emu, computer engineering department 1999/2000 academic year, spring semester

EMU, COMPUTER ENGINEERING DEPARTMENT

1999/2000 ACADEMIC YEAR, SPRING SEMESTER

C M P E 586Software Implementation of Fuzzy Systems

PREPARED BY DR. KONSTANTIN DEGTIAREVFEBRUARY/JUNE 2000Slides use the material of books and journal papers

CMPE 586

Reference G.J.Klir, U.H.St.Clair, Bo Yuan. Fuzzy Set Theory. Foundations &

Applications, Prentice Hall PTR, 1997

B.Kosko. Fuzzy Engineering, Prentice Hall, 1997

T.J.Ross. Fuzzy Logic with Engineering Applications, McGraw-Hill, 1995

J.Yen, R.Langari. Fuzzy Logic. Intelligence, Control, and Information, Prentice Hall, 1999

L.-X.Wang. A Course in Fuzzy Systems and Control, Prentice Hall, 1997

W.Pedrycz (ed.). Fuzzy Modelling. Paradigms and Practice (Int. Series in Intelligent Technologies), Kluwer Academic Publ., 1996

J.Yen. Fuzzy Logic - A Modern Perspective // IEEE Transactions on Knowledge and Data Engineering, vol.11, #1, January/February 1999

L.A.Zadeh. The Birth and Evolution of Fuzzy Logic // Int. Journal on General Systems, vol.17, 1990, pp.95-105

Software Implementation of Fuzzy Systems

1

Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 1

Section1 Outline

Introduction. Uncertainty, Imprecission and Vagueness Fuzzy Systems. Brief History of Fuzzy Logic. Foundation of

Fuzzy Theory. Fuzzy Sets and Systems. Fuzzy Systems in Commercial

Products Research fields in Fuzzy Theory

[ the discussion of these topics takes approximately 4 lecture hours. One example is explained (CubiCalc and fuzzyTECH software packages are used) ]

CMPE 586 Software Implementation of Fuzzy Systems

2


CMPE 586

Most of the phenomena we encounter everyday are impreciseimprecise - the imprecision may be associated with their shapes, position, color, texture, semantics that describe what they are

Fuzziness primarily describes uncertaintyuncertainty (partial truth) and imprecisionimprecision

The key idea of fuzziness comes from the multivalued logicmultivalued logic: Everything is a matter of degree

Imprecision raises in several faces, e.g. as a semantic ambiguitysemantic ambiguity

Software Implementation of Fuzzy Systems

3


By fuzzifying crisp datafuzzifying crisp data obtained from measurements, FL

enhances the robustness of a systemrobustness of a system Imprecision raises in several faces - for example, as

a semantic ambiguity

the statement “the soup is HOTthe soup is HOT” is ambiguous, but not fuzzy

e.g. [20º,80º]


The temperature of the soupHot

The amount of spices used

Definition of the domain

of discourse

Transaction to FuzzinessTransaction to Fuzziness

4


The word “fuzzy”“fuzzy” can be defined as “imprecisely defined, confused, vague”

Humans represent and manage natural language terms (data) which are vaguevague. Almost all answers to questions raised in everyday life are within some proximity of the absolute truth


empty half-full full? almost full?

or half-empty? nearly full? ……

Does itremainempty?

5


Probability theoryProbability theory is one of the most traditional theories for representing uncertainty in mathematical models

Nature of uncertaintyNature of uncertainty in a problem is a point which should be clearly recognized by engineer - there is uncertainty that arises from chance, from imprecision, from a lack of knowledge, from vagueness, from randomness…

probability theory deals with the expectation of an expectation of an eventevent (future event, its outcome is not known yet), i.e. it is a theory of random events


6


FuzzinessFuzziness deals with the impression of meaning of concepts expressed in natural language - it is not concerned with events at all

Fuzzy theory handles nonrandom uncertaintynonrandom uncertainty


Random

Uncertain Certain

Fuzzy, imprecise, vague

7


As it is stated by L.Zadeh, “in many cases there is more to be gained from cooperation than from arguments over which methodology is best…”

Many situations cover both kinds of uncertainty:

assume the weather forecast - “tomorrow “tomorrow slight rains are highly probable”slight rains are highly probable”


slight rains highly probable

includes both fuzziness and randomness ambiguious ?

8


The principle of incompatibilityprinciple of incompatibility (L.Zadeh, 1973):

“As the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics”


Math. equations

Model-free methods

Fuzzy systems

Complexity of a system

9


Intimate connectionconnection between fuzziness and complexity (L.A.Zadeh)

a new approach to system analysisnew approach to system analysis: approximate and yet effective means of describing the behavior of systems which are too complex or too ill-defined to admit of precise mathematical analysis


Cost

Utility (usefulness)

imprecise precise

10


A new approach to system analysisA new approach to system analysis: a departure from the conventional quantitative techniques of system analysis

A new paradigmA new paradigm: to develop approximate solutions that are both cost-effective and highly useful

a Fuzzy SystemFuzzy System (FS) is defined as a system with operating principles based on fuzzy information processing and decision making

There are several ways to represent knowledgerepresent knowledge, but the most commonly used has a form of rules:

IF (premise)A THEN (conclusion)B


11


From a knowledge representation viewpoint, a fuzzy IF-THEN rule is a schemescheme for capturing knowledge that involves imprecision - if we know a premise (fact), then we can infer another fact (conclusion)

A fuzzy system (FS) is constructed from a collection of fuzzy IF-THEN rules

Acquisition of knowledgeAcquisition of knowledge captured in IF-THEN rules is NOT a trivial task (expert knowledge, systems measurements, etc.)

The building blocks for fuzzy IF-THEN fuzzy IF-THEN rulesrules are FUZZY SETSFUZZY SETS


12


The rule

“IF the air is cool THEN set the motor speed to slow”

has a form:

IF x is A THEN y is B,

where fuzzy sets “cool”“cool” and “slow”“slow” are labeled by A and B, correspondingly

A and B characterize fuzzy propositionsfuzzy propositions about variables xx and yy

Most of the information involved in human communication uses natural language termsnatural language terms that are often vague, imprecise, ambiguous by their nature, and fuzzy setsfuzzy sets can serve as the mathematical foundation of natural language


13


A Fuzzy SetFuzzy Set is a set with a smooth boundaries Fuzzy Set TheoryFuzzy Set Theory generalizes classical set theory

to allow partial membership Fuzzy Set AA is a universal set UU is determined by a

membership function AA(x)(x) that assigns to each element xU a number A(x) in the unit interval [0,1]

Universal set UU (Universe of Discourse) contains all possible elements of concern for a particular application

Fuzzy set has a one-to-one correspondenceone-to-one correspondence with its membership function


14


Fuzzy set AA is defined as

A = { (x, A(x)) }, xU, A(x)[0,1]

A(x) = Degree(xDegree(xA)A) is a grade of membership of element xU in set A


X1 X2 X3 . 0 1/2 1

.

. unit interval

xN

.

. U (universe of discourse)

15


The membership functions themselves are NOT fuzzyNOT fuzzy - they are precise mathematical functions; once a fuzzy property is represented by a membership function, nothing is fuzzy nothing is fuzzy anymoreanymore

Suppose UU is the interval [0,85] representing the age of age of ordinary human beingsordinary human beings, and the linguistic term “young”“young” as a function of age (value of the variable age) can be defined as

[see the graphical representation on the next slide]

[ !! pay attention to the usage of the symbol “ / “ ]


16

x5

25-x1 x

1 young"" A

-185

25

225

0


If U is a set of integersset of integers from 1 to 10 ( U={1,2,…,10} ), then “small” is a fuzzy subset of U, and it can be defined using enumeration (summation notation):

A = “small” = 1/1+1/2+0.85/3+0.75/4+0.5/5+0.3/6+0.1/7


1

25

0 U (universe of discourse) 85

31

0.41

A

(x) Universe of discourse U is

continuos

17


In the previous example elements of U (universal universal setset) with zero membership degrees are not included into enumeration

A notion of a fuzzy set provides a convenient way of defining abstractiondefining abstraction - a process which plays a basic role in human thinking and communication

All theories that use the basic concept of fuzzy set can be called in a whole Fuzzy TheoryFuzzy Theory

Rough classification of Fuzzy Theory can be depicted as follows [note that dependencies between

the branches are not shown] :


18



Fuzzy Theory

Fuzzy Fuzzy Uncertainty & Mathematics Decision-Making Information

Fuzzy Systems Fuzzy Logic & AI

19


The idea of Fuzzy Sets appeared in 19641964 : L.A.Zadeh (Professor of the University of California at Berkeley): “We need a radically different kind of mathematics, the mathematics of fuzzy or cloudy quantities which are not described in terms of probability distributions…”

The paper “Fuzzy Sets” (Zadeh L.A., Information and Control, vol.8, pp.338-353, 1965) first used the word “fuzzy”“fuzzy” to mean “vague”“vague” in technical literature

criticized by academic community the idea caused a development of fuzzy set theory foundation (1965-19801965-1980)

academic research work stimulates first industrial applications of fuzzy systems (1977-19901977-1990) - cement kiln controller (Denmark), train control system (Sendai subway, Japan), digital and analog fuzzy chips (USA, Japan)


20


Currently, the application fields of fuzzy systems cover signal processing, communications, expert systems, medicine, business/finance, control (industrial processes and consumer electronics), …

widening of collaboration between universities and industry, “fuzzy boom” (1987-present1987-present): Japan Europe USA

19921992: 1st IEEE International Conference on Fuzzy Systems appearance of software companies (INFORMINFORM, AptronixAptronix,etc.) Fuzzy Logic ToolboxFuzzy Logic Toolbox for MATLAB was released in 1994 Courses on fuzzy sets and systems in Universities curricula

“Engineering consists largely of recommending decisions based on insufficient information.... It is essential that these students be exposed to ways of treating uncertainty and vagueness. This also requires that existing faculty utilize these methods…” (Colin Brown, conference of NAFIPS)


21


Appearance of the new computational paradigmsnew computational paradigms and intensification of research in certain areas (genetic algorithms/evolutionary strategies, neural networks)

L.A.Zadeh introduced a term soft computing (19921992)

------------ EXAMPLE 1 -------------

Fuzzy Toolbox Demo (MATLAB)by Dr.R.Babuška (Delft University of Technology, The Netherlands)

If-Then Rules. Fuzzy reasoning (example) Word 97 document (preliminary explanations)

* * * * * end of the Section 1 * * * * *


22



Section 2 Outline

Mathematical Background of Fuzzy Systems. Classical (crisp) vs. Fuzzy Sets. Representation of Fuzzy Sets

Types of Membership Functions. Basic concepts (support, singleton, height, -cut, convexity). Fuzzy Set Operations

S- and T-norms. Properties of Fuzzy Sets. Sets as points in Hypercubes. Cartesian Product. Crisp and Fuzzy Relations

Linguistic variables and hedges. Membership function design (shape analysis)

[ the discussion of these topics takes approximately 10 lecture hours. Examples are explained using CubiCalc, fuzzyTECH and FL Toolbox packages]

23


Fuzzy SystemsFuzzy Systems F: n p use m rules to map vector input x to vector or scalar outputs F(x)

Fuzzy (Rule-based) Systems make use of linguistic linguistic variablesvariables in their antecedents and consequents

Linguistic variables can be naturally represented by fuzzy fuzzy setssets and logical connectives of these sets


Function f : X Y

input X

24


A classical (crisp) set A in the universe of discourse U can be defined in three ways:

- by enumerating (listing) elements (often called listlist or extensionalextensional definition)

- by specifying the common properties of elements (intensionalintensional or rule rule definition)

the notation A = {x | P(x)} means that set A is composed of elements x such that every x has the property P(x)

- by introducing a zero-one membership function (characteristiccharacteristic or indicator indicator definition)


3 by divisiblenot is x and U,x if 0,

3 by divisible is x and U,x if , 1 (x)m )x(

3} by divisible is x | U{x A numbers), integer of(set U

AA

25


Crisp set is a set with precise boundary, and classical set theory is founded on the idea that we can make crisp, exact distinctions between two groups, i.e. between those individuals (elements) that are definitely in the result set (group 1), and those that are definitely outside it (group 2)

The basic operations on classical sets (A and B are crisp sets in the universe of discourse U):

complementcomplement divides the universal set U into 2 (two) parts


A\U AB}; x andA x | {x B\ A B-A

A}x and Ux | {x A A

B}x orA x | {x BA

B}x andA x | {x BA

C

C

26


Fundamental propertiesFundamental properties of the basic operations (these properties are also encountered in propositional logic):


A B)(A A A, B)(A A: absorption of Laws

set empty A A:ioncontradict ofLaw

U A A:middle excluded of Law

BA BA ,BA BA : laws sMorgan' De

U set empty set, empty U

n)(involutio A A : ationcomplement double

C)(AB)(A C)(B A

C)(AB)(A C)(B A: vitydistributi

CB)(A C)(B A

CB)(A C)(B A: ityassociativ

AA A A,A A: tautology) or cy(idempoten

AB B AA,B B A:itycommutativ

27


Fundamental properties satisfy to a principle of dualityprinciple of duality: replacing of empty set, U, , with U, empty set, , , respectively, brings again valid property

The notion of membership in fuzzy sets becomes a matter of degree (number in the closed interval [0,1])

Membership of an element from the universe in fuzzy set is measured by a function that attempts to describe vagueness and ambiguity


crisp (classical) set A A = set of TALL people fuzzy set A

1.0 1.0

0.65

0.0 0.0

1.75m height 1.75m

28


Membership functions can be represented (a) graphically, (b) in a tabular or list form, (c ) analytically and (d) geometrically (as a points in the unit cube)

Geometrical representation for two-element universal set U = ({x1,x2}) has a following vizualization:


membership values

1.0 (0,1) (1,1)

2

1

0.0 U (0,0)

x1 x2 1 (1,0)

graphical (standard) set of maximum “fuzziness” representation form

29


[see the previous figure] Vertices (0,0), (0,1), (1,0) and (1,1) represent all crisp sets that can be defined for the universal set U, e.g. the point (1,0) corresponds to the crisp set {x1} (element x2 has no membership)

Membership functions can be symmetrical or asymmetrical, and the most commonly used forms are triangulartriangular, trapezoidaltrapezoidal, GaussianGaussian and bellbell (the first two dominate in applications due to simplicity and computational efficiency)

Membership functions are typically defined on one-dimensional universes, and in most cases, the membership function appears in the continuos formcontinuos form

Fuzzy Toolbox Demo (MATLAB)by Dr.R.Babuška (Delft University of Technology)

FuzzyTECH and CubiCalc (explanations)


30


The height of a fuzzy set Aheight of a fuzzy set A is the highest (maximum) value of its membership function, i.e. height(A) =

If a fuzzy set has a height 1, then it is called a normal fuzzy normal fuzzy setset; in contrast, if height(A) < 1, the fuzzy set is said to be subnormalsubnormal

A subnormal set is a fuzzy set that contains only elements with partial (<1) membership

In most of applications fuzzy sets are normal, and during the reasoning process usually subnormal fuzzy sets are generated

A set of all elements of the universal set U whose degree of membership in a fuzzy set A is nonzero is called the support support of a fuzzy set Aof a fuzzy set A, i.e. supp(A) =


)x( max iAxi

})x( | Ux{ A 0

31


A set of all elements x of the universal set U with a property A(x) = 1 (A is a fuzzy set) is called the core of a fuzzy set Acore of a fuzzy set A (core(A))


1

0

a b U = [a,b]

core(A)

supp(A)

height(A) = 1 (normal fuzzy set)

Membershipfunction has a

trapezoidal form

32


A fuzzy set whose support is a single point in the universe of discourse U is called a fuzzy singletonfuzzy singleton

Each fuzzy set A is associated with a family of crisp subsetsfamily of crisp subsets of A - their elements have such membership degrees that they are restricted to a crisp subset of [0,1]

A crisp set A that contains those xU for which is called an -cut-cut of a fuzzy set Aof a fuzzy set A

The general property of -cuts: for any fuzzy set A and two values 1, 2 [0,1] that satisfy to the condition 1< 2 the following is true:


)x(A

121221

2

A A A A A A

result a as and , A A

1

33



Membership degree

1

2

1

0 U =[a,b] a b

A

1 U

A

2 U

Fuzzy sets may be completely characterized by their -cuts: (decomposition theorem of fuzzy sets)

Example (Lecture hours)34

[0,1]

A A


Consider a fuzzy set A which is represented analytically in the universe of discourse U = [5,15] as follows:


otherwise , 0

14 x 6 if , 4 / ) |10-x| ( - 1 )x(A

10.9

0.7

0.5

0.3

0.1

0 5 10 15

Triangularmembership function

0.3-cut (crisp set)

set 0.7-cut

Ex

amp

le

35


step 1: Several particular values of are chosen from the unit interval [0,1] - they are 0.1, 0.3, 0.5, 0.7 and 0.9

step 2: converting each of the -cuts A to fuzzy sets for each xU using the formula:

(fuz_set) = A(x)


36

Ex

amp

le 10.9

0.7

0.5

0.3

0.1

0 5 10 15

Sometimes the theorem is referred as resolution principle (approximate

representation of membership function):

Sometimes the theorem is referred as resolution principle (approximate

representation of membership function):

Ux

)],x([ (x)AA


Fuzzy set A is convexconvex if for any elements x1, x2 and x3 from the universal set U, the relation x1< x2< x3 implies that

General property: the intersection of two convex sets produces a convex set

Convexity and -cuts:


37

] )x( ),x(min[ )x( AAA 312

1 1

a0

universe of discourse U = [0,a]


Generalization of set operations to fuzzy sets is not obvious Operations on fuzzy sets are crucial to the fuzzy inference

process In the rule IF (A or B) THEN C the true value of C is

the true value of the disjunction (operation or) Assume two fuzzy sets A and B are defined on the universe

of discourse U - three basic operations can be represented as follows:

Fuzzy set A is equal toequal to fuzzy set B if and only if A(x) = B(x), xU


38

)complement fuzzy (standard )x( - 1 )x( (3)

on)intersecti fuzzy (standard ])x( ),x([ min )x( (2)

union) fuzzy (standard ])x( ),x([ max )x( )(

AA

BABA

BABA

1



39

1

0 A(x)

universe of discourse U B(x)

AB

(x)

AB

(x)

Fuzzy sets overlap with their complementsoverlap with their complements (an element may partially belong to both fuzzy set and set’s complement) In contrast, classical (crisp) sets never overlap with their complements


Two fundamental laws of Classical Set theory - law of Excluded Middle and law of Contradiction

are violated in Fuzzy Set Theory (!!) Standard fuzzy operations are quite adequate in many

practical applications of FS, but they do not utilize the real expressive power of fuzzy sets (what are the other possibilities that may satisfy the requirements of practice? )

In practice, algebraic sumalgebraic sum (1’) and algebraic productalgebraic product (2’) are used for a definition of union and intersection of two fuzzy sets, respectively:


40

)x((x) )x( )'(

)x((x) - )x()x( )x( )'(

BABA

BABABA

2

1

U AA AA


General notation:

Operator s is called an s-norms-norm if it satisfies to the following axioms for any x, y, z and w [0,1]:

Some of the operators (s-normss-norms) that “model” (i.e. extend) fuzzy union:

operator) nconjunctio (fuzzy )]x( ),x([t )x(

operator) ndisjunctio (fuzzy )]x( (x),[ s )x(

BABA

BABA


41

vity)(associati z] y],s[x,s[ ]z]s[y, s[x, [Ax.4]

vity)(commutati x]s[y, y]s[x, [Ax.3]

ity)(monotonic w yand z x if w]s[z, y]s[x, [Ax.2]

identity)-(zero x s[x,0] x]s[0, 1, s[1,1] [Ax.1]

see the next slide...


(S1) Drastic sum:

(S2) Hamacher sum:

(S3) Dubois-Prade class:

(S4) Yager class:

Note:Note: for arbitrary fuzzy sets A and B membership values x and y stand for A(x) and B(x), correspondingly


42

0 yand 0 x if 1,

0 y)min(x, if y),max(x, ]y,x[sD

yx - 1

y2x - y x y]s[x,

[0,1]

, y)- 1 x, - 1 ,max(

y)x, ,-min(1 - yx - y x ]y,x[s

[[0, ,) y (x 1,min ]y,x[s1

Continuation: t-norms (triangular norms)Continuation: t-norms (triangular norms)


Operator t is called an t-normt-norm (triangular norm) if it satisfies to the following axioms for any x, y, z and w [0,1]:

Some of the operators (t-normst-norms) that “model” (extend) fuzzy intersection:

(T1) Drastic product:

(T2) Hamacher product:


43

vity)(associati z] y],t[x,t[ ]z]t[y, t[x, [Ax.4]

vity)(commutati x]t[y, y]t[x, [Ax.3]

ity)(monotonic w yand z x if w]t[z, y]t[x, [Ax.2]

identity)-(one x t[x,1] x]t[1, 0, t[0,0] [Ax.1]

1 yand 1 x if 0,

1 y)max(x, if y),min(x, ]y,x[tD

yx - y x

yx y]t[x,

More still to come...More still to come...


(T3) Dubois-Prade class:

(T4) Yager class:

Self-studying exercise:Self-studying exercise: Prove that the Yager t-norm (class T4) converges to the min operator when the parameter is in the infinite limit :

An important properties of s-norms and t-norms can be summarized as follows:


44

[0,1] , y)x, ,max(

yx ]y,x[t

[[0,

, y)- (1 x)- (1 1,min - 1 ]y,x[t1

)]x( ),x(min[ (i.e. y]min[x, ]y,x[t lim BA


s-norms are bounded below by maxmax (standard fuzzy union) and bounded above by drastic sumdrastic sum (S1):

t-norms are bounded below by drastic productdrastic product (T1) and bounded above by minmin (standard fuzzy intersection):

s-norms (a set of fuzzy disjunction operators) are often called triangular conormstriangular conorms or shortly, t-conormst-conorms

The alternative forms of operators AND and OR are called compensatory operators (they compensate the

strictness of min and max operators proposed by L.A.Zadeh)


45

])x( ),x([ s ])x( ),x([ s ])x( ),x([ max BAD

BABA

])x( ),x([ min ])x( ),x([ t ])x( ),x([ t BABABAD


Operator c is called a fuzzy complementfuzzy complement if it satisfies to the following axioms for any x and y [0,1]:

Some of the operators that “model” (extend) fuzzy complement:

(C1) Sugeno’s complement:

(C2) Yager’s complement:

Demonstration (MATLAB environment)


46

n)(involutio x ]c[x]c[ [Ax.3]

ity)(monotonic y x if c[y] c[x] [Ax.2]

(boundary) 0 c[1] and 1 c[0] 1].Ax[

[ 1,( , x 1

x - 1 ]x[c

[ (0, ,)x - (1 ]x[c

1


Main types of membership functions (MF):

(a) Triangular MF is specified by 3 parameters {a,b,c}:

(b) Trapezoidal MF is specified by 4 parameters {a,b,c,d}:


47

c x if 0,

c x b if b),-(cx)-(c

b x a if a),-(ba)-(x

a x if , 0

)c,b,a:x(trn

d x if 0,

d x c if c),-(dx)-(d

c x b if 1,

b x a if a),-(ba)-(x

a x if 0,

d)c,b,a, : x(trp

More to come...More to come...


(c) Gaussian MF is specified by 2 parameters {a,}:

(d) Bell-shaped MF is specified by 3 parameters {a,b,}:

(e) Sigmoidal MF is specified by 2 parameters {a,b}:


48

2

2a) - (x-exp )a, : x(gsn

2b

a - x

1

1 )b,a, : x(bll

b)-a(x-e1

1 b)a, : x(sgm



49

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Assume X and Y are two arbitrary classical sets. The Cartesian product of sets X and Y is a set of all ordered pairs (xi,yj), xiX, yjY; that is

Suppose X = { x1, x2, x3, x4 }, Y = { y1, y2, y3 }; the set XxY consists of 12 ordered pairs (xi,yj), i=1,2,3,4, j=1,2,3 - in this case, a graphical representation (as nodes of a grid) is convenient…

Generalization of Cartesian product to n arbitrary classical sets X1, X2,…, Xn:


50

} m1, j ,n1, i ,Y y,Xx | )y,(x { YX jiji

} X x , ... ,X x ,X x | ) x , ... ,x ,x ( { X...XX nkn2

j21

i1

kn

j2

i1n21

nn2211 )card(X , ... , )card(X , )X(card



Y

y3

y2

y1

x1 x2 x3 x4 X

12 pairs – elements ofthe set XxY

51

If the cardinality of the set X is n(X) and the cardinality of the

set Y is n(Y), then the cardinality of the Cartesian product (set of elements) is

n(XxY) = n(X)*n(Y) Y

y2

y1 X =[x1,x2], Y=[y1,y2]

x1 x2 X

Set XxY (Cartesianproduct)

See the previous slide...


Omitting superscripts, an ordered sequence of n elements (x1, x2, …, xn) is called an ordered n-tupleordered n-tuple

A subset of the Cartesian product is called an n-ary relationn-ary relation built over domains (sets) X1,X2,…, Xn - if n=2, the binary relation on X1 and X2 (from X1 to X2) can be formally defined as a set of ordered pairs in X1

xX2 ; that is

where P(x1,x2) is a property to which each pair (x1,x2) satisfies

Example Suppose that both X1 and X2 are sets of real numbers [5,20], i.e. X1 = X2 = [5,20].

The binary relation (X1,X2) «less than» has


52

22112121 X x ,X x ),x ,P(x | )x ,(x

n21 X ... XX


the following formal analytical representation:

Graphical form of the binary relation :

A binary relation can be also represented by


53

X x ,X x ,x x | )x ,(x 22112121

20

X2

5

5 20 X1

Set is built on theCartesian product X1xX2

Points located onthe diagonal

(x1 = x2) of the

square do notbelong to the set.

The area is definedas follows

x1 5

x2 20x1 < x2


means of membership functionmembership function:

(arbitrary n-ary relation is a mapping:

(X1,X2,…, Xn) : X1xX2

x …xXn {0,1} )

If a set X1xX2 is finite, then the values of function can be

collected into a relational matrixrelational matrix Relations are intimately involved in logic, approximate

reasoning, rule-based systems, etc. A rule «IF x is A THEN y is B» describes a relation between the

variables x and y - as implication A B, rule expresses a mapping (subset of Cartesian product) between input and output domains


54

)x ,...,x,(x if 0,

)x ,...,x,(x if , 1

n21

n21)X ..., ,X,(X n1 2


A fuzzy relationfuzzy relation generalizes the concept of classical (crisp) relation introducing a degree of membership for each ordered n-tuple (x1, x2,…, xn) in

For 2D case it can be defined as follows:

Examples

a) x1 is close to x2 (both x1 and x2 are numbers)

b) if x1 is medium, then x2 is high (x1 is an observed state, whereas x2 is a result state or action)

c) x1 is similar to x2 (x1 and x2 can be objects, human beings, properties)


55

XX )x,(x | ) )x ,x( ),x,(x ( ~

212121~21

n21 X ... XX

relation fuzzy a denotes ~


Formally, fuzzy relation in can be defined as a fuzzy set

where is a mapping:

In different sources notations and can be used for crispcrisp and fuzzyfuzzy relations, correspondingly (if it is clear from the

contents which of two relations is used, sign ~ can be dropped)

means «membership degree of the ordered n-tuple in the fuzzy relation , where

», or «a degree to which fuzzy relation holds true for objects »

Example (Lecture hours)


56

X ...XX )x ..., ,x,(x | ))x..., ,x,x( ),x ..., ,x,((x ~

n21n21n21~n21

~ n21 X ... XX

~ [0,1] X ... XX n21

R R~

)x ..., ,x,x( n21~)X ..., ,X,X(

~n21

n1, i , X x ii ~)x, ... ,x,x( n21


Suppose that X1 and X2 are line segments [0,50] and [20,40], respectively, on the set of real numbers . A fuzzy relation «x1 is approximately equal to x2» may be defined by the membership function


57

2211)x(x-

)X,X(~ Xx ,Xx , e

221

21

)X,X(

~

~2111

Fuzzy relations enhance our capability to deal with

relational concepts expressed in a natural

language !

21 XX 2X

1X

Exa

mpl

e


A fuzzy relation «x1 is much larger than x2» may be defined by the membership function


58

)x(x0.5- )X,X(~

21212 e1

1

Membership values; line segment [0,1]

Fuzzy relations are also fuzzy sets, and

fundamental properties of fuzzy sets hold for fuzzy relations as well


Fuzzy set operations (complements, unions, intersections) are applicable to fuzzy relationsfuzzy relations too


59

Sugeno’s complement

(parameter = 2)Standard

complement


Operations on fuzzy sets defined on different universal domains produce a multidimensional fuzzy set (the following shows graphically the result of operation )


60result of intersection AB

1

Cartesianproduct XxY

Domain X

Domain Y

Membership values

Fuzzy set A

Fuzzy set B

YX )BA(



61


(MATLAB 5.2 environment)

Contour plot

Fuzzy sets Ai and Bi have Gaussian and bell-shaped forms

IF x is Ai ...

THEN y is Bi


A fuzzy rule is described formally by a fuzzy relationfuzzy relation between antecedent and consequent, and it defines a partition (fuzzy relation ) in the space (Cartesian product). The union of all partitions

forms a fuzzy graphfuzzy graph (term introduced by L.A.Zadeh) of a fuzzy model. The more partitions (patches) the more accurate description of the functional dependency Y=f(X), where f is a crisp function

Fuzzy system approximates a crisp function f by means of patches – the approximation is uniform, and it allows specification of error level (accuracy) in advance:


62

y)(x, B A R R in,1i

in1,i

i*

YXiR

set) (universal Xx , ε )x(F - )x(f *


A patch covering leads to major serious problem of fuzzy systems: exponential explosion of rulesexponential explosion of rules (number of rules) as a result of increasing the number of input variables

Fuzzy Approximation Theorem (B.Kosko, L.A.Zadeh)


63

A function can be approximated to «any prescribed accuracy

provided that sufficient … fuzzy rules are

available» [S.Wu and M.J.Er]

References: 1. L.Wang. Fuzzy Systems are Universal Approximators // Proc. Int.Conf.Fuzzy Syst., 1992

2. B.Kosko. Fuzzy Systems as Universal Approximators // IEEE Transactions on Computers, vol.43, #11, 1994

Domain X

Crisp function f

Fuzzy graph R* (fuzzy relation)


In a framework of fuzzy rules as conjunctions (IF x is Ai THEN y is Bi Ai Bi Ai Bi =

(functional relationshipfunctional relationship between input and output on a base of linguistic terms, i.e. Ai and Bi, i = 1,n)

(A) Mamdani implication: (correlation-minimum)

(B) Larsen implication: (correlation-product : dilution of membership values that are both small)

cases (A) and (B)

Compound fuzzy propositions (e.g. x1 is Ai1 and x2 is Ai2) are interpreted as fuzzy relations


64

]μ ,μ[ tii BA

])y(μ ),x(μ[ min )y,x(μiii BAR

)y(μ )x(μ )y,x(μiii BAR



(1) x1 is Ai1 and x2 is Ai2 : ,

where t : [0,1][0,1] [0,1] is a t-normt-norm

(2) x1 is Ai1 or x2 is Ai2 : ,

where s : [0,1][0,1] [0,1] is a s-norms-norm

As an implication, fuzzy rules “represents” human abilities of imprecise reasoning

Fuzzy rule is an implication between fuzzy propositions:

IF <fuzzy proposition>Ai THEN <fuzzy proposition>Bi (Ai Bi )

Fuzzy rule describes an entailmententailment of Bi by Ai (Ai entails Bi)

Fuzzy logic can be considered as a generalization of a classical binary and multivalued logic


65

])x(μ ),x(μ[ t )x,x(μ 2A1A21AA 2i

1i

2i

1i

])x(μ ),x(μ[ s )x,x(μ 2A1A21AA 2i

1i

2i

1i


“The foundation of a fuzzy implication rule is the narrow sense of fuzzy logic”

In the classical propositional calculuspropositional calculus (algebra of propositions), implication PQ, where P and Q are two simple propositions, is logically equivalent (gives the same truth table values) to

(1) and

(2)

As Ai and (or) Bi have fuzzy predicates,the implication becomes a fuzzy implication

Two-valued logic:


66

P Q P Q

1 1 1

1 0 0

0 1 1

0 0 1

Q) or P(not QP

Pnot or Q) and (P P)QP(

calculus) onal(propositi PQ)(P QP

n)implicatio (material QP QP

forms (1) and (2)


For the crisp propositions P and Q the implication PQ becomes global, i.e. the truth table covers all possible combinations of 0/1 values of P and Q)

Fuzzy implication can be considered as a local one, i.e. it gives a large truth value only when both and have large truth values

Having a rule IF x is Ai THEN y is Bi, we may propose a following interpretation:

IF x is Ai THEN y is Bi ELSE <nothing>

(each rule covers a local part of the whole “working space”) Implication as a conjunctionconjunction: Ai Bi Ai Bi

(Mamdani (min) and Larsen (product) [slide 64] implications are the most widely used in fuzzy systems and fuzzy control)


67

Q~

P~ P~ Q

~


Fuzzy rules can be considered as not being local as well, and this fact opens a way to classification of fuzzy rules as:

(A) fuzzy mapping rules (FMR)

(B) fuzzy implication rules (FIR)

Case (A): FMR describe a functional dependencyfunctional dependency between system’s inputs and outputs by means of linguistic terms (a totality of rules is represented by a fuzzy graph). A collection of fuzzy mapping rules are often called fuzzy model

Case (B): FIR describe an implication logic relationshipimplication logic relationship between two fuzzy propositions that use linguistic terms and hedges (generalization of two-valued logic)


68

Both types of rules:(1) are represented as a fuzzy relations between antecedent and

consequent parts(2) use compositional rule of inference as an inference engine

similarities


!

A major stages in fuzzy modelingfuzzy modeling are as follows:

1. fuzzy partition

2. mapping of regions to local models

3. fusing of local models into a global model

4. defuzzification (quantitative summary of possibility distribution of model’s output)

The representatives of fuzzy implication familiesfuzzy implication families (FI) are shown on the next slide (form FI1’ is also refered in

the literature as Dienes-Rescher implication)


69

But…(1) the semantics of relations is different(2) different operators are used in compositional rule of inference(3) two types of rules behave the same in the case when

antecedent parts of rules are satisfied, but the behavior is different if antecedents are not satisfied…

differences

Fuzzy Rule-Based Modeling

Fuzzy Rule-Based Modeling


Selected fuzzy implication (FI) operators:

(FI1) Zadeh’s classical maximum FI:

(FI1’) Zadeh’s classical maximum FI:

(FI2) Lukasiewicz’ implication:

(FI3) Godelian FI:

(FI4) Standard sequence implication: (it is shown on the next slide…)


70

])x(μ - 1 ],)y(μ ),x(μ[ min[ max )y,x(μiiii ABAR

])y(μ ),x(μ - 1[ max )y,x(μiii BAR

] ])y(μ ))x(μ - (1[ 1,[ min )y,x(μiii BAR

)y(μ )x(μ if ),y(μ

)y(μ (x)μ if , 1 )y,x(μ

iii

iii BAB

BAR

equivalent to (FI1) when

)x(μ )y(μii AB


Standard sequence implication:

“… Although some have been more extensively used, there is no such thing as the fuzzy implication operator and any practical application should consider different alternatives, checking the effectiveness of each one” (A.Kandel, R.Pacheco, A.Martins, S.Khator. “The foundations of rule-based computations in fuzzy models” )

(fuzzy implication operators: (1) Zadeh’s, (2) Lukasiewicz’, (3) Godelian)PS. Godelian fuzzy implication is also referred in the literature

as Brouwerian implication )


71

)y(μ )x(μ if , 0

)y(μ )x(μ if , 1 )y,x(μ

ii

iii BA

BAR



The relationship between implications:

Two questions of interest:

(1) What are the criteria for choosing a combination of fuzzy operators , and in order to obtain a certain form of fuzzy implication? For example, Lukasiewicz implication (FI2) is a member of a family that generalizes a material implication of a classical logic: (disjunction ) Yager’s s-norm with equal to 1 and (complement ) standard complementation “1-…”

(2) For the fuzzy implication what we mean by “equivalence”“equivalence” to and ?


72

YX y)(x, all for - FI2] n,implicatio cz[Lukasiewi )y,x(μ

]FI1' n,implicatio Rescher-[Dienes )y,x(μ

FI1] n,implicatio [Zadeh )y,x(μ

i

i

i

R

R

R

Q~

P~ QP P)QP(


When a fuzzy relation is NOT finite, and it is defined in n-dimensional space, we have to specify it analytically (through appropriate formula)

A fuzzy relation on a finite Cartesian product X1xX2 is usually

represented by a fuzzy relational matrixfuzzy relational matrix with elements taken from [0,1]

Example Assume X1 = { a1, a2, a3 }, X2 = { b1, b2, b3, b4 } are two sets of cities. The relational concept

«close» (distance expressed in Km) is represented by the (3,4)-matrix:


73

4321

3

2

1

bb b b

a

a

a

0.510.950.7

0.310.61

000.20.4

: R~

0.2 )b,a(close""

)b,a(

21

21R~


Two kinds of questions we can keep in mind:

a) what is the degreedegree that particular pair of cities are considered to be close to each other? (membership function of a fuzzy set)

b) what is a possibilitypossibility that a short distance (closeness to each other) corresponds to a specific pair of city ai ( ) and city bj ( )? (possibility distribution of a short distance (closeness))

Composition of relations

Composition of two crisp binary relations R1 and R2 requires their compatibilitycompatibility


74

1,3 i1,4 j

ZY R Y,X R 21 21 R R as denoted is

relations two of ncompositio the

Crisp case


Relation T=R1°R2 (composition of two crisp relationscomposition of two crisp relations) consists of those pairs (x,z) ,xX, zZ, of the Cartesian product XxZ that via the given relations R1 and R2 «share» at least one element yY

Example Assume that X = [20,40] , Y = [0,50] ,

Z = [10,40]; X,Y,Z (set of real numbers). Two crisp relations R1 XxY and R2 YxZ are defined (shown on the next slide), and it is required to find their composition:

- relation T(as a result of compositional operation) is a subsetof the Cartesian product XxZ


75

(see the next slide for the graphical representation)

[20,40] x x, z 10, z | z)(x, )z,x(T



76

5 0

4 0

1 0

2 0 4 0 X 5 0 Y

z" to equal or greater is y" y"to equal is x"

ZY R relation crisp YX R relation crisp 21

line z=yCartesian product

XxY

relation R2 YxZ

Both relations are defined in two-dimensional Euclidean space


Two common forms of composition operationcomposition operation:

a) max-min composition

or in terms of 0/1 membership functions:

b) max-product (in general, max-star) composition

For example, max-product composition has a following form:


77

])z,y(R ),y,x(R[ min max )z,x)(R(R )z,x(T 21Yy

21

) ( )z,x( )z,y(R)y,x(RYy

T 21

])z,y(R)y,x(R[ max )z,x)(R(R )z,x(T 21Yy

21

) ( )z,x( )z,y(R)y,x(RYy

T 21

sign

denotes any t-norm


It is stressed that “max-min method of composition effectively expresses the approximate and interpolative reasoning used by humans when they employ linguistic propositions for deductive reasoning” (T.Ross. “Fuzzy Logic with Engineering Applications”, McGraw-Hill, 1995; N.Vadiee. “Fuzzy rule based expert systems – I”, 1993)

A main goal of fuzzy logic is to form a foundation for reasoning (inference) with imprecise propositions; such reasoning is called approximate reasoning

Consider a given rule IF x is A THEN y is B, where A and B are fuzzy sets (fuzzy propositions, fuzzy predicates), and a

fact: x is A’ (A’ and A are not necessarily identical). The result produced by fuzzy inference engine:

y is B’ = A’R, where R is a fuzzy relation whichrepresents an implication (x is A) (y is B)


78


An ordinary-language term represented by a fuzzy set is called linguistic valuelinguistic value

A linguistic variable can be considered as a composition of a symbolic variable and a numeric variable

Linguistic variable is a fundamental element in human knowledge representation

Transition from crisp mathematics to fuzzy mathematics by means of fuzzy set theory has allowed mathematical representations to become compatible with expressions in natural language

Linguistic hedgeshedges are special linguistic terms by which other (primary) linguistic terms are modified

Linguistic hedge (modifier) may be interpreted as an unary unary operator operator that modifies the meaning of a fuzzy set


79


Some of the most commonly used operators (hedgeshedges) and their functions are as follows:


80

G r o u p 1 ( c o n c e n t r a t i o n h e d g e s )

v e r y ( A ) = 2)x(A e x t r e m e l y ( A ) = 3)x(A

v e r y , v e r y ( A ) = 4)x(A p l u s ( A ) = 251 .)x(A G r o u p 2 ( d i l a t i o n o r d i l u t i o n h e d g e s )

m o r e o r l e s s ( A ) = s l i g h t l y ( A ) = n o t s o ( A ) = )x(A

m i n u s ( A ) = 750 .)x(A s o m e w h a t ( A ) = 30 .)x(A G r o u p 3 ( i n t e n s i f i c a t i o n h e d g e s )

r e a l l y ( A ) =

1.0 A(x) 0.5 A(x))- (12 - 1

0.5 A(x) 0 A(x)22

2

Be cautious when treating the meaning

of NOT…

Functions of hedgesFunctions of hedges


Brief comments on the previously mentioned functions:

(A) Concentration functionConcentration function: keeping the original shape (form) of a membership function, “shrink” it over the universe of discourse, and a level of concentration can be adjusted by changing a value of power (>1) applied to membership values)

(B) Dilution functionDilution function: opposite to concentration function – it results in “spreading” of membership function over universal set through changing a power (<1) of MF values

(C) Contrast intensificationContrast intensification: changes slightly a shape of membership function, “widening” MF for possibility values > 0.5 and “narrowing” it when the latter is 0.5

(D) NegationNegation (not mentioned above): “mirrors” imaging

of MF with respect to (x)=0.5


81




82

Contrastintensification

Concentration & dilution

Negation


A definition of linguistic variablelinguistic variable proposed by L.A.Zadeh (“The concept of a linguistic variable and its application to approximate reasoning I,II”, Information Sciences,8,pp.199-251,301-357) can be formulated as follows:

A linguistic variable is characterized by (a) its name N (b) a set L of linguistic values it can take (c) universal set U (physical domain) in which it is defined (d) a rule R that associates each linguistic value of L with a fuzzy set in U

Some observations on MF shape analysisMF shape analysis:

(1) The location and granularity (number) of MFs are the two relatively more important (from the standpointof affecting performance of the fuzzy inference


83


algorithm) issues compared to the shape of each membership function

(2) The shape of MF characterizes uncertainty in the fuzzy variable – in general, a high level of detail in shape design is considered as a conceptual error

(3) Most of applications nowadays use simple convex membership functions (due to computational simplicity and relative easiness of implementation in hardware, the commonly practical are piece-wise-linearpiece-wise-linear forms, e.g. triangular and trapezoidal MFs)

(4) In most cases heights of membership functionsof antecedent variables are equal to 1.0 (normalnormalsetssets); if the heights of MFs in consequent part of the rules is less than 1.0, then it may cause …


84


so-called paralysis of implication resultsparalysis of implication results:


85

1 0 input domain X output domain Y xa xb

Dead zone ]b,1]

Interval of paralysis

height < 1

(5) OverlappingOverlapping is an important design consideration: each antecedent MF should overlap only with imme- diate neighboring membership functions, i.e.


formally it can be expressed as: . Overlap of MFs determines a degree of cooperationdegree of cooperation (or, switching degree) between corresponding rules

(6) Each consequent MF represents one rule; a heightheight of MF determines the strength of contribution from each rule, and MF’s locationMF’s location affects the actual decision value

(7) In general, shape modifications of antecedent MFantecedent MF (compared to those of consequent ones) produce more significant effects on the output behavior

(8) Adjustment (symmetric changes in overlap) of all consequent MFsconsequent MFs do not produce significant changes of output’s behavior

* * * * * end of the Section 2 * * * * *


86

1i ,i ,1ij , AA ji


Section 3 Outline

Basic Principles of Inference in Fuzzy Logic (Entailment, Conjunction, Generalized Modus Ponens, Generalized Modus Tollens). Fuzzy IF-THEN rules. Canonical form

Fuzzy Systems and Algorithms. Approximate Reasoning Fuzzy Inference Engines. Graphical Techniques of

Inference. Fuzzification/Defuzzification Fuzzy System Design and its Elements (conceptual model).

Design Options

[ the discussion of these topics takes approximately 810 lecture hours. Examples are explained using fuzzyTECH, FL Toolbox packages andMATLAB demonstrations]


87


«Contrariwise» , continued Tweedledee, «if it was so, it might be; and if it were so, it would be; but as it isn’t , it ain’t. That’s logic»

(Lewis Carroll, Through the Looking Glass)

In the propositional logicpropositional logic the inference can be depicted as follows:

If the resulting premises are both true, then the conclusion is also true (the truth conclusion is inferred or deduced from truth premises – we call it a valid deductionvalid deduction). For example, the following form is invalid:


88

Q

P

QP

• P and Q are propositions (variables)• sign represents the relation “if-then”“if-then”• symbol (therefore) is placed before conclusion

Q

P

QP

premise 1: IF P THEN Qpremise 2: not P


Among basic inference rules (the premises logically imply rules’ conclusions) we can mention the following:

(a) Modus Ponens (b) Modus Tollens (Lat. “method of affirming”): (Lat. “method of denying”):

(c) Hypothetical Syllogism (many mathematical arguments contain a chain of if-thenif-then statements):

A test for validity of Modus Tollens is as follows:


89

P

Q

QP

Q

P

QP

SP SQ

QP

The implication connective () is especially important as a basis of fuzzy implication rules


premises are conjuncted in the antecedent part of implication, and conclusion form its consequent part:


90

P)( )Q()QP(

P

Q

PQ

Q

(PQ)(Q)

0 0 1 1 1

1 0 0 1 0

0 1 1 0 0

1 1 1 0 0

How validity is checked?

How validity is checked?

Fuzzy Logic (FL) generalizes the notion of

truth values in classical logic, and provides a background for reasoningreasoning (inferencing) when …


corresponding conditions are only partially satisfied Approximate reasoningApproximate reasoning (original rule IF x is A THEN y is B):

1. Possibility distribution of the variable x2. Implication possibility from x to y

possibility distribution of y

Both fuzzy implicationfuzzy implication and fuzzy mapping rulesfuzzy mapping rules use a compositional rule of inference for calculation of output results, but there are still some differences…

If proposition P is described by set A X, and proposition Q is described by set B Y, then the classical implication PQ can be represented by the relation R as follows:

; its schematical representation (Venn diagram, Figure 1)


91

)YA()BA(R


is as follows:

A rule IF x is A THEN y is B can be expressed as a compound conditional statement IF x is A THEN y is B ELSE y is Nothing (no action). In general,expression IF x is A THEN y is B ELSE y is C


92

Y Y C B B A universal set X A universal set X Figure 1 Figure 2

See comments related to Figure 2 below…


can be expressed as a disjunction:

«IF x is A THEN y is B» «IF x is THEN y is C»

In the logic of compound statements this can be written as:

, where S is a proposition

described by set C Y (see Venn diagram, Figure 2)

Figure 1 (shaded area – truth domain of the implication PQ)

Figure 2 (shaded area – truth domain of the form , where )

Consider a given rule IF x is A THEN y is B, where A and B are fuzzy setsfuzzy sets (fuzzy propositions, fuzzy predicates) defined on universes X and Y, correspondingly,and a fact x is A’ (A’ and A are not necessarilyidentical)


93

)SP()QP(

A

)SP()QP( CB


Having a rule and a fact, what can we say about conclusion (consequent) B’ ?

Generalized Modus Ponens:

Rule (premise 1): x is A y is B

Fact (premise 2): x is A’

Infer (result produced by fuzzy inference enginefuzzy inference engine): y is B’

B’ = A’ R, where R is a fuzzy relationfuzzy relation (associations between the elements of two fuzzy sets) which represents an implication (x is A) (y is B)

Everyday (“reasonable”) intuitive criteria which relate Fact (premise 2) and the Conclusion (infer) can be summarized in the table form as follows:


94


In approximate reasoning, Generalized Modus Ponens is an inference mechanisminference mechanism that allows to obtain imprecise conclusion from imprecise (vague) fact

Classical (fundamental) Modus Ponens: The reasoning process when a given implication PQ istrue, and proposition P has a true value leads to the


95

Criterion

Fact: x is A’

Infer: y is B’

Comments

criterion 1 x is A y is B Fundamental MP

criterion 2 x is very A y is B Consider A’ A

criterion 3 x is very A y is very B Extension of hedges to fundamental MP

criterion 4 x is more or less A y is B Closeness of A’ to A

criterion 5 x is more or less A y is more or less B Extension of hedges to fundamental MP

criterion 6 x is not A y is not B Think about ELSE y is not B part in the rule (premise 1)

criterion 7 x is not A y is unknown Rule does not say anything about mismatch case


conclusion ( Q):

Compositional rule of inference (classicalcase):


96

10 10

00 proj

10

11

11

00 proj

Q)(PCylEx(P) proj )Q(tv

QQ

Q

repetition of all points x of P in the domain Q (variable y)

repetition of all points x of P in the domain Q (variable y)

101

110

10

P

Q

Implication PQ truth

table

{0,1} y

)y,x(R)y(tv )QP(PQ{0,1}x

P

relation R(x,y) relation R(x,y) max { min… } max { min… }


The inference process based on Generalized Modus Ponens (GMP) can be implemented differently depending on the calculus used, i.e. the inference process is not unique

Compositional operators (the most commonly used in practice):

a) max-min compositional operator

b) max-product compositional operator

In general, GMP states that for the premise x is A’ and the fuzzy relation R(x,y) A B (the rule IF x is A THEN y is B), the inferred set B’ (conclusion y is B’) iscalculated as follows: (see the next slide)


97

))y,x(μ ),x(μ min( max)y(μ BAA'Xx

'B

))y,x(μ)x(μ max)y(μ BA'AXx

'B


(see the previous slide)

where X is a domain of the variable x (antecedent domain), and the letter «t» denotes a t-norm. As a result, represents a possibility distribution over the domain of output variable

Similarly, Generalized Modus Tollens states the inferred set A’ (conclusion x is A’):

where Y is a domain of the variable y (B’Y, premise 2: y is B’) Important note: “Even though both fuzzy implications and fuzzy

mapping rules use the compositional rules of inference to compute their inference results, their usages differin two ways. First, the compositional rule of inference


98

]))y,x(μ ),x(μ[ t max)y(μ BAA'Xx

'B

]))y,x(μ ),y(μ[ t max)x(μ BAB'Yy

'A

)y(μ 'B


is applied to a set of fuzzy mapping rules that approximate a functional mapping. Second, the fuzzy relation of a fuzzy mapping rule is a Cartesian product of the rule’s antecedent and its consequent part. An entry in the fuzzy implication relation, however, is the possibility that a particular input value implies a particular output value” (J.Yen, R.Langari. “Fuzzy Logic. Intelligence, Control, and Information”, Prentice Hall, 1999)

Consider a fuzzy rule base (totality of n rules) : IF x is Ai THEN y is Bi, where Ai X and Bi Y are fuzzy setsfuzzy sets; x and y are input and output variables, respectively (it can be expanded to multiple input/output case)

Composition Inference (CI):

a) calculation of Ri(x,y) (Ai Bi) for each i-th rule

b) calculation of a single relation defined on XxY as

b1) Mamdani combination:


99n

1iiM )y,x(R )y,x(R


(constituent elements of fuzzy rule base are independent), or

b2) Gödel combination: (rules are dependent)

c) generation of a fuzzy output B’Y for a given input A’

Generalized Modus Ponens (GMPGMP), Mamdani combination:

One of the commonly used inference engines (algorithms) is a Product EngineProduct Engine which is based on:

a) Larsen’s implication (product operator)


100

)y,x(R )y,x(Rn

1iiG

)B(A , A't max μ ),x(μ t max)y(μn

1iii

Xx)y,x(RA'

Xx'B i

union of conjunctionsn

1iR )y,x(μ

i


b) Composition Inference (CI) with Mamdani combination (s-norm: standard maximum)

c) t-norm: algebraic product, i.e. formally, Product EngineProduct Engine has the following representation:

(Generalized Modus Ponens)

Individual Rules Inference (IRI):

a) calculation of Ri(x,y) (Ai Bi) for each i-th rule

b) calculation of Bi’Y (fired THEN-part set) as a composition of input A’ and implication Ai Bi using GMP

c) generation of a fuzzy output B’Y as a combi-


101

)y(μ)x(μ max)y,x(μ

))y,x(μ)x(μ ( max)y(μ

iiM

M

BAy,x

R

RA'Xx

'B

Larsen implicationLarsen implication

standard maximumstandard maximum


nation of separate Bi’ ,i.e

b1) Mamdani combination ,or

b2) Gödel combination

One of the commonly used inference engines (algorithms) is a Minimum EngineMinimum Engine which is based on:

a) Mamdani implication (minimum operator)b) Individual Rules Inference (IRI) with Mamdani combination (s-norm: standard maximum)

c) t-norm: standard minimum


102

)y(μ )y(μ

)y(μ )y(μ

n

1iB'B

n

1iB'B

'i

'i

Zadeh Engine Lukasiewicz Engine

a) Zadeh’s Implication a) Lukasiewicz Implication b) both Engines use Individual Rules Inference with Gödel combination c) both Engines use standard minimum as a t-norm

Zadeh Engine Lukasiewicz Engine

a) Zadeh’s Implication a) Lukasiewicz Implication b) both Engines use Individual Rules Inference with Gödel combination c) both Engines use standard minimum as a t-norm

Two other

Inference

Engines

Two other

Inference

Engines


Schematically, CI and IRI can be represented as follows: • Composition Inference

• Individual Rules Inference


103

Rule 1 Implication 1 input A’ Rule 2 Implication 2 Combination Inference …….. (Aggregation) Rule Defuzzification (Composition) Rule n Implication n GMP

Rule 1 Implication 1 Composition Rule 2 Implication 2 Composition Combination …….. Composition (Aggregation) Defuzzification Rule n Implication n Composition input A’


DefuzzificationDefuzzification is a process of conversion of fuzzy output (possibility distribution of the output) to precise (crisp) value

Among the major defuzzification techniques we can mention:

a) Mean of Maximum (Middle of Maxima) method (MoMMoM)

b) Centroid (Center of Area, Center of Gravity) method (CoACoA)

MoMMoM calculates the average of all variable values having maximum membership degree


104

1 y* - defuzzified output 0 a b output y y*

CoACoA calculates the weighted

average of the fuzzy output


If the inferred fuzzy output is denoted as B’(y), then the CoACoA defuzzified output is calculated as follows:

Which of defuzzification methods is the best? One of criteria is plausibilityplausibility – to be plausible, y* should lie approximately in the middle of the support region of B’(y) and have a high degree of membership in B’(y) (see also C.Thomas, H.Hellendoorn. “Defuzzification in fuzzy controllers”, Intelligent and Fuzzy Systems, vol.1, pp.109-123,1993)


105

iiB'

iiiB'

*

YB'

YB'

*

domainoutput discrete - )y(y)y( y

domainoutput continuous - dy)y(ydy)y( y




106

fuzzyTECH 5.31

CubiCalc 2.0

SINE 1.0

Fuzzy Logic Toolbox 2.0

emu, computer engineering department 1999/2000 academic year, spring semester

Documents

fuzzy engineering

fuzzy sets

word fuzzy

fuzzy modelling

evolution of fuzzy logic

foundation of fuzzy

fuzzy set theory

brief history of fuzzy