Upload
others
View
10
Download
0
Embed Size (px)
Citation preview
Survey on aggregation theory
Kira Adaricheva Agata Pilitowska*
Yeshiva University
Warsaw University of Technology
Workshop ”Algebra across the borders”Yeshiva University, New York, August 11, 2011
*The work has been supported by the European
Union in the framework of European Social Fund through the Warsaw
University of Technology Development Programme
Adaricheva, Pilitowska Survey on aggregation theory
Survey on aggregation theory
PART I - INTRODUCTION
Definition, basic examples, elementary property
Some classes of aggregation functions
MeansAssociative aggregation functionst-norms, t-conorms, uninormsFuzzy integrals
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation
The process of aggregation
An arbitrarily long vector of inputs x = (x1, . . . , xn)
⇉
A single output value A(x)
Example
Let {1, . . . , n} be a set of players of a cooperative game to whichsome judgements {x1, . . . , xn} are done.
In order to reach a consensus on these judgements, aggregationfunctions may be applied.
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation
The process of aggregation
An arbitrarily long vector of inputs x = (x1, . . . , xn)
⇉
A single output value A(x)
Example
Let {1, . . . , n} be a set of players of a cooperative game to whichsome judgements {x1, . . . , xn} are done.
In order to reach a consensus on these judgements, aggregationfunctions may be applied.
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation
The process of aggregation
An arbitrarily long vector of inputs x = (x1, . . . , xn)
⇉
A single output value A(x)
Example
Let {1, . . . , n} be a set of players of a cooperative game to whichsome judgements {x1, . . . , xn} are done.
In order to reach a consensus on these judgements, aggregationfunctions may be applied.
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation
The process of aggregation
An arbitrarily long vector of inputs x = (x1, . . . , xn)
⇉
A single output value A(x)
Example
Let {1, . . . , n} be a set of players of a cooperative game to whichsome judgements {x1, . . . , xn} are done.
In order to reach a consensus on these judgements, aggregationfunctions may be applied.
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation
The process of aggregation
An arbitrarily long vector of inputs x = (x1, . . . , xn)
⇉
A single output value A(x)
Example
Let {1, . . . , n} be a set of players of a cooperative game to whichsome judgements {x1, . . . , xn} are done.In order to reach a consensus on these judgements, aggregationfunctions may be applied.
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation functions
Aggregation functions are used in:
pure mathematics (theory of means and averages, measureand integration theory),
applied mathematics (probability, statistics),
computer and engineering sciences (artificial intelligence,information theory, automated reasoning),
economics and finance (game theory, voting theory, decisionmaking),
social sciences,
physics and natural sciences
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation functions
I - a nonempty real interval, bounded or not
f : In → ℝ
Requirements:
boundary conditions
nondecreasing monotonicity
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation functions
I - a nonempty real interval, bounded or not
f : In → ℝ
Requirements:
boundary conditions
nondecreasing monotonicity
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation functions
I - a nonempty real interval, bounded or not
f : In → ℝ
Requirements:
boundary conditions
nondecreasing monotonicity
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation functions
I - a nonempty real interval, bounded or not
f : In → ℝ
Requirements:
boundary conditions
nondecreasing monotonicity
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation functions
I - a nonempty real interval, bounded or not
f : In → ℝ
Requirements:
boundary conditions
nondecreasing monotonicity
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation functions
Definition
Let I ⊆ ℝ, n ∈ ℕ and let x := (x1, . . . , xn) ∈ In. An aggregationfunction with n variables is a function A : In → I such that
A is nondecreasing: for any x, x∗ ∈ In
x ≤ x∗ ⇒ A(x) ≤ A(x∗)
A fulfills the boundary conditions
infx∈In
A(x) = inf I and supx∈In
A(x) = sup I.
If I = [0, 1], the boundary conditions say:
A(0, . . . , 0) = 0 and A(1, . . . , 1) = 1.
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation functions
Definition
Let I ⊆ ℝ, n ∈ ℕ and let x := (x1, . . . , xn) ∈ In. An aggregationfunction with n variables is a function A : In → I such that
A is nondecreasing: for any x, x∗ ∈ In
x ≤ x∗ ⇒ A(x) ≤ A(x∗)
A fulfills the boundary conditions
infx∈In
A(x) = inf I and supx∈In
A(x) = sup I.
If I = [0, 1], the boundary conditions say:
A(0, . . . , 0) = 0 and A(1, . . . , 1) = 1.
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation functions
An extended aggregation function
The mapping
A :∪n∈ℕ
In → I
such thatA(n) := A∣In
is an n-ary aggregation function for any n ∈ ℕ.
A sequence of functions
(A(n) : In → I)n∈ℕ,
whose nth element is an n-ary function A(n) : In → I.
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation functions
An extended aggregation function
The mapping
A :∪n∈ℕ
In → I
such thatA(n) := A∣In
is an n-ary aggregation function for any n ∈ ℕ.
A sequence of functions
(A(n) : In → I)n∈ℕ,
whose nth element is an n-ary function A(n) : In → I.
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation functions - Basic examples
Let x := (x1, . . . , xn) ∈ In.
The arithmetic mean function
Aℳ(x) :=1
n
n∑i=1
xi
The geometric mean function (for I ⊆ [0,∞])
Gℳ(x) := (n∏
i=1
xi )1n
The k-projection function, for any k ∈ {1, . . . , n}
Pk(x) := xk
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation functions - Basic examples
Let � be a permutation on the set {1, . . . , n} such thatx�(1) ≤ . . . ≤ x�(n).
The k-order statistic function, for any k ∈ {1, . . . , n}
OSk(x) := x�(k)
The extreme order statistic functions
OS1(x) = min{x1, . . . , xn} = Min(x)
OSn(x) = max{x1, . . . , xn} = Max(x)
The median of values x1, . . . , xn
Med(x1, . . . , xn) =
{x�(k) if n = 2k − 1x�(k)+x�(k+1)
2 if n = 2k .
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation functions - Basic examples
Let w := (w1, . . . ,wn) ∈ In be any (weight) vector such thatn∑
i=1wi = 1.
The weighted arithmetic mean function
WAℳw(x) :=n∑
i=1
wixi
The ordered weighted averaging function
OWAw(x) :=n∑
i=1
wix�(i)
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation functions - Basic examples
Let S be a nonempty subset of {1, . . . , n}.The partial minimum function (associated with S)
MinS(x) := mini∈S
xi
The partial maximum function (associated with S)
MaxS(x) := maxi∈S
xi
The sum Σ : (0,∞)n → (0,∞)
Σ(x) :=n∑
i=1
xi
The product Π : (0,∞)n → (0,∞)
Π(x) :=n∏
i=1
xi
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation functions - Examples
Assume I = [a, b] is a closed interval.
The smallest aggregation function
A⊥(x) :=
{b, if xi = b for all i ∈ {1, . . . , n}a, otherwise
The greatest aggregation function
A⊤(x) :=
{a, if xi = a for all i ∈ {1, . . . , n}b, otherwise
Non aggregation function
The constant function Kc : In → I, given by
Kc(x) := c ,
where c ∈ I is a fixed constant (unless I = {c}).
Adaricheva, Pilitowska Survey on aggregation theory
Aggregation functions - Examples
Assume I = [a, b] is a closed interval.
The smallest aggregation function
A⊥(x) :=
{b, if xi = b for all i ∈ {1, . . . , n}a, otherwise
The greatest aggregation function
A⊤(x) :=
{a, if xi = a for all i ∈ {1, . . . , n}b, otherwise
Non aggregation function
The constant function Kc : In → I, given by
Kc(x) := c ,
where c ∈ I is a fixed constant (unless I = {c}).
Adaricheva, Pilitowska Survey on aggregation theory
Elementary properties
IMPORTANT
The choice of aggregation function should be based uponproperties dedicated by the framework in which the aggregation isperformed.
Example
If we consider the aggregation of opinions in voting procedures,and as usual, the voters are anonymous, the aggregation functionmust be symmetric.
Adaricheva, Pilitowska Survey on aggregation theory
Elementary properties
IMPORTANT
The choice of aggregation function should be based uponproperties dedicated by the framework in which the aggregation isperformed.
Example
If we consider the aggregation of opinions in voting procedures,and as usual, the voters are anonymous, the aggregation functionmust be symmetric.
Adaricheva, Pilitowska Survey on aggregation theory
Elementary properties
IMPORTANT
The choice of aggregation function should be based uponproperties dedicated by the framework in which the aggregation isperformed.
Example
If we consider the aggregation of opinions in voting procedures,and as usual, the voters are anonymous, the aggregation functionmust be symmetric.
Adaricheva, Pilitowska Survey on aggregation theory
Continuity
Example
The function A : (0,∞)n → (0,∞)
A(x) :=
(n∑
i=1xi )
2 −n∑
i=1x2i
n∑i=1
xi
is continuous aggregation function.
Adaricheva, Pilitowska Survey on aggregation theory
Idempotency
Definition
A : In → ℝ is an idempotent function if, for all x ∈ I,
A(x , . . . , x) = x .
Aℳ, WAℳw, OWAw, Min, Max and Med are idempotentaggregation functions.
Adaricheva, Pilitowska Survey on aggregation theory
Idempotency
Definition
A : In → ℝ is an idempotent function if, for all x ∈ I,
A(x , . . . , x) = x .
Aℳ, WAℳw, OWAw, Min, Max and Med are idempotentaggregation functions.
Adaricheva, Pilitowska Survey on aggregation theory
Idempotency
Definition
An element e ∈ I is idempotent for A : [a, b]n → ℝ if,
A(e, . . . , e) = e.
Let c ∈ (0, 1) be an arbitrarily chosen element. The aggregationfunction Ac : [0, 1]n → [0, 1]
Ac(x) := Med(0, c +n∑
i=1
(xi − c), 1)
is not idempotent but has a nonextreme idempotent element c .
Adaricheva, Pilitowska Survey on aggregation theory
Idempotency
Definition
An element e ∈ I is idempotent for A : [a, b]n → ℝ if,
A(e, . . . , e) = e.
Let c ∈ (0, 1) be an arbitrarily chosen element. The aggregationfunction Ac : [0, 1]n → [0, 1]
Ac(x) := Med(0, c +n∑
i=1
(xi − c), 1)
is not idempotent but has a nonextreme idempotent element c .
Adaricheva, Pilitowska Survey on aggregation theory
Idempotency
Definition
A : [a, b]n → ℝ is weakly idempotent function if a and b areidempotent elements for A.
Every aggregation function on I = [a, b] is weakly idempotent.
Adaricheva, Pilitowska Survey on aggregation theory
Idempotency
Definition
A : [a, b]n → ℝ is weakly idempotent function if a and b areidempotent elements for A.
Every aggregation function on I = [a, b] is weakly idempotent.
Adaricheva, Pilitowska Survey on aggregation theory
Symmetry
Definition
A : In → ℝ is symmetric function if, for all x ∈ In and anypermutation � on the set {1, . . . , n}
A(x) = A(x�(1), . . . , x�(n)).
Functions: Aℳ, Gℳ and OWAw are symmetric.Nonsymmetric aggregation function is WAℳw.
Adaricheva, Pilitowska Survey on aggregation theory
Symmetry
Definition
A : In → ℝ is symmetric function if, for all x ∈ In and anypermutation � on the set {1, . . . , n}
A(x) = A(x�(1), . . . , x�(n)).
Functions: Aℳ, Gℳ and OWAw are symmetric.
Nonsymmetric aggregation function is WAℳw.
Adaricheva, Pilitowska Survey on aggregation theory
Symmetry
Definition
A : In → ℝ is symmetric function if, for all x ∈ In and anypermutation � on the set {1, . . . , n}
A(x) = A(x�(1), . . . , x�(n)).
Functions: Aℳ, Gℳ and OWAw are symmetric.Nonsymmetric aggregation function is WAℳw.
Adaricheva, Pilitowska Survey on aggregation theory
Symmetry
Proposition [J.J.Rotman, 1995]
A : In → ℝ is symmetric function if and only if, for all x ∈ In wehave
A(x2, x1, x3, . . . , xn) = A(x1, x2, x3, . . . , xn)
A(x2, x3, . . . , xn, x1) = A(x1, x2, x3, . . . , xn)
Adaricheva, Pilitowska Survey on aggregation theory
Conjunction
Definition
A function A : In → ℝ is conjunctive if, for all x ∈ In
inf I ≤ A(x) ≤ Min(x).
The smallest aggregation function A⊥ is also the smallestconjunctive aggregation function.Min is the greatest conjunctive aggregation function.Min is also the only idempotent conjunctive aggregation function.
Adaricheva, Pilitowska Survey on aggregation theory
Conjunction
Definition
A function A : In → ℝ is conjunctive if, for all x ∈ In
inf I ≤ A(x) ≤ Min(x).
The smallest aggregation function A⊥ is also the smallestconjunctive aggregation function.
Min is the greatest conjunctive aggregation function.Min is also the only idempotent conjunctive aggregation function.
Adaricheva, Pilitowska Survey on aggregation theory
Conjunction
Definition
A function A : In → ℝ is conjunctive if, for all x ∈ In
inf I ≤ A(x) ≤ Min(x).
The smallest aggregation function A⊥ is also the smallestconjunctive aggregation function.Min is the greatest conjunctive aggregation function.
Min is also the only idempotent conjunctive aggregation function.
Adaricheva, Pilitowska Survey on aggregation theory
Conjunction
Definition
A function A : In → ℝ is conjunctive if, for all x ∈ In
inf I ≤ A(x) ≤ Min(x).
The smallest aggregation function A⊥ is also the smallestconjunctive aggregation function.Min is the greatest conjunctive aggregation function.Min is also the only idempotent conjunctive aggregation function.
Adaricheva, Pilitowska Survey on aggregation theory
Disjunction
Definition
A function A : In → ℝ is disjunctive if, for all x ∈ In
Max(x) ≤ A(x) ≤ sup I.
The greatest aggregation function A⊤ is also the greatestdisjunctive aggregation function.Max is the smallest conjunctive aggregation function.Max is the only idempotent disjunctive aggregation function.
Adaricheva, Pilitowska Survey on aggregation theory
Disjunction
Definition
A function A : In → ℝ is disjunctive if, for all x ∈ In
Max(x) ≤ A(x) ≤ sup I.
The greatest aggregation function A⊤ is also the greatestdisjunctive aggregation function.
Max is the smallest conjunctive aggregation function.Max is the only idempotent disjunctive aggregation function.
Adaricheva, Pilitowska Survey on aggregation theory
Disjunction
Definition
A function A : In → ℝ is disjunctive if, for all x ∈ In
Max(x) ≤ A(x) ≤ sup I.
The greatest aggregation function A⊤ is also the greatestdisjunctive aggregation function.Max is the smallest conjunctive aggregation function.
Max is the only idempotent disjunctive aggregation function.
Adaricheva, Pilitowska Survey on aggregation theory
Disjunction
Definition
A function A : In → ℝ is disjunctive if, for all x ∈ In
Max(x) ≤ A(x) ≤ sup I.
The greatest aggregation function A⊤ is also the greatestdisjunctive aggregation function.Max is the smallest conjunctive aggregation function.Max is the only idempotent disjunctive aggregation function.
Adaricheva, Pilitowska Survey on aggregation theory
Internality
Definition
A function A : In → ℝ is internal if, for all x ∈ In
Min(x) ≤ A(x) ≤ Max(x).
Proposition [B.de Finetti, 1931]
If A : In → ℝ is internal, then it is idempotent.
If A : In → ℝ is nondecreasing and idempotent then it is internal.
Adaricheva, Pilitowska Survey on aggregation theory
Internality
Definition
A function A : In → ℝ is internal if, for all x ∈ In
Min(x) ≤ A(x) ≤ Max(x).
Proposition [B.de Finetti, 1931]
If A : In → ℝ is internal, then it is idempotent.
If A : In → ℝ is nondecreasing and idempotent then it is internal.
Adaricheva, Pilitowska Survey on aggregation theory
Internality
Definition
A function A : In → ℝ is internal if, for all x ∈ In
Min(x) ≤ A(x) ≤ Max(x).
Proposition [B.de Finetti, 1931]
If A : In → ℝ is internal, then it is idempotent.If A : In → ℝ is nondecreasing and idempotent then it is internal.
Adaricheva, Pilitowska Survey on aggregation theory
Associativity
Min and Max are associative functions.
Functions Aℳ and Gℳ are not associative.
Extended associativity
A sequence of functions (A(n) : In → ℝ)n∈ℕ is associative if, forany x ∈ I, x1 ∈ Ip and x2 ∈ Ik with n = p + k ,
A(1)(x) = x , and
A(n)(x1, x2) = A(2)(A(p)(x1),A(k)(x2)).
The aggregation procedure can be decomposed into partialaggregation and we can start with the aggregation procedurebefore knowing all inputs to be aggregated.
Adaricheva, Pilitowska Survey on aggregation theory
Associativity
Min and Max are associative functions.Functions Aℳ and Gℳ are not associative.
Extended associativity
A sequence of functions (A(n) : In → ℝ)n∈ℕ is associative if, forany x ∈ I, x1 ∈ Ip and x2 ∈ Ik with n = p + k ,
A(1)(x) = x , and
A(n)(x1, x2) = A(2)(A(p)(x1),A(k)(x2)).
The aggregation procedure can be decomposed into partialaggregation and we can start with the aggregation procedurebefore knowing all inputs to be aggregated.
Adaricheva, Pilitowska Survey on aggregation theory
Associativity
Min and Max are associative functions.Functions Aℳ and Gℳ are not associative.
Extended associativity
A sequence of functions (A(n) : In → ℝ)n∈ℕ is associative if, forany x ∈ I, x1 ∈ Ip and x2 ∈ Ik with n = p + k ,
A(1)(x) = x , and
A(n)(x1, x2) = A(2)(A(p)(x1),A(k)(x2)).
The aggregation procedure can be decomposed into partialaggregation and we can start with the aggregation procedurebefore knowing all inputs to be aggregated.
Adaricheva, Pilitowska Survey on aggregation theory
Associativity
Min and Max are associative functions.Functions Aℳ and Gℳ are not associative.
Extended associativity
A sequence of functions (A(n) : In → ℝ)n∈ℕ is associative if, forany x ∈ I, x1 ∈ Ip and x2 ∈ Ik with n = p + k ,
A(1)(x) = x , and
A(n)(x1, x2) = A(2)(A(p)(x1),A(k)(x2)).
The aggregation procedure can be decomposed into partialaggregation and we can start with the aggregation procedurebefore knowing all inputs to be aggregated.
Adaricheva, Pilitowska Survey on aggregation theory
Means
Definition
An n-ary internal aggregation function A : In → I is called a mean.
Adaricheva, Pilitowska Survey on aggregation theory
Quasi-arithmetic mean
Definition
Let f : I→ ℝ be a continuous and strictly monotonic function.The n-ary quasi-arithmetic mean (generated by f ) is the functionAf : I n → I defined as
Af (x) := f −1(1
n
n∑i=1
f (xi )).
Remark
Quasi-arithmetic means are symmetric functions.
Adaricheva, Pilitowska Survey on aggregation theory
Quasi-arithmetic mean
Definition
Let f : I→ ℝ be a continuous and strictly monotonic function.The n-ary quasi-arithmetic mean (generated by f ) is the functionAf : I n → I defined as
Af (x) := f −1(1
n
n∑i=1
f (xi )).
Remark
Quasi-arithmetic means are symmetric functions.
Adaricheva, Pilitowska Survey on aggregation theory
Quasi-arithmetic means - Examples
Arithmetic mean
Aℳ(x) :=1
n
n∑i=1
xi , with f (x) = x
Quadratic mean
Qℳ(x) := (1
n
n∑i=1
x2i )12 , with f (x) = x2
Geometric mean
Gℳ(x) := (n∏
i=1
xi )1n , with f (x) = log x
Adaricheva, Pilitowska Survey on aggregation theory
Quasi-arithmetic means - Examples
Harmonic mean
ℋℳ(x) :=1
1n
n∑i=1
1xi
, with f (x) = x−1
Root mean power
ℳid�(x) := (1
n
n∑i=1
x�i )1� , with f (x) = x�, 0 ∕= � ∈ ℝ
Exponential mean
ℰℳ�(x) :=1
�ln(
1
n
n∑i=1
e�xi ), with f (x) = e�x , 0 ∕= � ∈ ℝ
Adaricheva, Pilitowska Survey on aggregation theory
Quasi-arithmetic means - Examples
A : (0, 1)n → (0, 1)
A(x) :=Gℳ(x)
Gℳ(x) + Gℳ(1− x),
where 1 := (1, . . . , 1).The function A is a quasi-arithmetic mean generated by theincreasing function f (x) = log x
1−x , whose inverse function is given
by f −1(x) = ex
1+ex .
Adaricheva, Pilitowska Survey on aggregation theory
Quasi-linear mean (weighted quasi-arithmetic mean)
Definition
Let f : I→ ℝ be a continuous, strictly monotonic function and let
w1, . . . ,wn > 0 be real numbers fulfillingn∑
i=1wi = 1. The n-ary
quasi-linear mean is the function Afw : I n → I defined as
Afw(x) := f −1(n∑
i=1
wi f (xi )).
Adaricheva, Pilitowska Survey on aggregation theory
Quasi-linear means - Examples
Weighted arithmetic mean
WAℳw(x) :=n∑
i=1
wixi , with f (x) = x
Weighted quadratic mean
WQℳw(x) := (n∑
i=1
wix2i )
12 , with f (x) = x2
Adaricheva, Pilitowska Survey on aggregation theory
Quasi-linear means - Examples
Weighted geometric mean
WGℳw(x) :=n∏
i=1
xwii , with f (x) = log x
Weighted root mean power
Wℳid�,w(x) := (n∑
i=1
wix�i )
1� , with f (x) = x�, 0 ∕= � ∈ ℝ
Adaricheva, Pilitowska Survey on aggregation theory
Associative aggregation functions
Theorem, [J.Aczel, 1948]
Let I be a real interval, which is open on one side. A : I2 → I iscontinuous, strictly increasing, and associative if and only if thereexists a continuous and strictly monotonic function f : I→ ℝ suchthat
A(x , y) = f −1(f (x) + f (y)).
Corollary
Every continuous, strictly increasing, and associative function issymmetric.
Adaricheva, Pilitowska Survey on aggregation theory
Associative aggregation functions
Theorem, [J.Aczel, 1948]
Let I be a real interval, which is open on one side. A : I2 → I iscontinuous, strictly increasing, and associative if and only if thereexists a continuous and strictly monotonic function f : I→ ℝ suchthat
A(x , y) = f −1(f (x) + f (y)).
Corollary
Every continuous, strictly increasing, and associative function issymmetric.
Adaricheva, Pilitowska Survey on aggregation theory
Associative aggregation functions
Theorem, [J.Aczel, 1948]
Let I be a real interval, which is open on one side. A : I2 → I iscontinuous, strictly increasing, and associative if and only if thereexists a continuous and strictly monotonic function f : I→ ℝ suchthat
A(x , y) = f −1(f (x) + f (y)).
Corollary
Every continuous, strictly increasing, and associative function issymmetric.
Adaricheva, Pilitowska Survey on aggregation theory
Associative aggregation functions
A(x , y) = f −1(f (x) + f (y))
Example
The sum Σ : (0,∞)2 → (0,∞)
Σ(x , y) = x + y , with f (x) = x
and the product Π : (0,∞)2 → (0,∞)
Π(x , y) = x ⋅ y = 10log(x)+log(y), with f (x) = log(x)
are continuous, strictly increasing, and associative functions.
Adaricheva, Pilitowska Survey on aggregation theory
Associative aggregation functions
A(x , y) = f −1(f (x) + f (y))
If x ∈ I then
A(x , x) = f −1(f (x) + f (x)) = f −1(2f (x)) = x ⇔ f (x) = 0.
Since f is strictly monotonic, A cannot be idempotent.
Corollary
There is no continuous, strictly increasing, idempotent, andassociative function.
Adaricheva, Pilitowska Survey on aggregation theory
Associative aggregation functions
A(x , y) = f −1(f (x) + f (y))
If x ∈ I then
A(x , x) = f −1(f (x) + f (x)) = f −1(2f (x)) = x ⇔ f (x) = 0.
Since f is strictly monotonic, A cannot be idempotent.
Corollary
There is no continuous, strictly increasing, idempotent, andassociative function.
Adaricheva, Pilitowska Survey on aggregation theory
Associative aggregation functions
A(x , y) = f −1(f (x) + f (y))
If x ∈ I then
A(x , x) = f −1(f (x) + f (x)) = f −1(2f (x)) = x ⇔ f (x) = 0.
Since f is strictly monotonic, A cannot be idempotent.
Corollary
There is no continuous, strictly increasing, idempotent, andassociative function.
Adaricheva, Pilitowska Survey on aggregation theory
Associative aggregation functions
Theorem [C.-H. Ling, 1965]
Let I = [a, b] be a real interval. A : I2 → I is continuous,nondecreasing, associative, and such that
A(b, x) = x , for x ∈ I (b is an identity for A)
A(x , x) < x , for x ∈ (a, b) (there is no idempotents for A in (a, b))
if and only if there exists a continuous and strictly decreasingfunction f : I→ [0,∞], with f (b) = 0, such that
A(x , y) = f −1(Min(f (x) + f (y), f (a))).
Adaricheva, Pilitowska Survey on aggregation theory
Associative aggregation functions
If A : I2 → I is an associative function then (I,A) is a semigroup.
Definition
Let {(Ik ,Ak) ∣ k ∈ K} be a collection of disjoint semigroupsindexed by a set K . The sum of {(Ik ,Ak) ∣ k ∈ K} is thesemigroup (
∪k∈K
Ik ,A) defined on the set-theoretic union∪
k∈KIk
under the following binary operation:
A(x , y) :=
{Ak(x , y), if exists k ∈ K such that x , y ∈ IkMin(x , y), if exist k1 ∕= k2 ∈ K with x ∈ Ik1 , y ∈ Ik2
Adaricheva, Pilitowska Survey on aggregation theory
Associative aggregation functions
If A : I2 → I is an associative function then (I,A) is a semigroup.
Definition
Let {(Ik ,Ak) ∣ k ∈ K} be a collection of disjoint semigroupsindexed by a set K . The sum of {(Ik ,Ak) ∣ k ∈ K} is thesemigroup (
∪k∈K
Ik ,A) defined on the set-theoretic union∪
k∈KIk
under the following binary operation:
A(x , y) :=
{Ak(x , y), if exists k ∈ K such that x , y ∈ IkMin(x , y), if exist k1 ∕= k2 ∈ K with x ∈ Ik1 , y ∈ Ik2
Adaricheva, Pilitowska Survey on aggregation theory
Associative aggregation functions
Definition
A semigroup (I = [a, b],A) is called Archimedean, if the functionA : I2 → I is continuous, nondecreasing, A(b, x) = x , for x ∈ I,and A(x , x) < x , for x ∈ (a, b) or A(a, x) = x , for x ∈ I, andA(x , x) < x , for x ∈ (a, b) (one endpoint of I is an identity for A,and there are no idempotents for A in (a, b)).
Adaricheva, Pilitowska Survey on aggregation theory
Associative aggregation functions
Theorem [P.S.Mostert and A.L.Shields, 1957]
A function A : [a, b]2 → [a, b] is continuous, associative, and suchthat A(a, x) = A(x , a) = a (a is a zero element) andA(b, x) = A(x , b) = x (b is an identity) if and only if
either A(x , y) = Min(x , y),
or ([a, b],A) is a conjunctive Archimedean semigroup,
or ([a, b],A) is the sum of conjunctive Archimedeansemigroups and one-point semigroups.
Adaricheva, Pilitowska Survey on aggregation theory
Associative aggregation functions
Theorem [J.C.Fodor, 1991]
A function A : I2 → I is continuous, nondecreasing, idempotent,and associative if and only if there exist �, � ∈ I such that for allx , y ∈ I
A(x , y) = Max(Min(�, x),Min(�, x),Min(x , y)).
Theorem [L.W.Fung and K.S.Fu, 1975]
A function A : I2 → I is symmetric, continuous, nondecreasing,idempotent, and associative if and only if there exists � ∈ I suchthat for all x , y ∈ I
A(x , y) = Med(x , y , �).
Adaricheva, Pilitowska Survey on aggregation theory
Associative aggregation functions
Theorem [J.C.Fodor, 1991]
A function A : I2 → I is continuous, nondecreasing, idempotent,and associative if and only if there exist �, � ∈ I such that for allx , y ∈ I
A(x , y) = Max(Min(�, x),Min(�, x),Min(x , y)).
Theorem [L.W.Fung and K.S.Fu, 1975]
A function A : I2 → I is symmetric, continuous, nondecreasing,idempotent, and associative if and only if there exists � ∈ I suchthat for all x , y ∈ I
A(x , y) = Med(x , y , �).
Adaricheva, Pilitowska Survey on aggregation theory
Associative aggregation functions
Theorem [E.Czogala and J.Drewniak, 1984]
If a function A : I2 → I is nondecreasing, idempotent, associative,and has an identity element e ∈ I, then there exists a decreasingfunction g : I→ I, with g(e) = e, such that for all x , y ∈ I
A(x , y) :=
⎧⎨⎩Min(x , y), if y < g(x),
Max(x , y), if y > g(x),
Max(x , y) or Min(x , y), if y = g(x).
If a function A : I2 → I is continuous, nondecreasing, idempotent,associative, and has an identity, then
A(x , y) = Min(x , y) or A(x , y) = Max(x , y).
Adaricheva, Pilitowska Survey on aggregation theory
Associative aggregation functions
Theorem [E.Czogala and J.Drewniak, 1984]
If a function A : I2 → I is nondecreasing, idempotent, associative,and has an identity element e ∈ I, then there exists a decreasingfunction g : I→ I, with g(e) = e, such that for all x , y ∈ I
A(x , y) :=
⎧⎨⎩Min(x , y), if y < g(x),
Max(x , y), if y > g(x),
Max(x , y) or Min(x , y), if y = g(x).
If a function A : I2 → I is continuous, nondecreasing, idempotent,associative, and has an identity, then
A(x , y) = Min(x , y) or A(x , y) = Max(x , y).
Adaricheva, Pilitowska Survey on aggregation theory
t-norms, t-conorms, uninorms
Definition
A t -norm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having 1 as the identity.
A t-conorm is a symmetric, nondecreasing, and associativefunction A : [0, 1]2 → [0, 1] having 0 as the identity.A uninorm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having any e ∈ (0, 1) as the identity.
t-norms are the most important class of conjunctive aggregationfunctions. (inf I ≤ A(x) ≤ Min(x))t-conorms are the most important class of disjunctive aggregationfunctions. (Max(x) ≤ A(x) ≤ sup I)
Adaricheva, Pilitowska Survey on aggregation theory
t-norms, t-conorms, uninorms
Definition
A t -norm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having 1 as the identity.
A t-conorm is a symmetric, nondecreasing, and associativefunction A : [0, 1]2 → [0, 1] having 0 as the identity.A uninorm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having any e ∈ (0, 1) as the identity.
t-norms are the most important class of conjunctive aggregationfunctions. (inf I ≤ A(x) ≤ Min(x))t-conorms are the most important class of disjunctive aggregationfunctions. (Max(x) ≤ A(x) ≤ sup I)
Adaricheva, Pilitowska Survey on aggregation theory
t-norms, t-conorms, uninorms
Definition
A t -norm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having 1 as the identity.A t-conorm is a symmetric, nondecreasing, and associativefunction A : [0, 1]2 → [0, 1] having 0 as the identity.
A uninorm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having any e ∈ (0, 1) as the identity.
t-norms are the most important class of conjunctive aggregationfunctions. (inf I ≤ A(x) ≤ Min(x))t-conorms are the most important class of disjunctive aggregationfunctions. (Max(x) ≤ A(x) ≤ sup I)
Adaricheva, Pilitowska Survey on aggregation theory
t-norms, t-conorms, uninorms
Definition
A t -norm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having 1 as the identity.A t-conorm is a symmetric, nondecreasing, and associativefunction A : [0, 1]2 → [0, 1] having 0 as the identity.A uninorm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having any e ∈ (0, 1) as the identity.
t-norms are the most important class of conjunctive aggregationfunctions. (inf I ≤ A(x) ≤ Min(x))t-conorms are the most important class of disjunctive aggregationfunctions. (Max(x) ≤ A(x) ≤ sup I)
Adaricheva, Pilitowska Survey on aggregation theory
t-norms, t-conorms, uninorms
Definition
A t -norm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having 1 as the identity.A t-conorm is a symmetric, nondecreasing, and associativefunction A : [0, 1]2 → [0, 1] having 0 as the identity.A uninorm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having any e ∈ (0, 1) as the identity.
t-norms are the most important class of conjunctive aggregationfunctions. (inf I ≤ A(x) ≤ Min(x))
t-conorms are the most important class of disjunctive aggregationfunctions. (Max(x) ≤ A(x) ≤ sup I)
Adaricheva, Pilitowska Survey on aggregation theory
t-norms, t-conorms, uninorms
Definition
A t -norm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having 1 as the identity.A t-conorm is a symmetric, nondecreasing, and associativefunction A : [0, 1]2 → [0, 1] having 0 as the identity.A uninorm is a symmetric, nondecreasing, and associative functionA : [0, 1]2 → [0, 1] having any e ∈ (0, 1) as the identity.
t-norms are the most important class of conjunctive aggregationfunctions. (inf I ≤ A(x) ≤ Min(x))t-conorms are the most important class of disjunctive aggregationfunctions. (Max(x) ≤ A(x) ≤ sup I)
Adaricheva, Pilitowska Survey on aggregation theory
t-norms, t-conorms, uninorms - Examples
t-norms
N : [0, 1]2 → [0, 1], N(x , y) := xy ,
N : [0, 1]2 → [0, 1], N(x , y) := Max(x + y − 1, 0).
t-conorms
N : [0, 1]2 → [0, 1], N(x , y) := x + y − xy ,
N : [0, 1]2 → [0, 1], N(x , y) := Min(x + y , 1).
Adaricheva, Pilitowska Survey on aggregation theory
t-norms, t-conorms, uninorms - Examples
Uninorms with neutral element e = 12
N : [0, 1]2 → [0, 1], N(x , y) :=
{2xy , if (x , y) ∈ [0, 12 ]2
Max(x , y), otherwise
N : [0, 1]2 → [0, 1], N(x , y) :=
⎧⎨⎩
Max(x , y), if (x , y) ∈ [12 , 1]2
4xy , if (x , y) ∈ [0, 14 ]2
14(4x − 1)(4y − 1) + 1
4 , if
(x , y) ∈ [14 ,12 ]2
Min(x , y), otherwise.
Adaricheva, Pilitowska Survey on aggregation theory
t-norms, t-conorms, uninorms
Theorem [J.-L.Marichal, 2009]
A function A : [0, 1]2 → [0, 1] is continuous, nondecreasing, weaklyidempotent, associative and has an identity in [0, 1] if and only ifA is a continuous t-norm or a continuous t-conorm.
Adaricheva, Pilitowska Survey on aggregation theory
Fuzzy integrals
Definition
A (discrete) fuzzy measure on N := {1, . . . , n} is a set function
� : 2N → [0, 1]
that is
monotonic (S ⊆ T ⇒ �(S) ≤ �(T )),
fulfills the boundary conditions
�(∅) = 0 and �(N) = 1.
ℱN := the set of all fuzzy measures on N
Adaricheva, Pilitowska Survey on aggregation theory
Fuzzy integrals
Definition
A (discrete) fuzzy measure on N := {1, . . . , n} is a set function
� : 2N → [0, 1]
that is
monotonic (S ⊆ T ⇒ �(S) ≤ �(T )),
fulfills the boundary conditions
�(∅) = 0 and �(N) = 1.
ℱN := the set of all fuzzy measures on N
Adaricheva, Pilitowska Survey on aggregation theory
The Choquet integrals
Definition
Let � ∈ ℱN . The (discrete) Choquet integral of x ∈ ℝn withrespect to � is defined by
C�(x) :=n∑
i=1
x�(i)(�(A�(i))− �(A�(i+1))),
where A�(i) := {�(i), . . . , �(n)} and A�(n+1) := ∅.(x�(1) ≤ . . . ≤ x�(n))
Example
If x3 ≤ x1 ≤ x2, we have C�(x1, x2, x3) =
x3(�({3, 1, 2})− �({1, 2})) + x1(�({1, 2})− �({2})) + x2�({2}).
Adaricheva, Pilitowska Survey on aggregation theory
The Choquet integrals
Definition
Let � ∈ ℱN . The (discrete) Choquet integral of x ∈ ℝn withrespect to � is defined by
C�(x) :=n∑
i=1
x�(i)(�(A�(i))− �(A�(i+1))),
where A�(i) := {�(i), . . . , �(n)} and A�(n+1) := ∅.(x�(1) ≤ . . . ≤ x�(n))
Example
If x3 ≤ x1 ≤ x2, we have C�(x1, x2, x3) =
x3(�({3, 1, 2})− �({1, 2})) + x1(�({1, 2})− �({2})) + x2�({2}).
Adaricheva, Pilitowska Survey on aggregation theory
The Choquet integral - properties
continuous
nondecreasing
idempotent
internal
comonotonic additive:
f (x1 + x ′1, . . . , xn + x ′n) = f (x1, . . . , xn) + f (x ′1, . . . , x′n),
for all vectors such that there is no i , j ∈ N such that xi > xjand x ′i < x ′jlinear with respect to the fuzzy measure:
C�1�1+�2�2 = �1C�1 + �2C�2 ,
for any �1, �2 ∈ ℱN and all �1, �2 ∈ ℝ such that�1�1 + �2�2 ∈ ℱN .
Adaricheva, Pilitowska Survey on aggregation theory
The Choquet integrals
Proposition [D.Schmeidler, 1986]
A function f : ℝn → ℝ is nondecreasing, comonotonic additive,and fulfills f (1) = 1 if and only if there exists � ∈ ℱN such thatf = C�.
Adaricheva, Pilitowska Survey on aggregation theory
The Choquet integrals
Proposition [T.Murofushi and M.Sugeno, 1993]
The Choquet integral C� : ℝn → ℝ is additive(f (x1 + x ′1, . . . , xn + x ′n) = f (x1, . . . , xn) + f (x ′1, . . . , x
′n), for all
vectors x, x′ ∈ ℝn) if and only if there exists w ∈ [0, 1]n withn∑
i=1wi = 1 such that C� = WAℳw =
n∑i=1
wixi .
Proposition [M.Grabisch, 1995]
The Choquet integral C� : ℝn → ℝ is a symmetric function if and
only if there exists a weight vector w ∈ [0, 1]n withn∑
i=1wi = 1 such
that
C� = OWAw =n∑
i=1
wix�(i).
Adaricheva, Pilitowska Survey on aggregation theory
The Sugeno integrals
The unit interval [0, 1] has naturally defined a structure of adistributive lattice (chain) with binary operations:
x ∧ y := min{x , y} and x ∨ y := max{x , y}.
Adaricheva, Pilitowska Survey on aggregation theory
The Sugeno integrals
Definition
Let � ∈ ℱN . The (discrete) Sugeno integral of x ∈ [0, 1]n withrespect to � is defined by
S�(x) :=n⋁
i=1
(x�(i) ∧ �(A�(i))),
where A�(i) := {�(i), . . . , �(n)} and A�(n+i) := ∅.(x�(1) ≤ . . . ≤ x�(n))
Example
If x3 ≤ x1 ≤ x2, then we have S�(x1, x2, x3) =
(x3 ∧ �({3, 1, 2})) ∨ (x1 ∧ �({1, 2})) ∨ (x2 ∧ �({2})).
Adaricheva, Pilitowska Survey on aggregation theory
The Sugeno integrals
Definition
Let � ∈ ℱN . The (discrete) Sugeno integral of x ∈ [0, 1]n withrespect to � is defined by
S�(x) :=n⋁
i=1
(x�(i) ∧ �(A�(i))),
where A�(i) := {�(i), . . . , �(n)} and A�(n+i) := ∅.(x�(1) ≤ . . . ≤ x�(n))
Example
If x3 ≤ x1 ≤ x2, then we have S�(x1, x2, x3) =
(x3 ∧ �({3, 1, 2})) ∨ (x1 ∧ �({1, 2})) ∨ (x2 ∧ �({2})).
Adaricheva, Pilitowska Survey on aggregation theory
The Sugeno integrals
S�(x) :=⋁T⊆N
(�(T ) ∧ (⋀i∈T
xi ))
Adaricheva, Pilitowska Survey on aggregation theory
The Sugeno integrals - properties
continuous
nondecreasing
idempotent
internal
comonotonic minitivity:
f (x1 ∧ x ′1, . . . , xn ∧ x ′n) = f (x1, . . . , xn) ∧ f (x ′1, . . . , x′n),
for all vectors such that there is no i , j ∈ N such that xi > xjand x ′i < x ′jcomonotonic maxitivity:
f (x1 ∨ x ′1, . . . , xn ∨ x ′n) = f (x1, . . . , xn) ∨ f (x ′1, . . . , x′n),
for all vectors such that there is no i , j ∈ N such that xi > xjand x ′i < x ′j
Adaricheva, Pilitowska Survey on aggregation theory
The Sugeno integrals
Proposition [J.-L.Marichal, 1998]
A function f : [0, 1]n → [0, 1] is nondecreasing, idempotent,comonotonic minitive and maxitive if and only if there exists� ∈ ℱN such that f = S�.
Adaricheva, Pilitowska Survey on aggregation theory
The Sugeno integrals
Theorem [J.-L.Marichal, 2001]
The set of Sugeno integrals coincides with the set of idempotentlattice polynomial functions.
Lattice term functions are Sugeno integrals defined from fuzzymeasures � : 2N → {0, 1} having their value in the set {0, 1}.
Adaricheva, Pilitowska Survey on aggregation theory
The Sugeno integrals
Theorem [J.-L.Marichal, 2001]
The set of Sugeno integrals coincides with the set of idempotentlattice polynomial functions.
Lattice term functions are Sugeno integrals defined from fuzzymeasures � : 2N → {0, 1} having their value in the set {0, 1}.
Adaricheva, Pilitowska Survey on aggregation theory
References
M. Grabisch, J.-L. Marichal, R. Mesiar, E. Pap,AggregationFunctions, Encyclopedia of Mathematics and Its Applicationsvol. 127, Cambridge University Press, Cambrige, 2009.
J.-L. Marichal, Aggregation functions for decision making,arXiv:0901.4232v1 [math.ST] 27 Jan 2009.
Adaricheva, Pilitowska Survey on aggregation theory