FLEA

1
FLEA

• Prof. dr. ir. J. Hellendoorn

2
Mamdani local, fast procedure I

• Find the degree of membership of x0 to Ai(x) (i
  1, , n)
• Example x0 x3

3
Mamdani local, fast procedure II

• Form the cuts of Bi as follows
• In our example

B1(y) (0.6, 0.6, 0.6, 0.5, 0.3, 0.0, 0.0)
B2 (y) (0.0, 0.5, 0.7, 0.8, 0.7, 0.5, 0.0)
B3 (y) (0.0, 0.0, 0.2, 0.2, 0.2, 0.2, 0.2)
4
Mamdani local, fast procedure III

• Form the union of Bi to obtain Bi
• In our example
• The same as local or global Mamdani

B(y) ? i Bi(y) (0.6, 0.6, 0.7, 0.8, 0.7,
0.5, 0.2)
5
Gödel, fast procedure

• There is no fast procedure for the Gödel
  inference method

6
Properties of a set of rules

• Completeness
• Consistency

7
Completeness of a set of rules

• Consider the rule set Ri if Ai(x) then Bi(y)
• The set of rules is complete if for each crisp
  input x0

8
Example I

• Consider two rules
• if A1(x) then B1(y)
• if A2(x) then B2(y)
• A1(x) 1/x1 .8/x2 .6/x3 .4/x4 .2/x5
  0/x6 0/x7
• A2(x) 0/x1 .4/x2 .8/x3 1/x4 .8/x5
  .4/x6 0/x7
• Then if x0 x7 we have thatA7(x) 0/x1 0/x2
  0/x3 0/x4 0/x5 0/x6 0/x7

9
Example II

• Then A7(x) o Rm 0, because x7 does not belong
  to A1(x) or A2(x).
• Hence the rule base is not complete.

10
Example of incomplete rulebase
11
Consistency of a set of rules

• A rule base is inconsistent if there exist at
  least two rules with the same if-parts and
  mutually exclusive then-parts.

m
m
A1
B1
0
X
0
Y
m
m
A1
B2
0
X
0
Y
x0
12
Defuzzification
Denormalization
Normalization
Inference
Defuzzification
Fuzzification
Rule base
Fuzzy Algorithm
Database
Algorithm
Rulebase
Data flow
Transition from syntax to semantics
13
Defuzzification methods

• Center-of-Area / Gravity Defuzzification
• Center-of-Sums Defuzzification
• Center of-Largest-Area Defuzzification
• First-of-Maxima Defuzzification
• Middle-of-Maxima Defuzzification
• Height Defuzzification

14
General problem I

• The output of the first rule is
• The output of the second rule is
• Take the union of and
• Calculate the mathematical center of gravity of
  with

15
General problem II
A
B
16
Center-of-Gravity defuzzification
A
B
single
17
Center-of-Gravity defuzzification

• Calculate
• The area of this output is
• The center of gravity is
• Problem computational complexity

18
Center-of-Sums defuzzification
A
B
double
19
Center-of-Sums defuzzification

• Calculate the center of gravity of (x1,y1)
• Calculate the center of gravity of (x2,y2)
• Calculate the mean of the centers of gravity
• The formula
  represents this

20
First-of-Maxima defuzzification
A
B
Middle-of-Maxima defuzzification
21
Height defuzzification
A
B
22
Criteria

• Continuity
• Disambiguity
• Plausibility
• Algorithmic complexity
• Multiple consideration of rules

23
Table
24
CoA and Quality defuzzification
1
0.8
0.2
0
QM
CoA
25
Qualitative results
26
Vague queries

• Which employment candidates are highly educated
  and moderately experienced?
• Which industries are forecasted to experience
  significant growth by a substantial number of
  experts?
• Which reasonably priced hotels are in close
  proximity to the city center?

27
Vague queries - fuzzy or detailed

• List the young salesmen who have a good selling
  record for households goods in the north of
  England!
• Not the same asList the salesmen under 25 years
  old who have sold more than EUR 1.000.000 of
  goods in the categories to shops in the regions
  ...

28
Crisp - imprecise

• Database
• Crisp
• Speed(Car) is 100
• Crisp
• Speed(Car) is 100
• Imprecise
• Speed(Car) 90,110

Query Crisp Speed(Car) 110? Imprecise
Speed(Car) 90,110? Crisp Speed(Car) 95?
Answer NO YES YES
29
No uncertainty

• Thus the answers in any one case can be either
  YES or NO. There is no uncertainty in the answer.
• Classical logic A fact can be either true or
  false and nothing else

30
Imprecise - imprecise

• Database
• Imprecise
• Speed(Car) 90,120
• Imprecise
• Speed(Car) 90,120
• Imprecise
• Speed(Car) 90,120

Query Imprecise Speed(Car) 95,100? Imprecise
Speed(Car) 50,85? Imprecise Speed(Car)
75,100?
Answer YES NO ??
31
Uncertainty

• The answers can be
• YES
• NO
• DONT KNOW, uncertainty
• Three valued logic
• A fact can be true, false, or one does not know
  anything as to whether it is true or false.

32
Crisp - fuzzy
Database Crisp Speed(Car) 95
Query Fuzzy Speed(Car) high?
Answer YES to a degree

• In this case a fact can be true (or false) to a
  certain degree.
• Two alternatives
• fuzzy logic or
• many-valued logic.

1
0.7
0
95
70
110
140
Speed
33
Imprecise - fuzzy

• Database
• Imprecise
• Speed(Car) 90,110
• Fuzzy
• Speed(Car) high
• Fuzzy
• Speed(Car) high
• Fuzzy
• Speed(Car) high

Query Fuzzy Speed(Car) high? Imprecise
Speed(Car) 80,95? Fuzzy Speed(Car) very
high? Crisp Speed(Car) 95?
Answer Possible to a degree Possible to a
degree Possible to a degree Possible to a degree
34
Possibilistic logic

• It is possible to a certain degree that a fact is
  true
• It is possible to a certain degree that the same
  fact is false

35
Degree of membership I

• Young(x) 1/20 .9/25 .7/30 .5/35 .3/40
  represents the meaning of the fuzzy concept
  young age.
• Someone who is 30 is compatible to a degree 0.7
  with my understanding of the meaning of young
  in terms of age.

36
Degree of membership II

• If I know that John is 30 years old and if I have
  to classify him as young, old, etc. I can say
  thatJohn is young to a degree 0.7 since the
  age of 30 is compatible with my understanding of
  what young means only to degree 0.7.

37
Degree of Possibility I

• Suppose that I already know that John is young
  (someone told me so).
• My understanding of young is Young(x) 1/20
  .9/25 .7/30 .5/35 .3/40
• In other words I know that John is either 20, or
  25, or 30, or 40 years old.
• Each of these ages is compatible with my idea of
  young to a certain degree.

38
Degree of Possibility II

• Someone asks me whether John is 25 years old, or
• I have to say how possible it is for John to be
  25 if I know that he is young

39
Answer I

• I know that young(John) is the case Young(John)
  1/20 .9/25 .7/30 .5/35 .3/40
• 25 is a possible candidate for Johns age,
  since it belongs to the domain of young age
• It is compatible with my idea of young age to a
  very high degree of 0.9.

40
Answer II

• So, one can say It is possible to a degree of
  0.9 (very high possibility) that John is 25 years
  old.
• But, then there is also the possibility that he
  can be 20, or 30, or 35, or 40 years of age,
  since these are also mentioned in the definition
  of young

41
Answer III
Where ?25 is defined by
42
Fuzzy logic

• The goal of fuzzy logic is to serve as
• a meaning representation language, and
• a meaning computation calculus

Question in natural language
Answer in natural language
Translation into fuzzy sets
Result in fuzzy sets
43
Fuzzy logic

• E.g., two natural language expressions
• If the speed is high and the turn-angle is small,
  then decrease the speed significantly
• The speed is very high and the turn-angle is
  small
• Express the required decrease in speed in natural
  language

44
Natural language expressions I

• Atomic expressions
• An object has a property
• An object is in relation to another object
• For example
• Pressure is high
• Pressure at time t1 is much higher than pressure
  at time t2

45
Natural language expressions II

• Composite expressions
• Pressure is high and temperature is low
• Speed is low or altitude is not high

46
Natural language expressions III

• Conditional expressions
• If speed is high and surface is very wet then
  braking distance is very long
• If speed is high and turn-angle is small then
  decrease speed significantly unless wind is
  strong
• If pressure is high then temperature is very high
  else temperature is moderate

47
Natural language expressions IV

• Quantified expressions
• Most Swedes are tall
• Many American cars are much bigger than most
  European cars

48
Natural language expressions V

• Qualified expressions
• Truth-qualified expression
• It is not very true that pressure is high
• Probability-qualified expression
• It is not very likely that pressure is high
• Possibility-qualified expression
• It is not very possible that pressure is high

49
Syllogistic reasoning

• Most US cars are bigA lot of big cars consume a
  lot of fuel? Q US cars consume quite a lot of
  fuel
• How to compute Q and the amount of fuel?
• General form Q1 As are Bs Q2 Bs are
  Cs? Q3 As are Cs

50
Quantifiers

• Q1 and Q2 are based on relative cardinalities of
  fuzzy sets
• Q3 is a function F(Q1, Q2) of Q1 and Q2

m
few
many
X
0
1
51
Example

• Twenty French people
• What is the truth value of Most Frenchmen
  are tall and Most Frenchmen are short?

52
Calculation

• Define tall tall G(x 175,195)
• Define short short L(x 160,180)
• SCount of tall Frenchmen is 7.4
• SCount of short Frenchmen is 6.25

53
Most Frenchmen are tall

• The relative SCount of tall Frenchmen is 0.37
• The relative SCount of short Frenchmen is
  0.3125
• Suppose most is defined by
• Truth value of Most Frenchmen are tall 0.14

54
Modification based reasoning

• Reasoning with qualified propositions
• It is not very true that temperature is high
• Temperature is ???
• It is not very likely that temperature is high
• The probability of this is ???
• It is very possible that temperature is high
• Temperature is ???

55
Reasoning w. truth-modification

• The typical rule of inference is (if x is A then
  y is B) is t1 x is A ? y is B
• How do we obtain the conclusion?

56
Truth-modification I

• Compute the truth t2 of TRUE(x is A x is A)
  t2
• Compute the degree to which y is B is true
  given t2 and the degree of truth for the whole
  rule, i.e., t1

57
Truth-modification II

• In other words (x is A ? y is B) t1 (x is A)
  t2 ? (y is B) t3
• where t3 min(1,1 - t1 t2)(one of the
  possible functions to compute t3)

58
Truth-modification III

• Once (y is B) t3 is is obtained, find the
  reference proposition B such that (y is B y
  is B) t3

59
Example I

• (if x is tall then y is short) is true x is very
  tall ? y is ???
• TRUE(x is tall x is very tall) very true
• x is tall ? y is short is truex is tall is very
  true ? y is short is fairly true
• fairly true(y is short) y is quite short

60
Example II

• Most Frenchmen are tall is fairly true
• fairly true
• Most Frenchmen are tall had truth value 0.14,
  hence Most Frenchmen are tall is fairly true
  has truth value 0.37

1
0
1