1. Intelligent Data Analysis
Introduction to Fuzzy Logic
Michael R. Berthold

2. Fuzzy Logic Fuzzy Sets Fuzzy Numbers Fuzzy Operators Fuzzy Rules
Fuzzy Inference

3. Types of Uncertainty Stochastic uncertainty Linguistic uncertainty
example: rolling a dice
Linguistic uncertainty
examples : low price, tall people, young age
Informational uncertainty
example : credit worthiness, honesty
Stochastic uncertainty an event occurs with a given probability
lexical or linguistic uncertainty
imprecise description of an object or concept
3. Informational uncertainty
uncertainty caused by missing or incomplete information

4. Classical Sets young = { x  P | age(x)  20 }
characteristic function:
young(x) =
1 : age(x)  20
0 : age(x) > 20
young(x)
A=“young” 1
x [years]

5. Fuzzy Sets Fuzzy Logic Classical Logic Element x belongs to set A
with a certain
degree of membership:
(x)[0,1]
Classical Logic
Element x belongs to set A
or it does not:
(x){0,1}
A(x)
A(x)
A=“young”
A=“young” 1 1
x [years]
x [years]

6. Fuzzy Set
Definition :
Fuzzy Set A = {(x, A(x)) : x  X,  A(x)  [0,1]}
a universe of discourse X : 0  x  100
a membership function A : X  [0,1]
A(x)
A=“young” 1
=0.8
Membership function also called characteristic function
x [years]
x=23

7. Definitions : Support of a fuzzy set A Core of a fuzzy set A
supp(A) = { x  X :  A(x) > 0 }
Core of a fuzzy set A
core(A) = { x  X :  A(x) = 1 }
=0.6
-cut of a fuzzy set A
A  = { x  X :  A(x)  }
(x) 1
Alpha cuts usual for implementation in a computer
Normalized fuzzy set A
sup x  X  A(x) = 1 a b c d x
Uni-modal fuzzy set
only one “peak”

8. Types of Membership Functions
Trapezoid: <a,b,c,d>
Gaussian: N(m,s)
(x)
(x) 1 1 s a b c d x m x
Triangular: <a,b,b,d>
Singleton: (a,1) and (b,0.5)
(x)
(x) 1 1 a b d x a b x

9. Membership functions core: CA:={x | A(x)=1} = [b,c] A(x) 1
-cut: A:={x | A(x)= }  a b c d x
support: SA:={x | A(x)>0} = (a,d)

10. Fuzzy Numbers Fuzzy Numbers have restricted fuzzy sets:
X = 1
their core is a point: 1 x : (x)=1
left and right flank of  are monotonically increasing, respectively decreasing
Common choices are triangular MS-functions  easy internal representation (a,b,c)
(x) 1 a b c x

11. Computing with Fuzzy Numbers
Addition: A+B(x) = max{A(y), B(z) | x=y+z}
(x)
A(x)
B(x)
A+B(x) 1 x
Multiplication: AB(x) = max{A(y), B(z) | x=yz} x
(x) 1
A(x)
B(x)
AB(x)

12. The Extension Principle
Assume a fuzzy set A and a function f:
How does the fuzzy set f(A) look like?
For arbitrary functions f:
f(A)(y) = max{A(x) | y=f(x)} f x
A(x) y
f(A)(y) f x
A(x) y
f(A)(y)
max

13. Operators on Fuzzy Sets
Union
Intersection
AB(x)=max{A(x),B(x)}
AB(x)=min{A(x),B(x)}
A(x)
B(x)
A(x)
B(x) 1 1 x x
AB(x)=min{1,A(x)+B(x)}
AB(x)=A(x)  B(x)
A(x)
B(x)
A(x)
B(x) 1 1 x x

14. T-Norms and T-Conorms
A function T : [0,1]x[0,1] => [0,1] is called a t-norm
iff for any u,v,w  [0,1]
T(u,1) = u as unit
u  v  T(u,w)  T(v,w) monotonicity
T(u,v) = T(v,u) commutativity
T(u,T(v,w)) = T(T(u,v),w) associativity
A t-norm is strict if it is strictly increasing in [0,1]x[0,1]
Bounded product
Examples: min(u,v) minimum
u•v algebraic product
max(0,u+v-1) bounded product

15. T-Conorms or S-norms
A function S : [0,1]x[0,1] => [0,1] is called a t-conorm
or s-norm iff for any u,v,w  [0,1]
S(u,0) = u as unit
u  v  S(u,w)  S(v,w) monotonicity
S(u,v) = S(v,u) commutativity
S(u,S(v,w)) = (S(u,v),w) associativity
A s-norm is strict if it is strictly increasing in [0,1]x[0,1]
Bounded sum
Examples: max(u,v) maximum
u+v-u•v algebraic sum
min(1,u+v) bounded sum

16. Complement Negation: A(x)= 1 - A(x)
Classical law does not always hold:
AA(x)  1
AA(x)  0
Example : A(x) =
A(x) = 1 - A(x) = 0.4
AA(x) = max(0.6,0.4) = 0.6  1
AA(x) = min(0.6,0.4) = 0.4  0

17. T-norms and S-norms S(u,v) = 1 – T(1-u,1-v) : De Morgan Law
T(u,v) = 1 – S(1-u,1-v)
max(u,v) = 1 – min(1-u,1-v)
u•v = 1 - (1-u)+(1-v) + (1-u)•(1-v)
max(0,u+v-1) = 1 - min(1,1-u+1-v)

18. { { Operator - Spectrum S-norms (T-conorms) T-norms drastic product
bounded
product
algebraic
product
min
max
algebraic
sum
bounded
sum
drastic
sum       
u if v=1
drastic product : T(u,v)= v if u=1
0 otherwise {
u if v=0
drastic sum : T(u,v)= v if u=0
1 otherwise {

19. Cylindrical Extension
Imbed a lower dimensional fuzzy set in a higher dimensional space
fuzzy set A with universe of discourse X1 and MSF mA and
additional universe X2
Definition: CX1xX2 A is called the cylindrical extension of A to
the universe X1x X2 iff
(x1,x2)  X1x X2 : m CX1xX2 A(x1,x2) = mA (x1)

20. Cartesian Product
fuzzy set A with universe of discourse X1 and MSF mA and
fuzzy set B with universe of discourse X2 and MSF mB and
Definition: AxB is called the cartesian product of A and B to
the universe X1x X2 iff
(x1,x2)  X1x X2 : m AxB (x1,x2) = min (mA (x1), mB (x2))
Notice: min can be replaced by any other t-norm

21. { Fuzzy Relations classical relation fuzzy relation
R  X x Y defined by mR(x,y) = iff (x,y)  R
0 iff (x,y)  R { |
fuzzy relation
R  X x Y defined by mR(x,y)  [0,1]
mR(x,y) describes to which degree x and y are related
It can also be interpreted as the truth value of the proposition x R y

22. Fuzzy Relations Example: X = { rainy, cloudy, sunny }
Y = { sun bathing, bicycling, camping, reading }
X/Y sun bathing bicycling camping reading
rainy
cloudy
sunny
0.0
0.2
0.0
1.0
0.0
0.8
0.3
0.3
1.0
0.2
0.7
0.0

23. Fuzzy Sets & Linguistic Variables
A linguistic variable combines several fuzzy sets.
example:
linguistic variable : temperature
linguistics terms (fuzzy sets) : { cold, warm, hot }
x [C]
(x) 1
cold
warm
hot 60 20

24. Fuzzy Rules
causal dependencies can be expressed in form of if-then-rules
general form: if <antecedent> then <consequence>
example: if temperature is cold and oil is cheap then heating is high
linguistic variables
linguistic values/terms (fuzzy sets)

25. Fuzzy Rulebase Heating Temperature : cold warm hot Oil price: cheap
normal
expensive
if temperature is cold and oil price is low then heating is high
high high medium
high medium low
medium low low
if temperature is hot and oil price is normal then heating is low

26. Fuzzy Knowledge-Base (x) cold warm hot 1 x [C] 20 60
Fuzzy Data-Base:
Definition of linguistic input and output variables
Definition of fuzzy membership functions
(x)
cold
warm
hot 1
x [C] 20 60
Fuzzy Rule-Base:
List of fuzzy rules
if temperature is cold and oil price is cheap
then heating is high ….

27. Fuzzy Inference 1. Fuzzification
Determine degree of membership for each term of an
input variable :
temperature : t=15 C oilprice : p=$13/barrel
cold(t)=0.5
cheap(p)=0.3 1 1
0.5
0.3 t p
15C
$13/barrel
if temperatur is cold ...
and oil is cheap ...

28. Fuzzy Inference
2. Combine the terms in one degree of fulfillment for the entire
antecedent using a fuzzy AND: min-operator t 1
cold(t)=0.5
if temperatur is cold ...
15C p
cheap(p)=0.3
and oil is cheap ...
$13/barrel
0.5
0.3
ante = min{cold(t), cheap(p)} = min{0.5,0.3} = 0.3

29. Fuzzy Inference
3. Inference step: Apply the degree of membership of the antecedent
to the consequent of the rule using a t-norm: min or prod operator
consequent(h)
ante =0.3
high(h) 1
...
min-inference:
cons. = min{ante , high } h
... then heating is high
ante =0.3
consequent(h)
high(h) 1
...
prod-inference:
cons. = ante • high h

30. Fuzzy Inference
4. Aggregate all the rules consequents using the max-operator
for union
... then heating is high
... then heating is medium
... then heating is low 1 h

31. Defuzzification
5. Determine crisp value from output membership function
for example using “Center of Gravity”-method:
consequent(h)
COG 1 h 73
Center of singletons is computationally easier
Center of singletons defuzzification:
mi = degree of membership fuzzy set i
Ai = area of fuzzy set i
ci = center of gravity of fuzzy set i
h = Si mi • Ai • ci
Si mi • Ai

32. Schema of a Fuzzy Decision
Fuzzification
Inference
Defuzzification
rule-base
if temp is cold then valve is open
cold
warm
hot
open
half
close
cold =0.7
0.7
0.7
if temp is warm then valve is half
0.2
0.2
warm =0.2 t v
measured
temperature
if temp is hot then valve is close
crisp output
for valve-setting
hot =0.0

33. Example Fuzzy Rule Base
1. If (temperature is cold) and (oilprice is normal) then (heating is high)
2. If (temperature is cold) and (oilprice is expensive) then (heating is medium)
3. If (temperature is warm) and (oilprice is cheap) then (heating is high)
4. If (temperature is warm) and (oilprice is normal) then (heating is medium)
5. If (temperature is cold) and (oilprice is cheap) then (heating is high)
6. If (temperature is warm) and (oilprice is expensive) then (heating is low)
7. If (temperature is hot) and (oilprice is cheap) then (heating is medium)
8. If (temperature is hot) and (oilprice is normal) then (heating is low)
9. If (temperature is hot) and (oilprice is expensive) then (heating is low)

36. Types of Fuzzy Rules Mamdani rule: Outputs are fuzzy sets B
if X1 is A1 and X2 is A2 … Xn is An then Y is B
Takagi-Sugeno-Kang (TSK)- rule: Outputs are functions f(xi)
if X1 is A1 and X2 is A2 …. Xn is An then Y=f(x1,x2,..xn)
1. Order TSK- rule: Outputs are linear functions of xi
if X1 is A1 and X2 is A2…Xn is An then Y=a0 + a1x1 … +anxn

38. Learning Fuzzy Rule Sets from Data
Adjustable parameters in a Fuzzy Rule Set:
Number,
Distribution,
and shape of membership functions for each attribute.
consequent for each rule
constructive or adaptive?

39. Dividing the Input Space
Grid based
Individual membership functions

40. Grid based Approaches
equidistant grid, assign output of each tile from nearest example (Wang&Mendel 91)
divide grid at point of largest error (Higgins&Goodman, 93)
initialize grid equidistant, move membership functions similar to BackProp (Nauck&Kruse, 91-95)

41. Individual Rule Approaches
Derive Fuzzy Rules from Radial Basis Function Network, using Gaussian membership functions (several authors)
Extract Rules directly from the data (Simpson 91, Berthold&Huber95)
Extract Rules from decision trees.

42. Finetuning Fuzzy Rule Sets
Heuristics that move membership functions to minimize error (similar to Backprop)
Convert Rules into Neural Network, then train Network and afterwards extract modified rules.
Use Genetic Algorithms to fine tune rules...

43. Resources
Web
Fuzzy & Neuro-Fuzzy Repository at the University of Southampton
NAFIPS:
Journals
Fuzzy Sets & Systems
IEEE Transactions on Fuzzy Systems
Journal of Approximative Reasoning
Intelligent Data Analysis (electronic)
Southampton has links to Software, Journals,

44. Resources Software / Shareware
Xfuzzy (fuzzy system design, verification, synthesis)
NEFCON (Matlab Simulink)
UNFUZZY (C / C++)
FOOL & FOX (Xwindows)
Commercial
Matlab Fuzzy Logic Toolbox
FuzzyTECH by Inform
DataEngine by MIT
SieFuzzy by SIEMENS
Xfuzzy : Sevilla, Spain (World Championships)
NEFCON : Braunschweig &Magdeburg Prof. Kruse
UNFUZZY : Colombia
FOOL&FOX Oldenburg