Fuzzy Systems
Overview:
  Literature:
Why Fuzzy
Fuzzy Logic and Sets
Fuzzy Reasoning
  chapter 4
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Why fuzzy thinking?
  Experts rely on common sense to solve problems
  Representation of vague, ambiguous terms
  If short means 300m or less, is 301m long?
  Fuzzy logic
  Not logic that is fuzzy
  But logic that describes fuzzyness = vagueness
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Vague concepts
-  at least 1.80 m?
-  could Frank be 1.78 m and be still tall?
-  if Frank is a basketball player
is 1.80m still tall?
-  if Frank is a kid of 10 years and 1.80 m
isn’t Frank very tall?
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Numerical vs. Logic uncertainty
reasoning
  Numerical systems: degree of belief
  Logical systems: degree of membership
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Fuzzy Logic - I
  Approach to formal treatment of uncertainty
  Relies on quantifying and reasoning through natural
language
  uses linguistic variables to describe concepts with vague
values
•  tall, large, small, heavy, ...
  Based on the idea that all things admit of degrees
•  Really hot, very tall, kind of small, light blue,…
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Fuzzy Logic - II
  Boolean logic uses sharp distinctions
  Fuzzy logic reflects how people think
  Fuzzy logic is a set of mathematical principles for
knowledge representation and reasoning based on degrees
of membership
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Fuzzy set
Crisp set (tall men):
  Is a function f: domain → [0,1]
Fuzzy set (tall men):
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Representing a domain
Fuzzy sets (men’s height):
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Membership function
  Categorization of element x into a set A
  described through a membership function µA(x)
  Formally, given a fuzzy set A of universe X
  µA(x): X → [0,1], where
µTall(200) = 1
•  µA(x) = 1 if x is totally in A
µTall(160) = 0
•  µA(x) = 0 if x is totally not in A
0 < µTall(180) <1
•  0 < µA(x) <1 if x is partially in A
  (Discrete) Fuzzy set A is represented as:
  A = {µA(x1)/x1, µA(x2)/x2, …, µA(xn)/xn}
  Tall = {0/160, 0.2/170, 0.8/180, 1/190}
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Operations on Fuzzy Sets
  Union: µA∪B(x) = max(µA(x), µB(x))
  Intersection: µA∩B(x) = min(µA(x), µB(x))
  Complement: µ
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¬A(x)
= 1 - µA(x)
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Fuzzy logic
  Based on fuzzy set properties
  Logic OR (∨) is set union
  A ∨ B = µA∪B(x) = max(µA(x), µB(x))
  Logical AND (∧) is set intersection
  A ∧ B = µA∩B(x) = min(µA(x), µB(x))
  Logical NOT (¬) is set complement
  ¬A = µ ¬A(x) = 1 - µA(x)
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Properties of fuzzy sets
  The same as for crisp sets
  Commutativity
  Associativity
  Distributivity
  Idempotency
  Identity
  De Morgan’s Laws
  …
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Linguistic variables
  A linguistic variable is a fuzzy variable
  The linguistic variable speed ranges between 0 and 300km/h
and includes the fuzzy sets slow, very slow, fast,…
  Fuzzy sets define the linguistic values
  Hedges are qualifiers of a linguistic variable
  All purpose: very, quite, extremely
  Probability: likely, unlikely
  Quantifiers: most, several, few
  Possibilities: almost impossible, quite possible
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Fuzzy set with hedge very
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Interpretation of hedges
  Hedges modify the shape of a fuzzy set
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Fuzzy rules
  A fuzzy rule is defined as a conditional statement of the
form
IF x is A
THEN y is B
  Where x and y are linguistic variables and A and B are
linguistic values determined by fuzzy sets on the universes
of discourse X and Y.
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Fuzzy rule example
IF height is tall
THEN weight is heavy.
  Where the fuzzy classes height and weight have a given
range (i.e. the universe of discourse).
  range(height) = [140, 220]
  range(weight) = [50, 250]
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Reasoning with fuzzy rules
  In classical systems, rules with true antecedents fire.
  In fuzzy systems, truth (i.e. membership in some class) is
relative – so all rules fire (to some extent).
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Applying fuzzy rules
  If the antecedent is true to some degree, the consequent is
true to the same degree.
  IF length is tall THEN weight is heavy
Tall men
Heavy men
  µTall(x) = 0.7 → µHeavy(y) = 0.7
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Multiple antecedents
  IF x is A AND y is B THEN z is C
  IF x is A OR y is B THEN z is C
  Use unification (OR) or intersection (AND) operations to
calculate a membership value for the whole antecedent.
  AND: µC(z) = min(µA(x), µB(y))
  OR: µC(z) = max(µA(x), µB(y))
  E.g. If rain is heavy AND wind is strong THEN weather is bad
  ((µheavy(rain) = 0.7)
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∧ (µstrong(wind) = 0.4)) → (µbad(weather) = 0.4)
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Multiple consequents
  IF x is A THEN y is B AND z is C
  Each consequent is affected equally by the membership in the
antecedent class(es).
  E.g. IF x is tall THEN x is heavy AND x has large feet
  µTall(x) = 0.7 →
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µHeavy(y) = 0.7 ∧ µLargeFeet(y) = 0.7
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Fuzzy Events
  Fuzzy probabilities:
  What is the probability of meeting a tall man?
  P(x is A) =
Where
  µA(x) is the membership function of elements in A
  P(x) is the probability of x
•  (Empirical evidence)
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Fuzzy probability
  What is the probability that temperature on July 16th is
around 25°C
  Based of average temperatures for July 16th:
°C
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
P
.002
.005
.005
.01
.04
.11
.15
.21
.16
.14
.11
.04
.01
.005
.002
P(temp is around 25°C) =
1
=
0
20
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Around 25°C
Σ µabout25(x)*P(x)
= µ (20)*P(20) + µ(21)*P(21) + … + µ(30)*P(30)
= 0*0.005 + 0.25*0.01 + 0.5*0.04 +… + 0*0.01
= 0.8125
≈ 0,8
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Using hedges
  The probability that temperature on July 16th is around
25°C is likely
  P(temp is about 25°C) = 0.8
Likely with probability 0.95
Very likely with prob. 0.32
0.8
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Fuzzy Inference
  How to reason using Fuzzy Logic?
1.  Fuzzify input variables
2.  Evaluate rules
3.  Aggregate rule outputs
4.  Defuzzify the output
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Rule 1
X is A1
w1
y is B1
Rule 2
X is A2
X
defuzzifier
y is B2
w2
Fuzzy
aggregator
…
y
Rule r
X is Ar
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wr
y is Br
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Step 1: Fuzzification
  Use crisp inputs from the user
  Determine membership values for all the relevant classes (i.e.
in right Universe of Discourse).
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Step 2: Rule evaluation
  Apply fuzzified inputs to all relevant rules
  Use union, intersection and complement operations to handle
composite antecedents
  Min-max method for composite antecedents produces clipped
functions
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Step 3: Result Aggregation
  Build membership function for each output (rule
consequents) by aggregating all the relevant classes
  Unification of outputs of all rules
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Step 4: Defuzzification (Mamdani)
  Definition: “It refers to the way a crisp value is extracted
from a fuzzy set as a representative value”
  There are five methods of defuzzifying a fuzzy set A of a
universe of discourse Z
  Centroid of area zCOA
  Bisector of area zBOA
  Mean of maximum zMOM
  Smallest of maximum zSOM
  Largest of maximum zLOM
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Step 4: Defuzzification (Mamdani)
  Use centre of area formula (centroid technique) to calculate crisp
output value
  Summation as approximation to integral value
≈
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Step 4: Defuzzification
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Sugeno Fuzzy Models
  “If x is A & y is B then z = f(x, y)”
Where A & B are fuzzy sets in the antecedent, while z =
f(x, y) is a crisp function in the consequent
  If f(.,.) is a constant then it is a zero-order Sugeno
fuzzy model (special case of Mamdani model)
  Often: Weighted average of singletons in each fuzzy set
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Exercise
  Suppose 3 fuzzy rules that determine heart attack risk,
based on:
  Three universes of discourse (UoD):
•  Diet (1000-4000 cal/day): with fuzzy classes poor, good
•  Exercise (1000-4000 cal/day): with fuzzy classes high, low
•  Risk (0-100 likelihood): with fuzzy classes high, low, medium
  Represent membership functions graphically
  Show fuzzy inference for following sample data
•  What is the risk of heart disease of Mary who has a diet of 2500
cal/day and spends 1000 cal/day on exercise
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Rules
  Rule1:
  Rule2:
  Rule3:
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IF diet is poor AND exercise is low
THEN risk is high
IF diet is good AND exercise is high
THEN risk is low
IF diet is good OR exercise is high
THEN risk is medium
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Membership function: Diet
Mary?
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Membership function: Exercise
Mary?
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Membership function: Risk
Mary?
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Mamdani inference
1.  Fuzzify input variables
  Determine membership values
2.  Evaluate rules
  Based on membership values of (composite) antecedents
3.  Aggregate rule outputs
  Unify all membership values for output from all rules
4.  Defuzzify the output
  COG: Center of gravity (approx. by summation)
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Step 1: fuzzification
  Fuzzify values for Mary:
  has a diet of 2500 cal/day and spends 1000 cal/day on
exercise
  µpoordiet(2500) = 0,6
  µgooddiet(2500) = 0,8
  µhighexer(1000) = 0
  µlowexer(1000) = 1
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Step 2: rule evaluation
  Rule1:
  IF diet is poor (0,6) AND exercise is low (1) THEN risk
is high
  µA∧B(x) = min(µA(x), µB(x)) = 0,6
  Consequent is true to the same degree as the composition
of antecedent
  µriskhigh(Mary) = 0,6
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Membership: Risk High
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Step 2 (continuation)
  RULE1: IF diet is poor (0,6) AND exercise is low (1)
THEN risk is high (0,6)
  RULE2: IF diet is good (0,8) AND exercise is high (0)
THEN risk is low (0)
  RULE3: IF diet is good (0,8) OR exercise is high (0)
THEN risk is medium (0,8)
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Step 3: Aggregation
Note that “low” has been clipped down to zero, and “average” hasn’t
been clipped, because the derived µ was higher than the original
membership function.
  The aggregate is simply the union of the three functions.
 
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Step 4: Defuzzification
  Centroid technique
  Example: sample every 12,5 units
  More points are better, integral is even better
  COG = 70
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Result
  The likelihood of heart disease for Mary is 70%
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Advantages of Fuzzy Logic
  general theory of uncertainty
  wide applicability, many practical applications
  natural use of vague and imprecise concepts
  helpful for commonsense reasoning, explanation
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Problems of Fuzzy Logic
  membership functions can be difficult to find
  multiple ways for combining evidence
  problems with long inference chains
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Coming up
  6 January 2008: Intelligent Interfaces (Agents and KS)
  13 January 2008: Knowledge Engineering (Evaluation)
  20 January 2008: KS in practice?
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