Research Seminar on

Aggregation Operators in Fuzzy Control

Andrea Zemankova

Aggregation operators - what they are?

Which car to choose? It should be fast, with low consumption,

nice design and cheap.

Usually no car on the market match all the criteria.

Speed Consumption Design PriceCar 1 0.3 0.4 0.9 0.8Car 2 0.9 0.5 0.5 0.4Car 3 0.2 0.9 0.8 0.3

How to decide which one is the best?

Which car is the best?

There are many way how to decide which one of these cars is thebest. Everything depends on the user preferences.

Example 1 If all criteria are equally important we can use the sum:

Car 1 0.3 + 0.4 + 0.9 + 0.8 = 2.4Car 2 0.9 + 0.5 + 0.5 + 0.4 = 2.3Car 3 0.2 + 0.9 + 0.8 + 0.3 = 2.2

or the product (in this case car which has middle score in all criteriais preferred to one which has half of the criteria with high score andhalf with low score)

Car 1 0.3 · 0.4 · 0.9 · 0.8 = 0.0864Car 2 0.9 · 0.5 · 0.5 · 0.4 = 0.09Car 3 0.2 · 0.9 · 0.8 · 0.3 = 0.0432

Example 2 If Consumption is the most important criterion, Designis the second most important and Speed and Price are less important,we can use weighted mean

Car 1 18 · 0.3 + 1

2 · 0.4 + 14 · 0.9 + 1

8 · 0.8 = 0.5625

Car 2 18 · 0.9 + 1

2 · 0.5 + 14 · 0.5 + 1

8 · 0.4 = 0.5375

Car 3 18 · 0.2 + 1

2 · 0.9 + 14 · 0.8 + 1

8 · 0.3 = 0.7125

Summary Every candidate is best in some criterion, but we need todecide which one is generally the best for us and order the cars fromthe best to the worst.

In other words we need aggregate scores for all criteria and assign toeach car just one number. Then we can see immediately which car isthe best.

Aggregation operators - what they are?

Definition

• n-ary aggregation operator is a non-decreasing function

A : [0,1]n −→ [0,1] such that

A(0, . . . ,0︸ ︷︷ ︸n−times

) = 0 and A(1, . . . ,1︸ ︷︷ ︸n−times

) = 1

• aggregation operator is a function A :⋃

n∈N[0,1]n −→ [0,1] such that

A(x) = x and A|[0,1]n is an n-ary aggregation operator for all n ∈ N.

Outline

• T-norms

• T-conorms, Uninorms

• Related operations - Implications, Negations

• Other types of aggregation operators (QAM,QWM,OWA,...)

Triangular norms

Notion ”t-norm” appeared first in 1942 in the paper of Karl Menger

and it was used in statistical metric spaces.

Later Schweizer and Sklar changed slightly the definition of a t-norm

and since then people find plenty of applications for t-norms in fuzzy

control, fuzzy measures and integrals, decision making, expert sys-

tems, . . .

In fuzzy control, t-norms are usually used when we need to find the

intersection of fuzzy sets, or when we need to model a conjunction

(AND connective).

Classical logic {0,1} Many-valued logics⇒ ÃLukasiewicz {0, 12,1}

1 0 1

0 0 0

0 1

1 0 12 1

12 0 ? 1

2

0 0 0 0

0 12 1

Example There are two sets: set of young people and set of tallpeople defined in ÃLukasiewicz three valued logic, i.e., every one eitheris definitely tall, or is definitely not tall or is something in between -i.e., he is tall to the degree 1

2 (similarly for the set of young people).

If somebody is young to degree 12 and is tall also to degree 1

2, whatis the truth value of the statement that he is young and tall?

In other words, to what degree does he belong to the intersection ofthe set of young people and the set of tall people?

Triangular norms

Fuzzy logic - truth values - [0,1] - AND operator - triangular norm

Intersection of fuzzy sets

Crisp set x ∈ A ∩B iff (x ∈ A&x ∈ B)

µA(x) =

1 if x ∈ A

0 elseµB(x) =

1 if x ∈ B

0 else

µA∩B(x) = AND(µA(x), µB(x)) =

1 if x ∈ A&x ∈ B

0 else

Fuzzy set µA∩B(x) = T (µA(x), µB(x))

Triangular norms

Function T : [0,1]2 −→ [0,1] is called a t-norm if it is

• Commutative, i.e., T (x, y) = T (y, x)

• Non-decreasing, i.e., T (x, y) ≤ T (x, z) whenever y ≤ z

• 1 is neutral element, i.e., T (1, x) = x

• Associative, i.e., T (x, T (y, z)) = T (T (x, y), z)

Basic t-norms

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

TP (x, y) = x · y

TL(x, y) = max(0, x + y − 1)

TD(x, y) = 0 (except the boundary, where TD(1, x) = x = TD(x,1))

TD ≤ TL ≤ TP ≤ TM

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Car example

First car

TM(0.3,0.4,0.9,0.8) = 0.3

TP(0.3,0.4,0.9,0.8) = 0.0864

TL(0.3,0.4,0.9,0.8) = 0

TD(0.3,0.4,0.9,0.8) = 0

Majority of non-continuous t-norms

is not suitable for fuzzy control

applications.

Class of continuous t-norms can be divided to two main groups.Archimedean and non-Archimedean. Archimedean are those t-normswhich except of 0 and 1 has no idempotent element.

If for x we have T (x, x) = x then x is called an idempotent elementof T.

For Archimedean T and x ∈ ]0,1[ there is T (x, x) < x

Ordinal sum Additive generator

T1

T2

T3

TM

T (x, y) = t−1(min(t(0), t(x)+ t(y)))

Isomorphism of t-norms

ϕ−1(T (ϕ(x), ϕ(y)))

Additive generator

Additive generator t(x) = − ln(x) then

T (x, y) = t−1(min(t(0), t(x) + t(y))) = e−(min(∞,− ln(x)+(− ln(y)))) =

e−(− ln(x·y)) = eln(x·y) = x · y

or if t(x) = 1− x then

T (x, y) = t−1(min(t(0), t(x) + t(y))) = 1− (min(1,1− x + (1− y))) =

1−min(1,2− x− y) = max(0,1− (2− x− y)) = max(0, x + y − 1)

Isomorphism

Isomorphism ϕ(x) = x2 then

Tϕ(x, y) = ϕ−1(T (ϕ(x), ϕ(y))) = (T (x2, y2))12

if T = TL we have TLϕ(x, y) = (max(0, x2+y2−1))12 - Schweizer-Sklar

t-norm with parameter p = 2

if T = TP we have TPϕ(x, y) = (x2 · y2)12 = x · y

Archimedean t-norms are isomorphic either to product TP (strict)

or to ÃLukasiewicz t-norm TL (nilpotent).

Ordinal sum

T = (〈TL,0, 12〉, 〈TP, 3

4,1〉)

T =

max(0, x + y − 12) if (x, y) ∈

]0, 1

2

[2,

4(x− 34)(y − 3

4) + 34 if (x, y) ∈

]34,1

[2,

min(x, y) else.

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Every continuous t-norm is an ordinal sum of Archimedean t-norms.

• conjunction in fuzzy logic, intersection of fuzzy sets

• aggregation

• fuzzy relations (f.e., T -equivalency (similarities), T -E-ordering, T -partition)

• addition of distribution functions, fuzzy numbers

• fuzzy integrals (f.e., Sugeno, Shilkret, Sugeno-Weber)

• copulas (1-Lipschitz t-norms) modelling of stochastic dependenceof random vectors

• we can derive dual t-conorm, residual implication, negation

t-conorm commutative, associative, non-decreasing, 0 is neutral ele-ment

t-conorm is used (besides other applications) to model OR in fuzzylogic

uninorm commutative, associative, non-decreasing, there is a neutralelement e ∈ ]0,1[

Aggregation operator is called compensatory if low score in one ofthe criteria (one small input) can be compensated by high score inanother of criteria (high input).

T-norms are non-compensatory operations since T ≤ min and thusone low input causes that the result will never be greater than thisinput.

T-conorms are fully compensatory operations since C ≥ max and thusone high input causes that the result will never be smaller than thisinput.

Aggregation operator A is

• conjunctive if A ≤ min (t-norm,x1x22x3

3 · · ·xnn)

• disjunctive if A ≥ max (t-conorm,x1x122x

133 · · ·x

1nn)

Uninorm U(x, y)

T

Cmin ≤ U ≤ max

min ≤ U ≤ max

e

e

dual t-conorm C(x, y) = 1− T (1− x,1− y)

CM(x, y) = max(x, y), CP (x, y) = x + y − xy, CL = min(x + y,1)

residual implication x →T y = sup{z ∈ [0,1] | T (x, z) ≤ y}

x →T y = 1 iff x ≤ y and otherwise

x →M y = y, x →P y = yx, x →L y = y − x + 1

negation nT (x) = x →T 0

nL(x) = 1− x, nP (x) = nM(x) = 0 for x > 0 and 1 for x = 0.

Note that residual implication and negation are important operatorsused in fuzzy logic, but they are not aggregation operators!

Idempotent (averaging) aggregation operators

Arithmetic mean x1+...+xnn f(x) = x

Geometric mean n√

x1 · · ·xn f(x) = ln(x)

Harmonic mean n1x1

+...+ 1xn

f(x) = 1x

Quasi-arithmetic mean f−1(f(x1)+...+f(xn)n )

f : [0,1] −→ R, f is continuous and monotone (f : S −→ R)

Exercise

What kind of mean is generated by f(x) = x2 for x ∈ [0,1]?

What kind of mean is generated by f(x) = 3x2 + 2 for x ∈ [0,1]?

Properties of quasi-arithmetic means

For (x, . . . , x) we have

Af(x, . . . , x) = f−1(f(x)+...+f(x)n ) = f−1(f(x)) = x

Thus quasi-arithmetic means are idempotent

For an idempotent aggregation operator A always

min ≤ A ≤ max

If g(x) = a · f(x) + c then g−1(x) = f−1(x−ca )

Ag(x1, . . . , xn) = g−1(g(x1)+...+g(xn)n ) = f−1(

(a·f(x1)+c+...+a·f(xn)+c

n )−ca ) =

f−1(f(x1)+...+f(xn)n ) = Af(x1, . . . , xn)

Quasi-arithmetic means are invariant wrt. offsets and scaling of f.

Exercise

There are 7 numbers to be aggregated by quadratic mean. However,

first three of them are lost, but we remember that quadratic mean of

these three numbers was 0.5. Compute the quadratic mean of all 7

numbers, where last four are 0.2,0.3,0.4,0.6.

Properties of quasi-arithmetic means

A(x1, . . . , xn) = A(x1, . . . , xk, xk+1, . . . , xn) = A(m, . . . , m︸ ︷︷ ︸k−times

, xk+1, . . . , xn),

where m = A(x1, . . . , xk)

Subsets of elements can be aggregated a priori, without altering the

mean, given that the multiplicity of elements is maintained (quasi-

arithmetic means are decomposable).

Weighted quasi-arithmetic means

Weighted mean w1x1+...+wnxnw1+...+wn

Weighted geometric mean (w1+...+wn)√

xw11 · · ·xwn

n

((xw1

1 · · ·xwnn )

1w1+...+wn

)

Weighted quasi-aritmetic mean f−1(w1f(x1)+...+wnf(xn)w1+...+wn

)

w1, . . . , wn ∈ [0,1]

Weighted means are used in TSK systems.

Exercise There are three students in the course. Their scores inMaths and Physics are

Maths PhysicsStudent A 20 1Student B 9 9Student C 0 22

Try to find a weighted mean W (x, y) = w1x + w2y with w1 + w2 = 1such that the overall ranking of students will be

1. A > B > C2. A > C > B3. C > B > A4. C > A > B5. B > A > C6. B > C > A

OWA operators

Ordered weighted average (OWA)w1xσ(1)+...+wnxσ(n)

w1+...+wn

where σ is a permutation such that xσ(1) ≤ · · · ≤ xσ(n)

Example If w1 = 1 and wi = 0 for i 6= 1 then

A(x1, . . . , xn) =w1xσ(1)+...+wnxσ(n)

w1+...+wn=

1·xσ(1)1 =

xσ(1) = min(x1, . . . , xn)

What operator do we get by setting wn = 1 and wi = 0 for i 6= n?

Example If n = 3, w1 = 0, w2 = 1, w3 = 0 then

A(x1, x2, x3) = xσ(2) = Med(x1, x2, x3)

is so-called Median (middle value).

We can obtain n-ary Median by putting wn+12

= 1 and wi = 0 for

i 6= n+12 in the case that n is odd and wn

2= 1

2, wn2+1 = 1

2 and wi = 0

for i 6= n2, n

2 + 1 in the case that n is even.

Choquet integral

Let σ be a permutation such that xσ(1) ≤ · · · ≤ xσ(n)

Let m : P({1, . . . , n}) −→ [0,1] be a fuzzy measure (monotone set

function with m({1, . . . , n}) = 1 and m(∅) = 0)

Then the Choquet integral for finite space X = {1, . . . , n} is defined

by

Ch(x1, . . . , xn) =n∑

i=1(xσ(i)−xσ(i−1))m(A(i)) =

n∑i=1

xσ(i)(m(A(i))−m(A(i+1)))

where xσ(0) = 0 and A(i) = {σ(i), . . . , σ(n)} with m(A(n+1)) = 0.

Choquet integral

A fuzzy measure m is said to be additive (on a finite space X) if

m(A) =∑

i∈A

m({i})

for all sets A ∈ X.

A fuzzy measure is said to be cardinal if it depends only on the

cardinality of sets, i.e., m(A) = m(B) whenever |A| = |B|.

Choquet integral

For additive fuzzy measure m(A(i)) = m({σ(i)})+ · · ·+m({σ(n)}) andthus

(m(A(i))−m(A(i+1))) = m({σ(i)})

Therefore Choquet integral is equal to

Ch(x1, . . . , xn) =n∑

i=1xσ(i)(m(A(i))−m(A(i+1))) =

n∑i=1

xσ(i)m({σ(i)})

Thus for an additive fuzzy measure Choquet integral is equal to a

weighted meann∑

i=1wixi with weights wi = m({i}).

Choquet integral

For cardinal fuzzy measure let us denote by m(k) measure of the set

with k elements. Then Choquet integral is equal to

Ch(x1, . . . , xn) =n∑

i=1xσ(i)(m(A(i))−m(A(i+1))) =

xσ(1)(1−m(n−1))+xσ(2)(m(n−1)−m(n−2))+ · · ·+xσ(n)(m(1)−0)

Thus for a cardinal fuzzy measure Choquet integral is equal to an

OWA operatorn∑

i=1wixσ(i) with weights wi = m(n− i + 1)−m(n− i).

Choquet integral

For infinite space X and f : X −→ [0,1] the Choquet integral is given

by

Ch(f) =∫ 1

0m({x ∈ X | f(x) ≥ t})dt

Ordinal scales

Sometimes we need to work on an ordinal scale, where weightedmeans (Choquet integral) cannot be used. Like for example scaleK = {weak,medium, excellent}.

In such a case addition and multiplication is replaced by maximumand minimum and thus instead of weighted mean here the weightedmaximum is used.

On [0,1] scale, weighted maximum operator is given by∨

i

wi ∧ xi,

where wi ∈ [0,1] are weights with∨i

wi = 1.

Weighted maximum is used in Mamdani systems.

Example On K = {bad,weak, fair,good, excellent} for n = 3 let

Design Consumption PriceCar 1 weak excellent fairCar 2 bad fair excellentImportance of the criteria excellent good fair

i.e., for Car 1 x1 = weak, x2 = excellent, x3 = fair with weightsw1 = excellent, w2 = good, w3 = fair. Then weighted maximum isgiven by

∨

i

wi ∧ xi = (w1 ∧ x1)∨

(w2 ∧ x2)∨

(w3 ∧ x3) =

(excellent ∧weak)∨

(good ∧ excellent)∨

(fair ∧ fair) =

weak ∨ good ∨ fair = good

Similarly weighted maximum for Car 2 is fair.

Sugeno integral

Let σ be a permutation such that xσ(1) ≤ · · · ≤ xσ(n)

Let m : {1, . . . , n} −→ [0,1] be a fuzzy measure (monotone set functionwith m({1, . . . , n}) = 1 and m(∅) = 0)

Then the Sugeno integral for finite space X = {1, . . . , n} is defined by

S(x1, . . . , xn) =n∨

i=1

xσ(i) ∧m(A(i))

where A(i) = {σ(i), . . . , σ(n)}

A fuzzy measure m is said to be maxitive (on a finite space X) if

m(A) =∨

i∈A

m({i})

for all sets A ∈ X.

For maxitive fuzzy measure, Sugeno integral is equal to

S(x1, . . . , xn) =n∨

i=1xσ(i) ∧m(A(i)) =

n∨i=1

xσ(i) ∧ (∨

j≥σ(i)m({j}))

After some computation we get

n∨i=1

xσ(i) ∧ (∨

j≥σ(i)m({j})) =

n∨i=1

xσ(i) ∧ (m({σ(i)}))

Thus for a maxitive fuzzy measure Sugeno integral is equal to a

weighted maximumn∨

i=1wi ∧ xi with weights wi = m({i}).

Sugeno integral

For cardinal fuzzy measure, Sugeno integral is equal to ordered weighted

maximumn∨

i=1wixσ(i) with weights wi = m(n− i + 1).

For infinite space X and f : X −→ [0,1] the Sugeno integral is givenby

S(f) =∨

t∈[0,1]

t ∧m({f(x) ≥ t})