Fuzzy Rule-Based Models

11 Fuzzy Rule-BasedModels

Fuzzy Systems EngineeringToward Human-Centric Computing

11.1 Fuzzy rules as a vehicle of knowledge representation

11.2 General categories of fuzzy rules and their semantics

11.3 Syntax of fuzzy rules

11.4 Basic functional modules

11.5 Types of rule-based systems and architectures

11.6 Approximation properties of fuzzy rule-based models

11.7 Development of rule-based systems

Contents

Pedrycz and Gomide, FSE 2007

11.8 Parameter estimation procedure for functionalrule-based systems

11.9 Design of rule-based systems: consistency,completeness and the curse of dimensionality

11.10 Course of dimensionality in rule-based systems

11.11 Development scheme of fuzzy rule-based models


11.1 Fuzzy rules as a vehicle ofknowledge representation


Rule conditional statement

If input variable is A then output variable is B

A and B: descriptors of pieces of knowledge

rule: expresses a relationship between inputs and outputs

Example

If the temperature is high then the electricity demand is high

If and then parts ....... formed by information granules

sets rough sets fuzzy sets


Rule-based system/model (FRBS)

FRBS is a family of rules of the form

If input variable is Ai then output variable is Bi

i = 1, 2,..., c

Ai and Bi are information granules

More complex rules

If input variable1 is Ai and input variable2 is Bi and ..... then output variable is Zi

multidimensional input space (Cartesian product of inputs) individual inputs aggregated by the and connective highly parallel, modular granular model


11.2 General categories of fuzzyrules and their semantics


Multi-input multi-output fuzzy rules

If X1 is A1 and X2 is A2 and ..... and Xn is Anthen Y1 is B1 and Y2 is B2 and ..... and Ym is Bm

Xi = variables whose values are fuzzy sets AiYj = variables whose values are fuzzy sets Bj

Ai on Xi, i = 1, 2,...,n

Bj on Yj, j = 1, 2,...,m

No loss of generality if we assume rules of the form

If X is A and Y is B then Z is C


Certainty-qualified rules

If X is A and Y is B then Z is C with certainty

[0,1]

: degree of certainty of the rule

= 1 rule is certain


Gradual rules

the more X is A the more Y is B

relationships between changes in X and Y

captures tendency between information granules

Examples:

the higher the income, the higher the taxes

the lower the temperature, the higher energy consumption


Functional fuzzy rules

If X is Ai then y = f (x,ai)

f : X Y

xRn

Rule: confines the function to the support of granule Ai

f : linear or nonlinear (neural nets, etc..)

Highly modular models


11.3 Syntax of fuzzy rules


Backus-Naur form (BNF)

If_then_rule ::= if antecedent then consequent{certainty}gradual_rule ::= word antecedentword consequent

word ::= more {less}antecedent ::= expressionconsequent ::= expressionexpression ::= disjunction{and disjunction}disjunction ::= variable{orvariable}

variable ::= attribute is valuecertainty ::= none{certainty [0,1]}


Construction of computable representations

Main steps:

1. specification of the fuzzy variables to be used

2. association of the fuzzy variables using fuzzy sets

3. computational formalization of each rule using fuzzyrelations and definition of aggregation operator to combinerules together


11.4 Basic functionalmodules of FRBS


I

n

p

u

t

I

n

t

e

r

f

a

c

e

Rule Base

Data Base

Fuzzy Inference O

u

t

p

u

t

I

n

t

e

r

f

a

c

e

X Y

General architecture of FRBS

Fuzzy if-then rules(input-output relationship)

Parametersof the FRBS

Process inputs and rules(approximate reasoning)


Input interface

(attribute) of (input) is (value)

the temperature of the motor is high

Canonical (atomic) form

p: X is A temperature (motor) is highX A

fuzzy set

Low Medium High

x (C)Pedrycz and Gomide, FSE 2007

Multiple fuzzy inputs: conjunctive canonical form

p : X1 is A1 and X2 is A2 and ..... and Xn is An conjunctive canonical form

Xi are fuzzy (linguistic) variables

Ai : fuzzy sets on Xi

i = 1, 2, ..., n

Compound proposition induces a fuzzy relation P on X1X1... Xn

)()()()()(1

221121 iin

innn xATxtAtxtAxAx,,x,xP

=

== KK

p : (X1, X2 , ....., Xn) is P

t (T) = t-norm


0 5 100

10

20

x

y

(b) Contours of P

Example Fuzzy relation associated with (X,Y) is P

Triangular fuzzy sets A1(x,4,5,6) = A, A2(y,8,10,12) = B

t-norm: algebraic product

P


q : X1 is A1 or X2 is A2 or ..... or Xn is An disjunctive canonical form

Xi are fuzzy (linguistic) variables

Ai : fuzzy sets on Xi

i = 1, 2, ..., n

Compound proposition induces a fuzzy relation Q on X1X1... Xn

)()()()()(1

221121 iin

innn xASxsAsxsAxAx,,x,xQ

=

== KK

q : (X1, X2 , ....., Xn) is Q

s (S) = t-conorm

Multiple fuzzy inputs: disjunctive canonical form


Example Fuzzy relation associated with (X,Y) is Q

Triangular fuzzy sets A1(x,4,5,6) = A, A2(y,8,10,12) = B

t-conorm: probabilistic sum

Q

0 5 100

10

20

x

y

(d) Contours of Q


Rule base

Fuzzy rule: If X is A then Y is B relationship between X and Y

Semantics of the rule is given by a fuzzy relation R on XY

R determined by a relational assignment

R(x,y) = f (A(x),B(y)) (x, y)XY

f : [0,1]2 [0,1]

In general f can be

fuzzy conjunction: ft fuzzy disjunction: fs fuzzy implication: fi


Choose a t-norm t and define:

R(x,y) ft (x,y) = A(x) t B(y) (x,y) XY

Examples:

t = min

Rc(x,y) fc (x,y) = min[A(x) t B(y)] (Mamdani)

t = algebraic product

Rp(x,y) fp (x,y) = A(x)B(y) (Larsen)

Fuzzy conjunction


Rc(x,y) = min {A(x), B(y)}

(A(x), B(y))[0,1]2Rc(x,y) = min {A(x), B(y)}

A(x) = A(x,4,5,6)B(y) = B(y,4,5,6)

Example: t = min


Rp(x,y) =A(x)B(y)

(A(x), B(y))[0,1]2Rp(x,y) = A(x)B(y)A(x) = A(x,4,5,6)B(y) = B(y,4,5,6)

Example: t = algebraic product


Choose a t-conorm s and define:

Rs(x,y) fs (x,y) = A(x) s B(y) (x,y) XY

Examples:

s = max

Rm(x,y) fm (x,y) = max[A(x) t B(y)]

s = Lukasiewicz t-conorm

Rl (x,y) fl (x,y) = min[1, A(x) + B(y)]

Fuzzy disjunction


Example: s = max

Rl (x,y) = max{A(x), B(y)}

(A(x), B(y))[0,1]2Rl (x,y) = max {A(x), B(y)}

A(x) = A(x,4,5,6)B(y) = B(y,4,5,6)


Example: s = Lukasiewicz

Rc(x,y) = min{1, A(x)+B(y)}

(A(x), B(y))[0,1]2Rc(x,y) = min{1, A(x)+B(y)}

A(x) = A(x,4,5,6)B(y) = B(y,4,5,6)


Fuzzy implication

Choose a fuzzy implication fi and define:

Ri(x,y) fi (x,y) (x,y) XY

fi : [0,1]2 [0,1] is a fuzzy implication if:

1. B(y1) B(y2) fi (A(x), B(y1)) fi (A(x), B(y2)) monotonicity 2nd argument

2. fi (0, B(y)) = 1 dominance of falsity

3. fi (1, B(y)) = B(y) neutrality of truth


Further requirements may include:

4. A(x1) A(x2) fi (A(x1), B(y)) fi (A(x2), B(y)) monotonicity 1st argument

5. fi (A(x1), fi (A(x2), B(y)) = fi (A(x2), fi (A(x1), B(y)) exchange

6. fi (A(x), A(x)) = 1 identity

7. fi (A(x), B(y)) = 1 A(x) B(y) boundary condition

8. fi is a continuous function continuity


+++

= )(1)()1()(11min))()((

xAyBxA

,yB,xAf

]))()(1(1[min))()(( 1 w/www yBxA,yB,xAf +=

=

otherwise0)()(if1))()(( yBxAyB,xAfa

=

otherwise)()()(if1))()((

yByBxA

yB,xAfg

=otherwise)(

)()()(if1

))()((xAyB

yBxAyB,xAfe

)]()(1[max))()(( yB,xAyB,xAfb =

Name Definition Comment

Lukasiewicz fl(A(x), B(y)) = min [1, 1 A(x) + B(y)]

Pseudo-Lukasiewicz > -1

Pseudo-Lukasiewicz w > 0

Gaines

Gdel

Goguen

Kleene

Reichenbach

Zadeh

Klir-Yuan

)]()()(1))()(( yBxAxAyB,xAfr +=

))]()((min)(1[max))()(( yB,xA,xAyB,xAfz =

)()()(1))()(( 2 yBxAxAyB,xAfk +=

Examples of fuzzy implications


Example: fl = Lukasiewicz

Rl (x,y) = min{1, 1 A(x)+B(y)}

(A(x), B(y))[0,1]2Rl (x,y) = min{1, 1 A(x)+B(y)}

A(x) = A(x,4,5,6)B(y) = B(y,4,5,6)


Example: fk = KlirYuan

Rk (x,y) = 1 A(x)+A(x)2B(y)

(A(x), B(y))[0,1]2Rk (x,y) = 1 A(x)+A(x)2B(y)

A(x) = A(x,4,5,6)B(y) = B(y,4,5,6)


Categories of fuzzy implications:

1. s-implications

zLukasiewic)}()(11min{))(),((

Kleene)]()(1max[))(),((

)()()())(),((

yBxA,yBxAf

yB,xAyBxAf

y,xysBxAyBxAf

g

b

is

+=

=

= YX

2. r-implications

Gdel)()()()()(if1))(),((

min

)()]()(|]10[sup[))(),((

>

=

=

=

yBxAyByBxA

yBxAf

t

y,xyBctxA,cyBxAf

g

ir YX


Semantics of gradual rules

x

A(x)

B(y)

y

1.0 1.0

BRd

the more X is A, the more Y is B B(y) A(x) xX and yY

BRd = {yY |B(y) A(x)} for each xX


Example: Rd = fa = Gaines

=

otherwise0)()(if1),( xAyByxRd

Rd(x,y) (A(x), B(y))[0,1]2

Rd (x,y)A(x) = A(x,3,5,7)B(y) = B(y,3,5,7)

00.2

0.40.6

0.81

0

0.5

10

0.2

0.4

0.6

0.8

1

A(x)

(a) Gradual rule Rd = fa

B(y)


Main types of rule bases

Fuzzy rule base {R1, R2,....,RN} finite family of fuzzy rules

Fuzzy rule base can assume various formats:

1. fuzzy graph

Ri: If X is Ai then Y is Bi is a fuzzy granule in XY, i = 1,...,N

2. fuzzy implication rule base

Ri: If X is Ai then Y is Bi is fuzzy implication, i = 1,...,N

3. functional fuzzy rule base

Ri: If X is Ai then y = fi(x) is a functional fuzzy rule, i = 1,...,N


Fuzzy graph

Fuzzy rule base R collection of rules R1, R2,....,RN

Each fuzzy rule Ri is a fuzzy granule (point)

Fuzzy graph R is a collection of fuzzy granules

granular approximation of a function

R = R1 or R2 or....or RN

general form

)(11

iN

ii

N

ii BARR ==

==

UU

)]()([),(1

ytBxASyxR iiN

i==


Point

0 2 4 6 8 100

2

4

6

8

10

x

(b)

P

0 2 4 6 8 100

1

x

A

(

x

)

(c)

A

0

2

4

6

8

10

01

(a)

B(y)

By

Point P in XYP = AB

A is a singleton in XB is a singleton in Y


Granule

0 2 4 6 8 100

2

4

6

8

10

xy

(b)

G

0 2 4 6 8 100

1

x

A

(

x

)

(c)

A

0

2

4

6

8

10

01

(a)

B(y)

B

Granule G in XYG = AB

A is an interval in XB is an interval in Y


Fuzzy granules fuzzy points

0 5 100

2

4

6

8

10

x

y

R

(b) Fuzzy granule R

0 2 4 6 8 100

1

x

A

(

x

)

(c)

A

0

2

4

6

8

10

01

(a)

B(y)

B

fuzzy granule R in XYR = AB

A is a fuzzy set on XB is a fuzzy set on Y


0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

x

(b) Fuzzy granules Ri

R1

R2

R3

R4

R5

y

0 2 4 6 8 100

1

x

(c)

A1 A2 A3 A4 A5

0

2

4

6

8

10

01

(a)

B(y)

B1

B2

B3

B4

B5

y

Fuzzy rule base as a set fuzzy granules

Ri = AiBi


0 2 4 6 8 10 120

2

4

6

8

10

12

x

y

(a) function y = f(x)

f

1 2 3 4 5 6 7 8 9 10 11 12

1

2

3

4

5

6

7

8

9

10

11

12

x

y

(b)Granular approximation of y = f(x)

R

R1

R2

R3R4

R5 R6 R7R8

Graph of a function f and its granular approximation R

f R

Ri = AiBi


0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

x

(b)Fuzzy rule base as a fuzzy graph (t = min)

R = Union Ri

y

Fuzzy rule base and fuzzy graphExample 1

Ri = AiBi Ri(x,y) = min [Ai(x), Bi(y)]

R = Ri R(x,y) = max [Ri(x,y), i = 1,..., N ]Pedrycz and Gomide, FSE 2007

Fuzzy rule base and fuzzy graphExample 2

Ri = Ai t Bi Ri(x,y) = Ai(x) Bi(y)

R = Ri R(x,y) = max [Ri(x,y), i = 1,..., N ]

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

x

(d) Fuzzy rule base as a fuzzy graph (t = product)

R = Union Ri

y


Fuzzy implication

Fuzzy rule base R collection of rules R1, R2,....,RN

Each fuzzy rule Ri is a fuzzy implication

Fuzzy rule base R is a collection of fuzzy relations

relation R is obtained using intersection

R = R1 and R2 and....and RN

general form

)(111

iN

ii

N

ii

N

ii BAfRR III

===

===

))()((1

yB,xAfTR iiiN

i==


Fuzzy rule as an implication

0 2 4 6 8 100

2

4

6

8

10

x

y

(b) Lukasiewicz implication R

R

0 2 4 6 8 100

1

x

A

(

x

)

(c)

A

0

2

4

6

8

10

01

(a)

B(y)

B

fuzzy rule R in XYR = fl (A,B)Lukasiewicz implication


Fuzzy rule base and fuzzy implicationExample 1a

Ri = fl (A,B) Ri(x,y) = min [1, 1 Ai(x) + Bi(y)] Lukasiewicz implication

R = Ri R(x,y) = min [Ri(x,y), i = 1,..., 5] min t-norm

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

x

y

(b) Fuzzy rule base as Lukasiewicz implication (t = min)


Fuzzy rule base and fuzzy implicationExample 1b

Ri = fl (A,B) Ri(x,y) = min [1, 1 Ai(x) + Bi(y)] Lukasiewicz implication

R = Ri R(x,y) = R1(x,y) tl R2(x,y) tl .... tl Ri(x,y) Lukasiewicz t-norm

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

x

y

(b) Fuzzy rule base as Lukasiewicz implication (t = min)


Fuzzy rule base and fuzzy implicationExample 2a

Ri = fz (A,B) Ri(x,y) = max [1 Ai(x), min(Ai(x), Bi(y)] Zadeh implicationR = Ri R(x,y) = min [Ri(x,y), i = 1,..., 5] min t-norm

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

x

y

(b) Fuzzy rule base as Zadeh implication (t = min)


Fuzzy rule base and fuzzy implicationExample 2b

Ri = fz (A,B) Ri(x,y) = max [1 Ai(x), min(Ai(x), Bi(y)] Zadeh implicationR = Ri R(x,y) = R1(x,y) tl R2(x,y) tl .... tl Ri(x,y) Lukasiewicz t-norm

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

x

y

(d) Fuzzy rule base as Zadeh implication (t = Lukasiewicz)


Data base

Data base contains definitions of: universes scaling functions of input and output variables granulation of the universes membership functions

Granulation granular constructs in the form of fuzzy points granules along different regions of the universes

Construction of membership functions expert knowledge learning from data


Granulation

X X

granular constructs inthe form of fuzzy points

granules along differentregions of the universes


Fuzzy inference

Basic idea of inference

0 2 4 6 8 10 120

2

4

6

8

10

12

x

y

(a)

I

ac

f

a

b

x = a

y = f (x) y = b

b = ProjY (ac f )

b = ProjY ( I )


Inference involves operations with sets

x = A y = f (x) B = f (A) ={f (x), xA}

B = ProjY (Ac f )

B = ProjY ( I ) 0 2 4 6 8 10 1202

4

6

8

10

12

x

y

(b)

I

Ac

A

a b

B c

d

f


Inference involving sets and relations

x is A(x,y) is Ry is B

B = ProjY (Ac R )

B = ProjY ( I )

f

0 2 4 6 8 10 120

2

4

6

8

10

12

x

y

(a)

I

Ac

R

A

B


Fuzzy inference ands operations with fuzzy sets and relations

X is A (fuzzy set on X)(X,Y) is R (fuzzy relation on XY)Y is B (fuzzy set on Y)

B = ProjY (Ac R )

B = ProjY ( I )

f

2 4 6 8 10 12

2

4

6

8

10

12

x

y

(b)

A

B R

I

Ac

)}()({sup)( y,xtRxAyBx X

=


Fuzzy inference

Compositional rule of inference

X is A (X,Y) is RY is B

X is A(X,Y) is RY is AoR

RAB o=


procedure FUZZY-INFERENCE (A, R) returns a fuzzy setinput : fuzzy relation: R

fuzzy set: Alocal: x, y: elements of X and Y

t: t-norm

for all x and y doAc(x,y) A(x)

for all x and y doI(x,y) Ac(x,y) t R(x,y)

B(y) supx

I(x,y) return B

Fuzzy inference procedure


Example: compositional rule of inference


Example: fuzzy inference with fuzzy graph


11.5 Types of rule-basedsystems and architectures


Linguistic fuzzy models

P: X is A and Y is B input

R1: If X is A1 and Y is B1 then Z is C1......................

Ri: If X is Ai and Y is Bi then Z is Ci rule base.......................

RN: If X is AN and Y is BN then Z is CN

Z: Z is C output

all fuzzy sets A, B, Ai,s and Bi,s are given rule and connectives (and, or) with known semantics membership function of fuzzy set C = ??


min-max fuzzy models

Assume

P: X is A and Y is B P(x,y) = min{A(x), B(y)}

Ri: If X is Ai and Y is Bi then Z is Ci Ri(x,y,z) = min{Ai(x), Bi(y), Ci(z)}

i = 1,..., N

)]}1),(max(),({min[sup)(

1

,...,Niz,y,xRy,xPzC

RPRPC

iy,x

N

ii

==

==

=

Uoo

Using the compositional rule of inference (t = min)


rulethiofactivationofdegreetheis

}1),(max{}1)max{()(

)()(

)Poss()]()([sup

)Poss()]()([sup

)]}()()()()({sup)]}(),({min[sup)(

)(111

====

=

==

==

==

=

====

===

i

iiiii

iiii

iiiy

iiix

iiiy,x

iy,x

i

ii

N

ii

N

ii

N

ii

N,,izCN,,i,CnmzC

zCnmzC

nB,ByByB

mA,AxAxA

zCyBxAyBxAz,y,xRy,xPzC

RPC

CRPRPRPC

KK

o

ooo UUU


procedure MIN-MAX-MODEL (A,B) returns a fuzzy setlocal: fuzzy sets: Ai, Bi, Ci, i =1,.., N

activation degrees: i

Initialization C =

for i = 1: N domi = max (min (A, Ai))ni = max (min (B, Bi))i = min (mi, ni)if i 0 then Ci = min (i , Ci) and C = max(C, Ci)

return C

min-max fuzzy model processing


Example: min-max fuzzy model processing

Ai Aj Bi Bj A B

mi

mj

ni

nj

Ci Cj 1 1 1

nj

mi Ci

Cj

x y z


min-max fuzzy model architecture

Poss 1 Min

Max

A1,B1 C1

Poss i Min

Ai,Bi Ci

Poss N Min

AN,BN CN

C (A,B)

Ci

CN

C1


Special case: numeric inputs

=

=

=

=

otherwise0if1)(

otherwise0if1)( oo yyyBandxxxA

Numeric output

iN

iii

N

iiii

v

nm

vnm

z

dzzCdzzzC

z

valuesmodalaverageweighted)(

)(

ationdefuzzificcentroid)()(

1

1

=

=

=

=

Z

Z


ExampleP: X is xo and Y is yo inputs (xo, yo), xo, yo [-2, 2]

R1: If X is A1 and Y is B1 then Z is C1rules

R2: If X is A2 and Y is B2 then Z is C2

N = 2, centroid defuzzification

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.5

1

x

A

i

(

x

)

A1 A2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.5

1

y

B

i

(

y

)

B2 B2

(a) Input and output fuzzy sets

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.5

1

z

C

i

(

z

)

C1 C2

-2-1

01

2

-2

-1

0

1

2

-0.5

0

0.5

x

(b) Input-output mapping

y

z


min-sum fuzzy models

Assume

P: X is A and Y is B P(x,y) = min{A(x), B(y)}

Ri: If X is Ai and Y is Bi then Z is Ci Ri(x,y,z) = min{Ai(x), Bi(y), Ci(z)}

i = 1,..., N

=

=

=

N

iii

iiiy,x

i

CwzC

zCyBxAyBxAzC

1)(

)]()()()()([sup)(

Using the compositional rule of inference (t = min)

Additive fuzzy models(Kosko, 1992)


Example: min-sum fuzzy model processing

Ai Aj Bi Bj A B

mi

mj

ni

nj

Ci Cj 1 1 1

nj

mi Ci Cj

x y z

Ci


min-sum fuzzy model architecture

Poss 1 Min

A1,B1 C1

Poss i Min

Ai,Bi Ci

Poss N Min

AN,BN CN

C (A,B) wi

w1

wN

CN

C1

Ci


ExampleP: X is xo and Y is yo inputs (xo, yo), xo, yo [-2, 2]

R1: If X is A1 and Y is B1 then Z is C1rules

R2: If X is A2 and Y is B2 then Z is C2

N = 2 w1 = w2 = 1, centroid defuzzification

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.5

1

x

A

i

(

x

)

A1 A2

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.5

1

y

B

i

(

y

)

B2 B2

(a) Input and output fuzzy sets

-2 -1.5 -1 -0.5 0 0.5 1 1.5 20

0.5

1

z

C

i

(

z

)

C1 C2

-2-1

01

2

-2

-1

0

1

2

-0.5

0

0.5

x

(b) Input-output mapping

y

z


product-sum fuzzy models

1- Productprobabilistic sum

)()(

)()(

1zCSzC

zCnmzC

iN

ip

iiii

=

=

=

2- Productsum

=

=

=

N

ii

iiii

zCzC

zCnmzC

1)()(

)()(

-2-1

01

2

-2

-1

0

1

2

-0.5

0

0.5

x

(b) Input-output mapping of product-probabilistic sum model

y

z

-2-1

01

2

-2

-1

0

1

2

-0.5

0

0.5

x

(d) Input-output mapping of product-sum model

y

z


3 - Bounded product-bounded sum

]10[

}1min{

}10max{

)()(

)()(

1

,a,b

ba,ba

ba,ba

zCzC

zCnmzC

iN

i

iiii

+=

+=

=

=

=

-2-1

01

2

-2

-1

0

1

2

-1

-0.5

0

0.5

1

x

(c) Input-output mapping of bounded product-bounded sum model

y

z


Functional fuzzy models

P: X is x and Y is y input

R1: If X is A1 and Y is B1 then z = f1 (x,y)......................

Ri: If X is Ai and Y is Bi then z = fi (x,y) rule base.......................

RN: If X is AN and Y is BN then z = fN (x,y)

i(x,y) = Ai(x) t Bi(y) t = t-norm degree of activation

output

=

=

== N

ii

ii

N

iii

y,x

w,y,xfy,xwz

1

1 )()()(


Functional fuzzy model architecture

w1

f1(x,y)

A1,B1

wi

Ai,Bi

wN

AN,BN

z (x,y)

f2(x,y)

fN(x,y)


Example 1P: X is x inputs x [0, 3]

R1: If X is A1 then z = xrules

R2: If X is A2 then z = x + 3

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

x

A

i A1 A2

(a) Antecedent fuzzy sets

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

f1f2

x

y

(b) Consequent functions


+

++

=

)32[if3]21[if)3)(()(]10(if

21,xx

,xxxAxxA,xx

z

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

y

(b) Output of the functional model

output



R1: If X is A1 then y = sin(2x)

R2: If X is A2 then y = 0.5x rules

R3: If X is A3 then y = sin(3x)

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

A1 A2 A3


x

y

-5 -4 -3 -2 -1 0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

1

1.5(b) output of the functional fuzzy model

y

x

output



R1: If X is A1 then y = 1

R2: If X is A2 then y = x rules

R3: If X is A3 then y = 1

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

A1 A2 A3


x

y

-5 -4 -3 -2 -1 0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

1

1.5

x

y

(c) output of the functional fuzzy model

output


Gradual fuzzy models

Ri: The more X is Ai the more Z is Ci

i = 1,..., N

IIN

i

N

ii

iii

ii CCC

xAzCyxR

11)(

otherwise0)()(if1),(

=

=

==

=


Gradual fuzzy model architecture

Poss 1 C1

Min

A1 C1

Poss i

Ai Ci

Poss N

AN CN

C x

Ci

CN

C1

C1

C1


x z

A1 A2

2

1

C1 C2 1 1

1

2 C1 C2

Example: gradual fuzzy model processing


ExampleP: X is x inputs x [0, 3]

R1: The more X is A1 the more Z is C1rules

R2: The more X is A1 the more Z is C1

1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

x

A

i

A1 A2

1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

z

C

i

C1 C2

1 1.5 2 2.5 3 3.5 41

1.5

2

2.5

3

3.5

4

x

z

output


11.6 Approximation propertiesof fuzzy rule-based models


FRBS uniformly approximates continuous functions any degree of accuracy closed and bounded sets

Universal approximation with (Wang & Mendel, 1992): algebraic product t-norm in antecedent rule semantics via algebraic product rule aggregation via ordinary sum Gaussian membership functions sup-min compositional rule of inference pointwise inputs centroid defuzzification


Universal approximation when (Kosko, 1992): min t-norm in antecedent rule aggregation via ordinary sum symmetric consequent membership functions sup-min compositional rule of inference pointwise inputs centroid defuzzification

(additive models)


Universal approximation with (Castro, 1995): arbitrary t-norm in antecedent rule semantics: r-implications or conjunctions triangular or trapezoidal membership functions sup-min compositional rule of inference pointwise inputs centroid defuzzification


11.7 Development of rule-basedsystems


Expert-based development

Knowledge provided by domain experts basic concepts and variables links between concepts and variables to form rules

Reflects existing knowledge can be readily quantified short development time


Example: fuzzy control

FuzzyController Processr

e u y+

Ri: If Error is Ai and Change of Error is Bi then Control is Ci

Ri: If e is Ai and de is Bi then u is Ci


Ri: If e is Ai and de is Bi then u is Ci

Change of Error (de) / Error (e) NM NS ZE PS PMNB PM NB NB NB NM

NM PM NB NS NM NM

NS PM NS Z NS NM

Z PM NS Z NS NM

PS PM PS Z NS NM

PM PM PM PS PM NM

PB PM PM PM PM NM

r

y

t

e


Data-driven development

Given a finite set of input/output pairs{(xk, yk), k = 1,..., M}xk = [x1k, x2k,...., xnk] Rn

zk = [xk, yk] Rn+1, k = 1,..., M

Clustering zk = [xk, yk] Rn+1, k = 1,..., M (e.g. using FCM)v1, v2,....,vN prototypes/cluster centersvi Rn+1, i = 1,..., N

Idea: fuzzy clusters fuzzy rules


Example

R1v1

v2v3

v4

R2R3

R4


Projecting the prototypes in the input and output spaces

v1[y], v2[y],....,vN [y] projections of prototypes in Y

v1[x], v2[x],....,vN [x] projections of prototypes in X

Ri: If X is Ai then Y is Ci, i = 1,..., N

x

y

x

y

x

y


11.8 Parameter estimation forfunctional rule-basedsystems


Functional fuzzy rules

Ri: If Xi1 is Ai1 and ... and Xin is Ain then z = aio + ai1x1 + ....+ ainxni = 1,..., N

Given input/output data: {(x1, y1), (x2, y2),....,(xM, yM)}

Let ai = [aio, ai1, ai2,...., ain]T

Output of functional models

Output for linear consequents

=

=

== N

iki

kiik

N

iikiikk

x

xw,,fwy

11 )(

)()( ax

TT

1

T ]1[ kikikN

iiikk w,,y xzaz ==

=


[ ]

=

=

N

Nkkkk

N

y

a

a

a

zzz

a

a

a

aM

LM

21

TT2

T1

21

=

=

TT2

T1

T2

T22

T12

T1

T12

T11

21

NMMM

N

N

M

Z

y

yy

zzz

zzz

zzz

y

L

MOMM

L

L

M

Let

and

then y = Za


Global least squares approach

Mina JG(a) = || y Za||2

|| y Za||2 = (y Za)T (y Za)Solution

aopt = Z# y

Z# = (ZT)1ZT


Local least squares approach

Solution

aiopt = Zi# y

Zi# = (ZiT)1ZiT

=

= =

T

T2

T1

1

2)(JMin

iM

i

i

i

N

iiiL

Z

Z

z

z

z

ayaa

M


11.9 Design issues of FRBS:Consistency andcompleteness


Given input/output data: {(x1, y1), (x2, y2),....,(xM, yM)}

dataconsistencycompletenessaccuracy

rules

Issue: quality of the rules


Completeness of rules

All data points represented through some fuzzy setmaxi = 1,..., M Ai(xk) > 0 for all k = 1,2,..., M

Input space completely covered by fuzzy setsmaxi = 1,..., M Ai(xk) > for all k = 1,2,..., M


Consistency of rules

Rules in conflict similar or same conditions completely different conclusions

Conditions and Conclusions

Similar Conclusions

Different Conclusions

Similar Conditions rules are redundant rules are in conflict

Different Conditions

different rules; could be eventually

mergeddifferent rules


|})()(||)()({|)cons(1

kjkikjM

iki xAxAyByBj,i =

=

Ri: If X is Ai then Y is BiRj: If X is Aj then Y is Bj

is an implication induced by some t-norm (r-implication)

))}()(Poss())()({Poss()cons(1

kjkikjM

iki yB,yBxA,xAj,i =

=

Alternatively

=

=

M

jj,i

Mi

1)cons(1)cons(


11.10 The curse of dimensionalityin rule-based systems


Curse of dimensionality number of variables increase exponential growth of the number of rules

Example n variables each granulated using p fuzzy sets number of different rules = pn

Scalability challenges


11.11 Development scheme offuzzy rule-based models


Accuracy Stability

Interpretability Knowledge

Representation

Spiral model of development incremental design, implementation and testing multidimensional space of fundamental characteristics


Fuzzy Rule-Based Models

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