.02.01.01.01.0
5.0)100908070(2.0)60504030(1.0)20100(
COG
1.0
0.0
0.2
0.4
0.6
0.8
0 20 30 40 5010 70 80 90 10060
Z
DegreeofMembership
67.4
Applications of Fuzzy Logic
Why use fuzzy expert systems or fuzzy control systems?Apply fuzziness (and therefore accuracy) to
linguistically defined terms and rulesLack of crisp or concrete mathematical models
exist
When do you avoid fuzzy expert systems?Traditional approaches produce acceptable resultsCrisp or concrete mathematical models exist and
are easily implemented