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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
Pedrycz and Gomide, FSE 2007
11.1 Fuzzy rules as a vehicle ofknowledge representation
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
11.2 General categories of fuzzyrules and their semantics
Pedrycz and Gomide, FSE 2007
Multi-input multi-output fuzzy rules
• If X1 is A1 and X2 is A2 and ..... and Xn is An
then Y1 is B1 and Y2 is B2 and ..... and Ym is Bm
Xi = variables whose values are fuzzy sets Ai
Yj = 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
Pedrycz and Gomide, FSE 2007
Certainty-qualified rules
• If X is A and Y is B then Z is C with certaintyµ
µ ∈[0,1]
µ : degree of certainty of the rule
µ = 1 rule is certain
Pedrycz and Gomide, FSE 2007
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 higherthe income, the higherthe taxes
the lower the temperature, the higherenergy consumption
Pedrycz and Gomide, FSE 2007
Functional fuzzy rules
• If X is Ai then y = f (x,ai)
f : X → Y
x∈Rn
• Rule: confines the function to the support of granule Ai
f : linear or nonlinear (neural nets, etc..)
• Highly modular models
Pedrycz and Gomide, FSE 2007
11.3 Syntax of fuzzy rules
Pedrycz and Gomide, FSE 2007
Backus-Naur form (BNF)
⟨ If_then_rule⟩ ::= if ⟨antecedent⟩ then⟨consequent⟩{ ⟨certainty⟩}⟨gradual_rule ⟩ ::= ⟨word⟩ ⟨antecedent⟩⟨word⟩ ⟨consequent⟩
⟨word⟩ ::= ⟨more⟩ { ⟨less⟩}⟨antecedent⟩ ::= ⟨expression⟩⟨consequent⟩ ::= ⟨expression⟩⟨expression⟩ ::= ⟨disjunction⟩{and ⟨disjunction⟩}⟨disjunction⟩ ::= ⟨variable⟩{or⟨variable⟩}
⟨variable⟩ ::= ⟨attribute⟩ is ⟨value⟩⟨certainty⟩ ::= ⟨none⟩{certainty µ∈[0,1]}
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
11.4 Basic functionalmodules of FRBS
Pedrycz and Gomide, FSE 2007
Inpu
t Int
erf
ace Rule Base
Data Base
Fuzzy Inference O
utp
ut I
nte
rfac
e
X Y
General architecture of FRBS
Fuzzy if-then rules(input-output relationship)
Parametersof the FRBS
Process inputs and rules(approximate reasoning)
Pedrycz and Gomide, FSE 2007
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 X1×X1×... Xn
)()()()()(1
221121 ii
n
innn xATxtAtxtAxAx,,x,xP
=== KK
p : (X1, X2 , ....., Xn) is P
t (T) = t-norm
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
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 X1×X1×... Xn
)()()()()(1
221121 ii
n
innn xASxsAsxsAxAx,,x,xQ
=== KK
q : (X1, X2 , ....., Xn) is Q
s (S) = t-conorm
Multiple fuzzy inputs: disjunctive canonical form
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
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 X×Y
• R determined by a relational assignment
R(x,y) = f (A(x),B(y)) ∀(x, y)∈X×Y
f : [0,1]2 → [0,1]
• In general f can be
– fuzzy conjunction: ft– fuzzy disjunction: fs– fuzzy implication: fi
Pedrycz and Gomide, FSE 2007
Choose a t-norm t and define:
R(x,y) ≡ ft (x,y) = A(x) t B(y) ∀(x,y) ∈ X×Y
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
Pedrycz and Gomide, FSE 2007
Rc(x,y) = min {A(x), B(y)}
∀ (A(x), B(y))∈[0,1]2
Rc(x,y) = min {A(x), B(y)}A(x) = A(x,4,5,6)B(y) = B(y,4,5,6)
Example: t = min
Pedrycz and Gomide, FSE 2007
Rp(x,y) =A(x)B(y)
∀ (A(x), B(y))∈[0,1]2
Rp(x,y) = A(x)B(y)A(x) = A(x,4,5,6)B(y) = B(y,4,5,6)
Example: t = algebraic product
Pedrycz and Gomide, FSE 2007
Choose a t-conorm s and define:
Rs(x,y) ≡ fs(x,y) = A(x) s B(y) ∀(x,y) ∈ X×Y
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
Pedrycz and Gomide, FSE 2007
Example: s = max
Rl (x,y) = max{A(x), B(y)}
∀ (A(x), B(y))∈[0,1]2
Rl (x,y) = max {A(x), B(y)}A(x) = A(x,4,5,6)B(y) = B(y,4,5,6)
Pedrycz and Gomide, FSE 2007
Example: s = Lukasiewicz
Rc(x,y) = min{1, A(x)+B(y)}
∀ (A(x), B(y))∈[0,1]2
Rc(x,y) = min{1, A(x)+B(y)}A(x) = A(x,4,5,6)B(y) = B(y,4,5,6)
Pedrycz and Gomide, FSE 2007
Fuzzy implication
• Choose a fuzzy implication fi and define:
Ri(x,y) ≡ fi (x,y) ∀(x,y) ∈ X×Y
• 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
Pedrycz and Gomide, FSE 2007
• 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
Pedrycz and Gomide, FSE 2007
λ++λ+−=λ )(1
)()1()(11min))()((
xA
yBxA,yB,xAf
]))()(1(1[min))()(( 1 w/www yBxA,yB,xAf +−=
≤
=otherwise0
)()(if1))()((
yBxAyB,xAfa
≤
=otherwise)(
)()(if1))()((
yB
yBxAyB,xAfg
≤
= otherwise)(
)()()(if1
))()((xA
yByBxA
yB,xAfe
)]()(1[max))()(( yB,xAyB,xAfb −=
Name Definition Comment
Lukasiewicz fl(A(x), B(y)) = min [1, 1 − A(x) + B(y)]
Pseudo-Lukasiewicz λ > -1
Pseudo-Lukasiewicz w > 0
Gaines
Gödel
Goguen
Kleene
Reichenbach
Zadeh
Klir-Yuan
)]()()(1))()(( yBxAxAyB,xAfr +−=
))]()((min)(1[max))()(( yB,xA,xAyB,xAfz −=
)()()(1))()(( 2 yBxAxAyB,xAfk +−=
Examples of fuzzy implications
Pedrycz and Gomide, FSE 2007
Example: fl = Lukasiewicz
Rl (x,y) = min{1, 1–A(x)+B(y)}
∀ (A(x), B(y))∈[0,1]2
Rl (x,y) = min{1, 1–A(x)+B(y)}A(x) = A(x,4,5,6)B(y) = B(y,4,5,6)
Pedrycz and Gomide, FSE 2007
Example: fk = Klir–Yuan
Rk (x,y) = 1–A(x)+A(x)2B(y)
∀ (A(x), B(y))∈[0,1]2
Rk (x,y) = 1–A(x)+A(x)2B(y)A(x) = A(x,4,5,6)B(y) = B(y,4,5,6)
Pedrycz and Gomide, FSE 2007
• Categories of fuzzy implications:
1. s-implications
zLukasiewic)}()(11min{))(),((
Kleene)]()(1max[))(),((
)()()())(),((
yBxA,yBxAf
yB,xAyBxAf
y,xysBxAyBxAf
g
b
is
+−=
−=
×∈∀= YX
2. r-implications
Gödel)()()(
)()(if1))(),((
min
)()]()(|]10[sup[))(),((
>≤
=
=
×∈∀≤∈=
yBxAyB
yBxAyBxAf
t
y,xyBctxA,cyBxAf
g
ir YX
Pedrycz and Gomide, FSE 2007
Semantics of gradual rules
x
A(x)
B(y)
y
1.0 1.0
BRd
the more X is A, the moreY is B ⇒ B(y) ≥ A(x) ∀x∈X and ∀y∈Y
BRd= { y∈Y |B(y) ≥ A(x)} for each x∈X
Pedrycz and Gomide, FSE 2007
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)
Pedrycz and Gomide, FSE 2007
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 X×Y, 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
Pedrycz and Gomide, FSE 2007
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 ii
N
i ==
Pedrycz and Gomide, FSE 2007
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 X×YP = A×B
A is a singleton in XB is a singleton in Y
Pedrycz and Gomide, FSE 2007
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 X×YG = A×B
A is an interval in XB is an interval in Y
Pedrycz and Gomide, FSE 2007
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 X×YR = A×B
A is a fuzzy set on XB is a fuzzy set on Y
Pedrycz and Gomide, FSE 2007
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 = Ai×Bi
Pedrycz and Gomide, FSE 2007
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
R3
R4
R5 R6R7
R8
Graph of a function f and its granular approximation R
f R
Ri = Ai×Bi
Pedrycz and Gomide, FSE 2007
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 graph
Example 1
Ri = Ai×Bi ⇒ 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 graph
Example 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
Pedrycz and Gomide, FSE 2007
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 iii
N
i==
Pedrycz and Gomide, FSE 2007
Fuzzy rule as an implication
0 2 4 6 8 100
2
4
6
8
10
xy
(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 X×YR = fl (A,B)Lukasiewicz implication
Pedrycz and Gomide, FSE 2007
Fuzzy rule base and fuzzy implication
Example 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)
Pedrycz and Gomide, FSE 2007
Fuzzy rule base and fuzzy implication
Example 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)
Pedrycz and Gomide, FSE 2007
Fuzzy rule base and fuzzy implication
Example 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)
Pedrycz and Gomide, FSE 2007
Fuzzy rule base and fuzzy implication
Example 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)
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
Granulation
X X
granular constructs inthe form of fuzzy points
granules along differentregions of the universes
Pedrycz and Gomide, FSE 2007
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 )
Pedrycz and Gomide, FSE 2007
Inference involves operations with sets
x = A y = f (x) B = f (A) ={ f (x), x∈A}
B = ProjY (Ac ∩ f )
⇓
B = ProjY ( I ) 0 2 4 6 8 10 12
0
2
4
6
8
10
12
x
y
(b)
I
Ac
A
a b
B
c
d
f
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
Fuzzy inference ands operations with fuzzy sets and relations
X is A (fuzzy set on X)(X,Y) is R (fuzzy relation on X×Y)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∈
=⇒
Pedrycz and Gomide, FSE 2007
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=
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
Example: compositional rule of inference
Pedrycz and Gomide, FSE 2007
Example: fuzzy inference with fuzzy graph
Pedrycz and Gomide, FSE 2007
11.5 Types of rule-basedsystems and architectures
Pedrycz and Gomide, FSE 2007
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,sare given• rule and connectives (and, or) with known semantics• membership function of fuzzy set C = ??
Pedrycz and Gomide, FSE 2007
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)
Pedrycz and Gomide, FSE 2007
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
λ
N,,izCλN,,i,CnmzC
zCnmzC
nB,ByByB
mA,AxAxA
zCyBxAyBxAz,y,xRy,xPzC
RPC
CRPRPRPC
KK
o
ooo UUU
Pedrycz and Gomide, FSE 2007
procedure MIN-MAX-MODEL ( A,B) returns a fuzzy setlocal: fuzzy sets: Ai, Bi, Ci, i =1,.., N
activation degrees: λi
Initialization C = ∅
for i = 1: N domi = max (min (A, Ai))ni = max (min (B, Bi))λi = min (mi, ni)if λi ≠ 0 then Ci
’ = min (λi , Ci) and C = max(C, Ci’)
return C
min-max fuzzy model processing
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
min-max fuzzy model architecture
Poss λ1 Min
Max
A1,B1 C1
Poss λi Min
Ai,Bi Ci
Poss λN Min
AN,BN CN
C (A,B)
Ci’
CN’
C1’
Pedrycz and Gomide, FSE 2007
Special case: numeric inputs
=
= =
=otherwise0
if1)(
otherwise0
if1)( oo yy
yBandxx
xA
Numeric output
iN
iii
N
iiii
v
nm
vnm
z
dzzC
dzzzCz
valuesmodalaverageweighted
)(
)(
ationdefuzzificcentroid)(
)(
1
1
∑
∑
∫∫
=
=
∧
∧=
=Z
Z
Pedrycz and Gomide, FSE 2007
Example
P: 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
Ai(
x)
A1 A2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 20
0.5
1
y
Bi(
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
Ci(
z)
C1 C2
-2-1
01
2
-2
-1
0
1
2
-0.5
0
0.5
x
(b) Input-output mapping
y
z
Pedrycz and Gomide, FSE 2007
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)
Pedrycz and Gomide, FSE 2007
Example: min-sum fuzzy model processing
A i A j Bi Bj A B
mi
mj
ni nj
Ci Cj 1 1 1
nj
mi Ci
’ Cj’
x y z
∑Ci’
Pedrycz and Gomide, FSE 2007
min-sum fuzzy model architecture
Poss λ1 Min
∑
A1,B1 C1
Poss λi Min
Ai,Bi Ci
Poss λN Min
AN,BN CN
C (A,B)
wi
w1
wN
CN’
C1’
Ci’
Pedrycz and Gomide, FSE 2007
Example
P: 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
Ai(
x)
A1 A2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 20
0.5
1
y
Bi(
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
Ci(
z)
C1 C2
-2-1
01
2
-2
-1
0
1
2
-0.5
0
0.5
x
(b) Input-output mapping
y
z
Pedrycz and Gomide, FSE 2007
product-sum fuzzy models
1- Product–probabilistic sum
)()(
)()(
1
zCSzC
zCnmzC
i
N
ip
iiii
′=
=′
=
2- Product–sum
∑=
′=
=′
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
Pedrycz and Gomide, FSE 2007
3 - Bounded product-bounded sum
]10[
}1min{
}10max{
)()(
)()(
1
,a,b
ba,ba
ba,ba
zCzC
zCnmzC
i
N
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
Pedrycz and Gomide, FSE 2007
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)
λi(x,y) = Ai(x) t Bi(y) t = t-norm degree of activation
output∑
∑
=
===
N
ii
ii
N
iii
y,xλ
λw,y,xfy,xwz
1
1 )(
)()(
Pedrycz and Gomide, FSE 2007
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)
⊗
⊗
×
×
×
Pedrycz and Gomide, FSE 2007
Example 1
P: 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
Ai
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
Pedrycz and Gomide, FSE 2007
∈+−∈+−+∈
=)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
Pedrycz and Gomide, FSE 2007
Example 2
P: X is x inputs x∈ [0, 3]
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
(a) Antecedent fuzzy sets
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
Pedrycz and Gomide, FSE 2007
Example 2
P: X is x inputs x∈ [0, 3]
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
(a) Antecedent fuzzy sets
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
Pedrycz and Gomide, FSE 2007
Gradual fuzzy models
Ri: The more Xis Ai the more Zis Ci
i = 1,..., N
IIN
i
N
ii
iii
iiCCC
xAzCyxR
11)(
otherwise0
)()(if1),(
=αα
==′=
≥
=
Pedrycz and Gomide, FSE 2007
Gradual fuzzy model architecture
Poss α1 Cα1
Min
A1 C1
Poss αi
Ai Ci
Poss αN
AN CN
C x
Ci’
CN’
C1’
Cα1
Cα1
Pedrycz and Gomide, FSE 2007
x z
A1 A2
α2
α1
C1 C2 1 1
α1
α2 C1
’ C2’
Example: gradual fuzzy model processing
Pedrycz and Gomide, FSE 2007
Example
P: X is x inputs x ∈ [0, 3]
R1: The more Xis A1 the more Zis C1
rulesR2: The more Xis A1 the more Zis C1
1 1.5 2 2.5 3 3.5 40
0.2
0.4
0.6
0.8
1
x
Ai
A1 A2
1 1.5 2 2.5 3 3.5 40
0.2
0.4
0.6
0.8
1
z
Ci
C1 C2
1 1.5 2 2.5 3 3.5 41
1.5
2
2.5
3
3.5
4
x
z
output
Pedrycz and Gomide, FSE 2007
11.6 Approximation propertiesof fuzzy rule-based models
Pedrycz and Gomide, FSE 2007
• 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
Pedrycz and Gomide, FSE 2007
• 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)
Pedrycz and Gomide, FSE 2007
• 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
Pedrycz and Gomide, FSE 2007
11.7 Development of rule-basedsystems
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
Example: fuzzy control
FuzzyController
Processre 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
Pedrycz and Gomide, FSE 2007
Ri: If e is Ai and de is Bi then u is Ci
Change of Error (de) / Error (e) NM NS ZE PS PM
NB 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
Pedrycz and Gomide, FSE 2007
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 centers
vi ∈ Rn+1, i = 1,..., N
• Idea: fuzzy clusters ≡ fuzzy rules
Pedrycz and Gomide, FSE 2007
Example
R1
v1
v2v3
v4
R2R3
R4
Pedrycz and Gomide, FSE 2007
• 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
Pedrycz and Gomide, FSE 2007
11.8 Parameter estimation forfunctional rule-basedsystems
Pedrycz and Gomide, FSE 2007
• Functional fuzzy rules
• Ri: If Xi1 is Ai1 and ... and Xin is Ain then z= aio + ai1x1 + ....+ainxn
i = 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
xλ
xλw,,fwy
1
1 )(
)()( ax
TT
1
T ]1[ kikikN
iiikk w,,y xzaz == ∑
=
Pedrycz and Gomide, FSE 2007
[ ]
=
=
N
Nkkkk
N
y
a
a
a
zzz
a
a
a
aM
LM
2
1
TT2
T1
2
1
=
=
TT2
T1
T2
T22
T12
T1
T12
T11
2
1
NMMM
N
N
M
Z
y
y
y
zzz
zzz
zzz
y
L
MOMM
L
L
M
Let
and
then y = Za
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
Local least squares approach
Solution
aiopt= Zi# y
Zi# = (Zi
T)–1ZiT
=
−=∑=
T
T2
T1
1
2)(JMin
iM
i
i
i
N
iiiL
Z
Z
z
z
z
ayaa
M
Pedrycz and Gomide, FSE 2007
11.9 Design issues of FRBS:Consistency andcompleteness
Pedrycz and Gomide, FSE 2007
Given input/output data: {( x1, y1), (x2, y2),....,(xM, yM)}
dataconsistencycompletenessaccuracy
rules
Issue: quality of the rules
Pedrycz and Gomide, FSE 2007
Completeness of rules
• All data points represented through some fuzzy set
maxi = 1,..., M Ai(xk) > 0 for all k = 1,2,..., M
• Input space completely covered by fuzzy sets
maxi = 1,..., M Ai(xk) > δ for all k = 1,2,..., M
Pedrycz and Gomide, FSE 2007
Consistency of rules
• Rules in conflict
– similar or same conditions
– completely different conclusions
Conditions and Conclusions
Similar Conclusions
Different Conclusions
Similar Conditions rules are redundantrules are in
conflict
Different Conditions
different rules; could be eventually
mergeddifferent rules
Pedrycz and Gomide, FSE 2007
|})()(||)()({|)cons(1
kjkikjM
iki xAxAyByBj,i −⇒−=∑
=
Ri: If X is Ai then Y is Bi
Rj: 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(
Pedrycz and Gomide, FSE 2007
11.10 The curse of dimensionalityin rule-based systems
Pedrycz and Gomide, FSE 2007
• 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
Pedrycz and Gomide, FSE 2007
11.11 Development scheme offuzzy rule-based models
Pedrycz and Gomide, FSE 2007
Accuracy
Stability
Interpretability Knowledge
Representation
• Spiral model of development
– incremental design, implementation and testing
– multidimensional space of fundamental characteristics
Pedrycz and Gomide, FSE 2007