universitÄt linz johannes kepler · johannes kepler universitÄt linz netzwerk für forschung,...

U N I V E R S I T Ä T L I N ZN e t z w e r k f ü r F o r s c h u n g , L e h r e u n d P r a x i s

J O H A N N E S K E P L E RU N I V E R S I T Ä T L I N Z

N e t z w e r k f ü r F o r s c h u n g , L e h r e u n d P r a x i s

J O H A N N E S K E P L E R

Knowledge-Based Methods in Image Processing and Pattern Recognition; Ulrich Bodenhofer 11

Unit 2

A Brief Introduction to Fuzzy Logic andFuzzy Systems






Motivation

In our everyday life, we use vague, qualitative, im-

precise linguistic terms like “small”, “hot”, “around two

o’clock”

Even very complex and crucial human actions are de-

cisions are based on such concepts, e.g. in

Process control

Driving

Financial/business decisions

Law






Motivation (cont’d)

These imprecise terms and the way they are pro-

cessed, therefore, play a crucial role in everyday life

To have a mathematical model which is able to ex-

press the complex semantics of such terms, hence,

would lead to more intelligent systems and open com-

pletely new opportunities

Concepts in classical mathematics and technology

are inadequate to provide such models






L. A. Zadeh (1965)

“More often than not, the classes of objects encountered

in the real physical world do not have precisely defined cri-

teria of membership. [. . . ] Yet, the fact remains that such

imprecisely defined “classes” play an important role in hu-

man thinking, particularly in the domains of pattern recog-

nition, communication of information, and abstraction.”






Fuzzy Logic and Fuzzy Sets

Fuzzy logic is a generalized kind of logic

Fuzzy sets are the key to the semantics of vague lin-

guistic terms

Fuzzy sets are based on fuzzy logic






What are Fuzzy Sets?

The idea behind fuzzy logic is to replace the set of truth

values {0,1} by the entire unit interval [0,1].

A fuzzy set on a universe X is represented by a function

which maps each element x ∈ X to a degree of member-

ship from the unit interval [0,1]. These so-called member-

ship functions are direct generalizations of characteristic

functions.






What Do We Need?

In order to be able to proceed IF-THEN rules involving

vague linguistic expressions which are modeled by fuzzy

sets, we need to have proper generalizations of logical

operations and an inference scheme.

Let us start with the first question: How can we extend

the classical logical operations ∧,∨,¬ to the unit interval

[0,1]?






Triangular Norms

A mapping T : [0,1]2 → [0,1] is a triangular norm (t-norm) ifit has the following properties (for all x, y, z ∈ [0,1]):

Commutativity: T (x, y) = T (y, x)

Associativity: T (x, T (y, z)) = T (T (x, y), z)

Non-decreasingness: x ≤ y ⇒ T (x, z) ≤ T (y, z)

Neutral element: T (x,1) = x






The Four Standard t-Norms

TM(x, y) = min(x, y)

TP(x, y) = x · y

TL(x, y) = max(x + y − 1,0)

TD(x, y) =

x if y = 1

y if x = 1

0 otherwise






The Four Standard t-Norms (Cont’d)

TM

0 0.2 0.4 0.6 0.8 10.5

10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10.5

10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

TP

TL

0 0.2 0.4 0.6 0.8 10.5

10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10.5

10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

TD






Some Observations

For all x, y ∈ [0,1], we have:

TD(x, y) ≤ TL(x, y) ≤ TP(x, y) ≤ TM(x, y)

It is easy to check that TM is the largest possible t-normand that TD is the smallest possible t-norm

TM is the only t-norm fulfilling idempotence (T (x, x) = x)

All except TD are continuous

TP is the only differentiable one

TP is the only one that is strictly non-decreasing






Examples of Intersections

TM 0.5

1

0.5

1

TP 0.5

1

0.5

1

TL 0.5

1

0.5

1

TD 0.5

1

0.5

1






Fuzzy Negations

A mapping N : [0,1] → [0,1] is called a negation if

it is non-increasing and fulfills the boundary conditions

N(0) = 1 and N(1) = 0. A negation is called strong

if it is continuous and strictly decreasing (therefore, bijec-

tive). A strong negation is called strict if it is involutive,

i.e.

N(N(x)) = x

for all x ∈ [0,1]. The most common negation is NS(x) =

1− x.






Triangular Conorms

A mapping S : [0,1]2 → [0,1] is a triangular conorm (t-conorm) if it has the following properties (for all x, y, z ∈ [0,1]):

Commutativity: S(x, y) = S(y, x)

Associativity: S(x, S(y, z)) = S(S(x, y), z)

Non-decreasingness: x ≤ y ⇒ S(x, z) ≤ S(y, z)

Neutral element: S(x,0) = x






The Four Standard t-Conorms

SM(x, y) = max(x, y)

SP(x, y) = x + y − x · y

SL(x, y) = min(x + y,1)

SD(x, y) =

x if y = 0

y if x = 0

1 otherwise






The Four Standard t-Conorms (Cont’d)

SM

0 0.2 0.4 0.6 0.8 10.510

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10.510

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

SP

SL

0 0.2 0.4 0.6 0.8 10.510

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10.510

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

SD






Some Observations

For all x, y ∈ [0,1], we have:

SM(x, y) ≤ SP(x, y) ≤ SL(x, y) ≤ SD(x, y)

It is easy to check that SD is the largest possible t-conormand that SM is the smallest possible t-conorm

SM is the only t-conorm that is idempotent (S(x, x) = x)

All except SD are continuous

SP is the only differentiable one

SP is the only one that is strictly non-decreasing






Examples of Unions

SM 0.5

1

0.5

1

SP 0.5

1

0.5

1

SL 0.5

1

0.5

1

SD 0.5

1

0.5

1






Implications

S-implication:For a t-conorm S and a negation N , we define

IS,N(x, y) = S(N(x), y)

Residual implication ( R-implication):For a (left-)continuous t-norm T , we define

T→

(x, y) = sup{u ∈ [0,1] | T (x, u) ≤ y}






Examples

ISM,NS(x, y) = max(1− x, y) T→

M(x, y) =

1 if x ≤ y

y otherwise

ISP,NS(x, y) = 1− x + x · y T→

P(x, y) =

1 if x ≤ y

yx

otherwise

ISL,NS(x, y) = min(1− x + y,1) T→

L(x, y) = min(1− x + y,1)






Examples (cont’d)

ITM,NSITP,NS

ITL,NS

00.25

0.50.75

1

0

0.25

0.5

0.75

1

0

0.25

0.5

0.75

1

00.25

0.50.75

1

00.25

0.50.75

1

0

0.25

0.5

0.75

1

0

0.25

0.5

0.75

1

00.25

0.50.75

1

00.25

0.50.75

1

0

0.25

0.5

0.75

1

0

0.25

0.5

0.75

1

00.25

0.50.75

1

00.25

0.50.75

1

0

0.25

0.5

0.75

1

0

0.25

0.5

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1

00.25

0.50.75

1

00.25

0.50.75

1

0

0.25

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1

0

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0.5

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00.25

0.50.75

1

00.25

0.50.75

1

0

0.25

0.5

0.75

1

0

0.25

0.5

0.75

1

00.25

0.50.75

1

T→M T

→P T

→L






Aggregation Operators

A function

A :⋃n∈N

[0,1]n → [0,1]

is called an aggregation operator if it has the following proper-ties:

1. A(x1, . . . , xn) ≤ A(y1, . . . , yn) whenever xi ≤ yi for alli ∈ {1, . . . , n}

2. A(x) = x for all x ∈ [0,1]

3. A(0, . . . ,0) = 0 and A(1, . . . ,1) = 1






Aggregation Operators: Examples

All t-norms and t-conorms are aggregation operators (withthe conventions T (x) = x and S(x) = x)

All weighted arithmetic and geometric means are aggrega-tion operators






Linguistic Variables

A linguistic variable is a quintuple of the form

V = (N, G, T, X, M),

where N , T , X, G, and M are defined as follows:

1. N is the name of the linguistic variable V

2. G is a grammar

3. T is the so-called term set, i.e. the set linguistic expressions re-sulting from G

4. X is the universe of discourse

5. M is a T → F(X) mapping which defines the semantics—a fuzzyset on X—of each linguistic expression in T






The Basic Setup

Let us in the following consider a system with n inputs and one output.Assume that we have n linguistic variables

v1 = (N1, G1, T1, X1, M1),

... =...

vn = (Nn, Gn, Tn, Xn, Mn),

associated to the n inputs of the system and one linguistic variableassociated to the output:

vy = (Ny , Gy , Ty , Xy , My)






Fuzzy Rule Base with m Rules

IF cond1 THEN action1

......

......

IF condm THEN actionm

The conditions condi and the actions actioni are ex-pressions built up according to an appropriate syn-tax.






Example

Consider a system with two inputs and one output:

v1 = (N1 = “ϕ”, G1, T1 = {“nb”, “ns”, “z”, “ps”, “pb”},

X1 = [−30,30], M1),

v2 = (N2 = “ϕ”, G2, T2 = {“nb”, “ns”, “z”, “ps”, “pb”},

X2 = [−30,30], M2),

vy = (Ny = “f ”, Gy, Ty = {“nb”, “ns”, “z”, “ps”, “pb”},

Xy = [−100,100], My)






Example (cont’d)

IF (ϕ is z and ϕ is z) THEN f is z

IF (ϕ is ns and ϕ is z) THEN f is ns

IF (ϕ is ns and ϕ is ns) THEN f is nb

IF (ϕ is ns and ϕ is ps) THEN f is z...

......

...

How can we define a control function from these rules?[go to fuzzy sets]






What Do We Need?

1. We have to feed our input values into the system

2. We have to evaluate the truth values of the conditions

3. We have to come to some conclusions/actions for each rule

4. We have to come to an overall conclusion/action for thewhole set of rules

5. We have to get an output value

Steps 3 and 4 are usually considered the steps of actual infer-ence.






Steps 1 and 2

Assume we are given n crisp input values xi ∈ Xi (i = 1, . . . , n) andassume we have fixed a De Morgan triple (T, S, N).

Then we can compute the truth value t(condi) of each condition condi

recursively in the following way:

t(Ni is lij) = µMi(l

ij)

(xi)

t(a and b) = T (t(a), t(b))

Trivial extensions are necessary if the language allows more than onlyconjunctions.






Steps 3 and 4: Basic Remarks

1. It may happen that the conditions of two or more rules

are fulfilled with a non-zero truth value

2. It may even happen that this is true for two or more

rules with different (conflicting?) actions

3. This is not at all a problem, but a great advantage!

4. In any case, the following basic requirement is obvi-

ous: The higher the truth value of a rule’s condition,

the higher its influence on the output should be






Steps 3 and 4: Two Fundamental Approaches

Deductive interpretation: Rules are considered as logical de-duction rules (implications)

Assignment interpretation: Rules are considered as condi-tional assignments (like in a procedural programming lan-guage)

Both approaches have in common that separate output/actionfuzzy sets are computed for each rule. Finally, the output fuzzysets of all rules are aggregated into one global output fuzzy set.






Step 3 in the Assignment Interpretation

We fix a t-norm T in advance. Assume that we consider

the i-th rule which looks as follows:

IF condi THEN Ny is lyj

Assume that the condition condi is fulfilled with a degree

of ti. Then the output fuzzy set Oi is defined in the follow-

ing way:

µOi(y) = T

(ti, µM(lyj )

(y))






An Example

µA(x)

1 2 3 4 5

0.2

0.4

0.6

0.8

1






An Example

µA(x)

1 2 3 4 5

0.2

0.4

0.6

0.8

1

TM(0.4, µA(x))

1 2 3 4 5

0.2

0.4

0.6

0.8

1






An Example

µA(x)

1 2 3 4 5

0.2

0.4

0.6

0.8

1

TM(0.4, µA(x))

1 2 3 4 5

0.2

0.4

0.6

0.8

1

TP(0.4, µA(x))

1 2 3 4 5

0.2

0.4

0.6

0.8

1






An Example

µA(x)

1 2 3 4 5

0.2

0.4

0.6

0.8

1

TM(0.4, µA(x))

1 2 3 4 5

0.2

0.4

0.6

0.8

1

TP(0.4, µA(x))

1 2 3 4 5

0.2

0.4

0.6

0.8

1

TL(0.4, µA(x))

1 2 3 4 5

0.2

0.4

0.6

0.8

1






Step 4 in the Assignment Interpretation

We fix an aggregation operator A in advance. Assume

that the output fuzzy sets Oi of all rules (i = 1, . . . , m) have

been computed. Then the output fuzzy set O is computed

in the following way:

µO(y) = A(µO1

(y), . . . , µOm(y))






Some Remarks

The assignment interpretation is by far the more commonone in practice. There is only one package that seriouslyoffers the deductive interpretation (LFLC). It uses I = T

→L

and T = TM.

The most common variant of the assignment-based ap-proach is T = TM and A = SM. This classical variantis better known as Mamdani/Assilian inference or max-mininference. Another common variant uses T = TP and thesum/arithmetic mean as aggregation A. This variant is of-ten called sum-prod inference.






Example

We consider the rule base from the previous example.[go back]We define the following fuzzy sets for variables with names ϕ

and ϕ (left) and f (right):

-30 -20 -10 10 20 30 -100-80 -40-20 20 40 80 100






Step 5: Defuzzification

In many applications, we need a crisp value as output. The followingvariants are common:

Mean of maximum (MOM): The output is computed as the center ofgravity of the area where µO takes the maximum, i.e.

ξMOM(O) :=

∫Ceil(O)

y dy∫Ceil(O)

1 dy,

where

Ceil(O) := {y ∈ Xy | µO(y) = {µO(z) | z ∈ Xy}}






Step 5: Defuzzification (cont’d)

Center of gravity (COG): The output is computed as the cen-ter of gravity of the area under µO:

ξCOG(O) :=

∫Xy

y · µO(y) dy∫Xy

µO(y) dy

Center of area (COA): The output is computed as the pointwhich splits the area under µO into two equally-sized parts.






Fuzzy Classification Systems

Fuzzy classification systems are ordinary fuzzy systems,

however, with the important difference that the universe of

the output variable Xy and the term set Ty coincide and

are finite sets of class labels.






Example

Ty = Xy = {“dog”, “horse”, “fish”}

IF (no-of-legs is four and height is tall) THEN class is horse

IF (no-of-legs is four and height is short) THEN class is dog

IF no-of-legs is zero THEN class is fish






How to Process Fuzzy Classification Rules?

IF condi THEN Ny is lyi

Assume that the conditions condi are fulfilled with degrees ti.We compute individual output fuzzy sets Oi for each rule in thefollowing way:

µOi(x) =

ti if x = lyi

0 otherwise

The individual output fuzzy sets are then aggregated by meansof the aggregation operator A. Most often, the maximum t-conorm SM is used.






Defuzzification

In many cases, we need a crisp decision to which class

the object belongs.

The almost only way to do this defuzzification on a finite

universe of class labels is to use that class the member-

ship to which is maximal.






Summary

1. Feed our input values into the system: evaluate the truthdegrees to which the inputs belong to the fuzzy sets asso-ciated to the linguistic terms

2. Evaluate the truth values of the conditions using fuzzy log-ical operations (e.g. a De Morgan triple (T, S, N))

3. Compute the conclusions/actions for each rule by connect-ing the truth value of the condition with the output fuzzy setusing a t-norm T






Summary (cont’d)

4. Compute the overall conclusion/action for the whole set ofrules by aggregating the output fuzzy sets with an aggre-gation operator A (most often a t-conorm)

5. Use defuzzification to get a crisp output value (optional)

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