11 Fuzzy Rule-Based Models Fuzzy Systems Engineering Toward - - PowerPoint PPT Presentation
11 Fuzzy Rule-Based Models Fuzzy Systems Engineering Toward Human-Centric Computing Contents 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
Fuzzy Systems Engineering Toward 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
Pedrycz and Gomide, FSE 2007
11.8 Parameter estimation procedure for functional rule-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
Pedrycz and Gomide, FSE 2007
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
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
Pedrycz and Gomide, FSE 2007
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
If X is A and Y is B then Z is C with certainty µ µ ∈[0,1] µ : degree of certainty of the rule µ = 1 rule is certain
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
〈 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
Main steps:
relations and definition of aggregation operator to combine rules together
Pedrycz and Gomide, FSE 2007
Pedrycz and Gomide, FSE 2007
Input Interface Rule Base Data Base Fuzzy Inference Output Interface X Y
Fuzzy if-then rules (input-output relationship) Parameters
Process inputs and rules (approximate reasoning)
Pedrycz and Gomide, FSE 2007
(attribute) of (input) is (value) the temperature of the motor is high Canonical (atomic) form p: X is A temperature (motor) is high X A fuzzy set
Low Medium High x (°C)
Pedrycz and Gomide, FSE 2007
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 2 2 1 1 2 1 i i n i n n n
x A T x tA t x tA x A x , , x , x P
=
= =
t (T) = t-norm
Pedrycz and Gomide, FSE 2007
5 10 10 20
x y (b) Contours of P
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 2 2 1 1 2 1 i i n i n n n
x A S x sA s x sA x A x , , x , x Q
=
= =
s (S) = t-conorm
Pedrycz and Gomide, FSE 2007
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
5 10 10 20
x y (d) Contours of Q
Pedrycz and Gomide, FSE 2007
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:
Rc(x,y) ≡ fc (x,y) = min[A(x) t B(y)] (Mamdani)
Rp(x,y) ≡ fp (x,y) = A(x)B(y) (Larsen)
Pedrycz and Gomide, FSE 2007
Rc(x,y) = min {a, b} ∀ (a, b)∈[0,1]2 Rc(x,y) = min {A(x), B(y)} ∀ (A(x), B(y))∈[0,1]2
Pedrycz and Gomide, FSE 2007
A(x) = A(x,4,5,6), B(y) = B(y,4,5,6)
Rp(x,y) =ab ∀ (a, b)∈[0,1]2 Rp(x,y) = A(x)B(y) ∀ (a, b)∈[0,1]2
Pedrycz and Gomide, FSE 2007
A(x) = A(x,4,5,6), B(y) = B(y,4,5,6)
Choose a t-conorm s and define: Rs(x,y) ≡ fs (x,y) = A(x) s B(y) ∀(x,y) ∈ X×Y Examples:
Rm(x,y) ≡ fm (x,y) = max[A(x), B(y)]
R (x,y) ≡ f (x,y) = min[1, A(x) + B(y)]
Pedrycz and Gomide, FSE 2007
Rm (x,y) = max{A(x), B(y)} ∀ (A(x), B(y))∈[0,1]2 Rm (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
R (x,y) = min{1, A(x)+B(y)} ∀ (A(x), B(y))∈[0,1]2 R (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
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:
monotonicity 2nd argument
dominance of falsity
neutrality of truth
Pedrycz and Gomide, FSE 2007
Further requirements may include:
monotonicity 1st argument
exchange
identity
boundary condition
continuity
Pedrycz and Gomide, FSE 2007
λ + + λ + − =
λ
) ( 1 ) ( ) 1 ( ) ( 1 1 min )) ( ) ( ( x A y B x A , y B , x A f ] ) ) ( ) ( 1 ( 1 [ min )) ( ) ( (
1 w / w w w
y B x A , y B , x A f + − = ≤ =
) ( ) ( if 1 )) ( ) ( ( y B x A y B , x A fa ≤ =
) ( ) ( ) ( if 1 )) ( ) ( ( y B y B x A y B , x A fg ≤ =
) ( ) ( ) ( ) ( if 1 )) ( ) ( ( x A y B y B x A y B , x A fe )] ( ) ( 1 [ max )) ( ) ( ( y B , x A y B , x A fb − = Name Definition Comment Lukasiewicz f(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 )) ( ) ( ( y B x A x A y B , x A fr + − = ))] ( ) ( ( min ) ( 1 [ max )) ( ) ( ( y B , x A , x A y B , x A fz − = ) ( ) ( ) ( 1 )) ( ) ( (
2
y B x A x A y B , x A fk + − =
Pedrycz and Gomide, FSE 2007
R (x,y) = min{1, 1– A(x)+B(y)} ∀ (A(x), B(y))∈[0,1]2 R (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
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:
z Lukasiewic )} ( ) ( 1 1 min{ )) ( ), ( ( Kleene )] ( ) ( 1 max[ )) ( ), ( ( ) ( ) ( ) ( )) ( ), ( ( y B x A , y B x A f y B , x A y B x A f y , x y sB x A y B x A f
g b is
+ − = − = × ∈ ∀ = Y X
Gödel ) ( ) ( ) ( ) ( ) ( if 1 )) ( ), ( ( min ) ( )] ( ) ( | ] 1 [ sup[ )) ( ), ( ( > ≤ = = × ∈ ∀ ≤ ∈ = y B x A y B y B x A y B x A f t y , x y B c t x A , c y B x A f
g ir
Y X
Pedrycz and Gomide, FSE 2007
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) ∀x∈X and ∀y∈Y BRd = {y∈Y |B(y) ≥ A(x)} for each x∈X
Pedrycz and Gomide, FSE 2007
≥ =
) ( ) ( if 1 ) , ( x A y B y x Rd
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)
0.2 0.4 0.6 0.8 1 0.5 1 0.2 0.4 0.6 0.8 1
A(x) (a) Gradual rule Rd = fa B(y)
Pedrycz and Gomide, FSE 2007
Fuzzy rule base ≡ {R1, R2,....,RN} ≡ finite family of fuzzy rules Fuzzy rule base can assume various formats:
Ri: If X is Ai then Y is Bi is a fuzzy granule in X×Y, i = 1,...,N
Ri: If X is Ai then Y is Bi is fuzzy implication, i = 1,...,N
Ri: If X is Ai then y = fi(x) is a functional fuzzy rule, i = 1,...,N
Pedrycz and Gomide, FSE 2007
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
) (
1 1 i N i i N i i
B A R R × = =
= =
( ) ( [ ) , (
1
y tB x A S y x R
i i N i=
=
Pedrycz and Gomide, FSE 2007
2 4 6 8 10 2 4 6 8 10
x (b) P
2 4 6 8 10 1
x A(x) (c) A
2 4 6 8 10 1
(a) B(y) B y
Point P in X×Y P = A×B A is a singleton in X B is a singleton in Y
Pedrycz and Gomide, FSE 2007
2 4 6 8 10 2 4 6 8 10
x y (b) G
2 4 6 8 10 1
x A(x) (c) A
2 4 6 8 10 1
(a) B(y) B
Granule G in X×Y G = A×B A is an interval in X B is an interval in Y
Pedrycz and Gomide, FSE 2007
5 10 2 4 6 8 10
x y R (b) Fuzzy granule R
2 4 6 8 10 1
x A(x) (c) A
2 4 6 8 10 1
(a) B(y) B
fuzzy granule R in X×Y R = A×B A is a fuzzy set on X B is a fuzzy set on Y
Pedrycz and Gomide, FSE 2007
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 x (b) Fuzzy granules Ri R1 R2 R3 R4 R5 y 2 4 6 8 10 1 x (c) A1 A2 A3 A4 A5 2 4 6 8 10 1 (a) B(y) B1 B2 B3 B4 B5 y
Ri = Ai×Bi
Pedrycz and Gomide, FSE 2007
2 4 6 8 10 12 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 R6 R7 R8
f R
Ri = Ai×Bi
Pedrycz and Gomide, FSE 2007
1 2 3 4 5 6 7 8 9 10 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
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
Ri = Ai t Bi ⇒ Ri(x,y) = Ai(x) Bi(y) R = ∪ Ri ⇒ R(x,y) = max [Ri(x,y), i = 1,..., N ]
1 2 3 4 5 6 7 8 9 10 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
) (
1 1 1 i N i i N i i N i i
B A f R R
= =
⇒ = = = )) ( ) ( (
1
y B , x A f T R
i i i N i=
=
Pedrycz and Gomide, FSE 2007
2 4 6 8 10 2 4 6 8 10
x y (b) Lukasiewicz implication R R
2 4 6 8 10 1
x A(x) (c) A
2 4 6 8 10 1
(a) B(y) B
fuzzy rule R in X×Y R = f (A,B) Lukasiewicz implication
Pedrycz and Gomide, FSE 2007
Ri = f (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
1 2 3 4 5 6 7 8 9 10 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
Ri = f (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
1 2 3 4 5 6 7 8 9 10 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
Ri = fz (A,B) ⇒ Ri(x,y) = max [1 – Ai(x), min(Ai(x), Bi(y)] Zadeh implication R = ∩ Ri ⇒ R(x,y) = min [Ri(x,y), i = 1,..., 5] min t-norm
1 2 3 4 5 6 7 8 9 10 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
Ri = fz (A,B) ⇒ Ri(x,y) = max [1 – Ai(x), min(Ai(x), Bi(y)] Zadeh implication R = ∩ Ri ⇒ R(x,y) = R1(x,y) tl R2(x,y) tl .... tl Ri(x,y) Lukasiewicz t-norm
1 2 3 4 5 6 7 8 9 10 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 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
X X
granular constructs in the form of fuzzy points granules along different regions of the universes
Pedrycz and Gomide, FSE 2007
Basic idea of inference
2 4 6 8 10 12 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 )
2 4 6 8 10 12 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 R y is B B = ProjY (Ac ∩ R ) ⇓ B = ProjY ( I )
f
2 4 6 8 10 12 2 4 6 8 10 12
x y (a) I Ac R A B
Pedrycz and Gomide, FSE 2007
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 , x tR x A y B
x X ∈
=
⇒
Pedrycz and Gomide, FSE 2007
Compositional rule of inference X is A (X,Y) is R Y is B X is A (X,Y) is R Y is AoR
Pedrycz and Gomide, FSE 2007
procedure FUZZY-INFERENCE (A, R) returns a fuzzy set input : fuzzy relation: R fuzzy set: A local: x, y: elements of X and Y t: t-norm for all x and y do Ac(x,y) ← A(x) for all x and y do I(x,y) ← Ac(x,y) t R(x,y) B(y) ← supx I(x,y) return B
Pedrycz and Gomide, FSE 2007
Pedrycz and Gomide, FSE 2007
Pedrycz and Gomide, FSE 2007
Pedrycz and Gomide, FSE 2007
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
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 = ??
Pedrycz and Gomide, FSE 2007
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
,...,N i z , y , x R y , x P z C R P R P C
i y , x N i i
= = = =
=
Pedrycz and Gomide, FSE 2007
rule th i
activation
degree the is } 1 ), ( max{ } 1 ) max{( ) ( ) ( ) ( ) Poss( )] ( ) ( [ sup ) Poss( )] ( ) ( [ sup )]} ( ) ( ) ( ) ( ) ( { sup )]} ( ), ( {min[ sup ) ( ) (
1 1 1
− = ∧ = = ∧ = ∧ ∧ = ′ = = ∧ = = ∧ ∧ ∧ ∧ ∧ = = ′ = ′ ′ = = = =
= = = i i i i i i i i i i i i i y i i i x i i i y , x i y , x i i i N i i N i i N i i
λ N , , i z C λ N , , i , C n m z C z C n m z C n B , B y B y B m A , A x A x A z C y B x A y B x A z , y , x R y , x P z C R P C C R P R P R P C
procedure MIN-MAX-MODEL (A,B) returns a fuzzy set local: fuzzy sets: Ai, Bi, Ci, i =1,.., N activation degrees: λi Initialization C = ∅ for i = 1: N do mi = 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
Pedrycz and Gomide, FSE 2007
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
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
= = = =
if 1 ) (
if 1 ) (
y y B and x x x A
Numeric output
i N i i i N i i i i
v n m v n m z dz z C dz z zC z values modal average weighted ) ( ) ( ation defuzzific centroid ) ( ) (
1 1
∑ ∑ ∫ ∫
= =
∧ ∧ = =
Z Z
Pedrycz and Gomide, FSE 2007
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 C1 rules R2: If X is A2 and Y is B2 then Z is C2 N = 2, centroid defuzzification
0.5 1 1.5 2 0.5 1
x Ai(x) A1 A2
0.5 1 1.5 2 0.5 1
y Bi(y) B2 B2 (a) Input and output fuzzy sets
0.5 1 1.5 2 0.5 1
z Ci(z) C1 C2
1 2
1 2
0.5
x (b) Input-output mapping y z
Pedrycz and Gomide, FSE 2007
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 i i i i i i y , x i
C w z C z C y B x A y B x A z C
1
) ( )] ( ) ( ) ( ) ( ) ( [ sup ) (
Using the compositional rule of inference (t = min) Additive fuzzy models (Kosko, 1992)
Pedrycz and Gomide, FSE 2007
Ai Aj 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
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
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 C1 rules R2: If X is A2 and Y is B2 then Z is C2 N = 2 w1 = w2 = 1, centroid defuzzification
0.5 1 1.5 2 0.5 1
x Ai(x) A1 A2
0.5 1 1.5 2 0.5 1
y Bi(y) B2 B2 (a) Input and output fuzzy sets
0.5 1 1.5 2 0.5 1
z Ci(z) C1 C2
1 2
1 2
0.5
x (b) Input-output mapping y z
Pedrycz and Gomide, FSE 2007
1- Product–probabilistic sum
) ( ) ( ) ( ) (
1
z C S z C z C n m z C
i N i p i i i i
′ = = ′
=
2- Product–sum
∑
=
′ = = ′
N i i i i i i
z C z C z C n m z C
1
) ( ) ( ) ( ) (
1 2
1 2
0.5 x (b) Input-output mapping of product-probabilistic sum model y z
1 2
1 2
0.5 x (d) Input-output mapping of product-sum model y z
Pedrycz and Gomide, FSE 2007
3 - Bounded product-bounded sum
] 1 [ } 1 min{ } 1 max{ ) ( ) ( ) ( ) (
1
, a,b b a , b a b a , b a z C z C z C n m z C
i N i i i i i
∈ + = ⊕ − + = ⊗ ′ ⊕ = ⊗ ⊗ = ′
=
1 2
1 2
0.5 1
x (c) Input-output mapping of bounded product-bounded sum model y z
Pedrycz and Gomide, FSE 2007
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
∑ ∑
= =
= =
N i i i i N i i i
y , x λ λ w , y , x f y , x w z
1 1
) ( ) ( ) (
Pedrycz and Gomide, FSE 2007
⊗ 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
P: X is x inputs x ∈ [0, 3] R1: If X is A1 then z = x rules R2: If X is A2 then z = – x + 3
0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1.2 x Ai A1 A2 (a) Antecedent fuzzy sets 0.5 1 1.5 2 2.5 3 0.5 1 1.5 2 2.5 3 f1 f2 x y (b) Consequent functions
Pedrycz and Gomide, FSE 2007
∈ + − ∈ + − + ∈ = ) 3 2 [ if 3 ] 2 1 [ if ) 3 )( ( ) ( ] 1 ( if
2 1
, x x , x x x A x x A , x x z
0.5 1 1.5 2 2.5 3 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
Pedrycz and Gomide, FSE 2007
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)
1 2 3 4 5 0.2 0.4 0.6 0.8 1 1.2 A1 A2 A3 (a) Antecedent fuzzy sets x y
1 2 3 4 5
0.5 1 1.5 (b) output of the functional fuzzy model y x
Pedrycz and Gomide, FSE 2007
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
1 2 3 4 5 0.2 0.4 0.6 0.8 1 1.2 A1 A2 A3 (a) Antecedent fuzzy sets x y
1 2 3 4 5
0.5 1 1.5 x y (c) output of the functional fuzzy model
Pedrycz and Gomide, FSE 2007
Ri: The more X is Ai the more Z is Ci i = 1,..., N
i N i i i i i
i i
C C C x A z C y x R
1 1
) (
) ( ) ( if 1 ) , (
= α α =
= ′ = ≥ =
Pedrycz and Gomide, FSE 2007
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
’
Pedrycz and Gomide, FSE 2007
P: X is x inputs x ∈ [0, 3] R1: The more X is A1 the more Z is C1 rules R2: The more X is A1 the more Z is C1
1 1.5 2 2.5 3 3.5 4 0.2 0.4 0.6 0.8 1
x Ai A1 A2
1 1.5 2 2.5 3 3.5 4 0.2 0.4 0.6 0.8 1
z Ci C1 C2
1 1.5 2 2.5 3 3.5 4 1 1.5 2 2.5 3 3.5 4
x z
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
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
Fuzzy Controller Process r 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
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
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
R1 v1 v2 v3 v4 R2 R3 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
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 i k i k i ik N i i k i ik k
x λ x λ w , , f w y ˆ
1 1
) ( ) ( ) ( a x
T T 1 T
] 1 [
k ik ik N i i ik k
w , , y ˆ x z a z = = ∑
=
Pedrycz and Gomide, FSE 2007
[ ]
= =
N Nk k k k N
y ˆ a a a z z z a a a a
1 T T 2 T 1 2 1
= =
T T 2 T 1 T 2 T 22 T 12 T 1 T 12 T 11 2 1 NM M M N N M
Z y ˆ y ˆ y ˆ z z z z z z z z z y
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)–1Zi T
= − = ∑
= T T 2 T 1 1 2
) ( J Min
iM i i i N i i i L
Z Z z z z a y a
a
Pedrycz and Gomide, FSE 2007
Given input/output data: {(x1, y1), (x2, y2),....,(xM, yM)}
data consistency completeness accuracy rules Issue: quality of the rules
Pedrycz and Gomide, FSE 2007
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
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 merged different rules
Pedrycz and Gomide, FSE 2007
|} ) ( ) ( | | ) ( ) ( {| ) cons(
1 k j k i k j M i k i
x A x A y B y B j , 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 k j k i k j M i k i
y B , y B x A , x A j , i ⇒ = ∑
=
Alternatively
∑
=
=
N j
j i N i
1
) , cons( 1 ) cons(
Pedrycz and Gomide, FSE 2007
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
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