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Aggregation Functions. Part I: Means
Michel GRABISCH
University of Panth´eon-Sorbonne
Paris, France
Jean-Luc MARICHAL
University of Luxembourg
Luxembourg
Radko MESIAR
Slovak University of Technology
Bratislava, Slovakia,
Endre PAP
University of Novi Sad
Novi Sad, Serbia
November 13, 2009
Abstract The two-parts state-of-art overview of aggregation theory summarizes the essential information concerning aggregation issues. Overview of ag- gregation properties is given, including the basic classification of aggrega- tion functions. In this first part, the stress is put on means, i.e., averaging aggregation functions, both with fixed arity (n-ary means) and with open arity (extended means).
1 Introduction
Aggregation functions play an important role in many of the technological tasks scientists are faced with nowadays. They are specifically important in many problems related to the fusion of information. More generally, aggregation func- tions are widely used in pure mathematics (e.g., functional equations, theory of means and averages, measure and integration theory), applied mathemat- ics (e.g., probability, statistics, decision mathematics), computer and engineer- ing sciences (e.g., artificial intelligence, operations research, information theory, engineering design, pattern recognition and image analysis, data fusion, au- tomated reasoning), economics and finance (e.g., game theory, voting theory, decision making), social sciences (e.g., representational measurement, mathe- matical psychology) as well as many other applied fields of physics and natural sciences. Thus, a main characteristic of the aggregation functions is that they are used in a large number of areas and disciplines. The essence of aggregation is that the output value computed by the ag- gregation function should represent or synthesize “in some sense” all individual inputs, where quotes are put to emphasize the fact that the precise meaning
of this expression is highly dependent on the context. In any case, defining or choosing the right class of aggregation functions for a specific problem is a dif- ficult task, considering the huge variety of potential aggregation functions. In this respect, one could ask the following question: Consider a set of n values, lying in some real interval [a, b] to be aggregated. Is any function from [a, b]n to R a candidate to be an aggregation function? Obviously not. On the other hand, it is not that easy to define the minimal set of properties a function should fulfill to be an aggregation function. A first natural requirement comes from the fact that the output should be a synthetic value. Then, if inputs are supposed to lie in the interval [a, b], the output should also lie in this interval. Moreover, if all input values are identical to the lower bound a, then the output should also be a, and similarly for the case of the upper bound b. This defines a bound- ary condition. A second natural requirement is nondecreasing monotonicity. It means that if some of the input values increase, the representative output value should reflect this increase, or at worst, stay constant. These two requirements are commonly accepted in the field, and we adopt them as the basic definition of an aggregation function. Thus defined, the class of aggregation functions is huge, making the problem of choosing the right function (or family) for a given application a difficult one. Besides this practical consideration, the study of the main classes of aggregation functions, their properties and their relationships, is so complex and rich that it becomes a mathematical topic of its own. A solid mathematical analysis of aggregation functions, able to answer both mathematical and practical concerns, was the main motivation for us to prepare a monograph [?]. From related recent monographs, recall the handbook [?] and [?]. The aim of these two-parts invited state-of-art overview is to summarize the essential information about aggregation functions for Information Sciences readers, to open them the door to the rich world of tools important for informa- tion fusion. With a kind permition of the publisher, some parts of [?] were used in this manuscript. Moreover, to increase the transparentness, proofs of several introduced results are not included (or sketched only), however, for interested readers always an indication where the full proofs can be found is given. The paper is organized as follows. In the next section, basic properties of aggregation functions and several illustrative examples are given. Section 3 is devoted to means related to the arithmetic mean and means with some special properties. In Section 4, non-additive integral - based aggregation functions are discussed, stressing a prominent role of the Choquet and Sugeno integrals. Part I ends with some concluding remarks. In Part II, Section 2 deals with conjunctive aggregation functions, especially with triangular norms and copulas. In Section 3, disjunctive aggregation functions are summarized, exploring their duality to conjunctive aggregation functions. Moreover, several kinds of aggregation functions mixing conjunctive and disjunctive aggregation functions, are also included (uninorms, nullnorms, gamma operators, etc.). Several construction methods for aggregation functions are introduced in Section 4. These methods are not only a summarization showing how several kinds of aggregation functions were introduced from simpler ones, but they allow to readers to taylor their aggregation model when solving nonstandard problems resisting to standard aggregation functions being used and fit to real data constraints. Finally, several concluding remarks are included.
Definition 2. Consider an (extended) aggregation function A : In^ → I ( A : ∪n∈NIn^ → I). Then
(i) A is called conjunctive whenever A 6 Min, i.e., A(x) 6 Min(x) for all x ∈ In^ (x ∈ ∪n∈NIn).
(ii) A is called disjunctive whenever A > Max.
(iii) A is called internal whenever Min 6 A 6 Max.
(iv) A is called mixed if it is not conjunctive neither disjunctive nor internal.
In particular case I = [0, 1] (or I = [a, b] ⊂ R), the standard duality of aggregation functions is introduced.
Definition 3. Let A : [0, 1]n^ → [0, 1] be an aggregation function. Then the function Ad^ : [0, 1]n^ → [0, 1] given by
A(x) = 1 − A(1 − x 1 ,... , 1 − xn) (3)
is called a dual aggregation function (to A).
Evidently, Ad^ given by (3) is an aggregation function on [0, 1]. Similarly, dual extended aggregation function Ad^ to A acting on [0, 1] can be introduced. If I = [a, b] ⊂ R, then (3) should be modified into
Ad(x) = a + b − A(a + b − x 1 ,... , a + b − xn).
It is evident that dual to a conjunctive (respectively disjunctive, mixed) ag- gregation function is disjunctive (respectively conjunctive, mixed) aggregation function. Note that many properties defined for n-ary functions can be naturally adapted to extended functions. For instance, with some abuse of language, the extended function F : ∪n∈NIn^ → R is said to be continuous if, for any n ∈ N, the corresponding n-ary function F(n)^ = F|In is continuous. These adaptations are implicitly assumed throughout, for example in sections 2.2, 2.3. Proper- ties defined for extended aggregation functions only will be stressed explicitly (note that these properties make an important link between aggregation func- tions with fixed but different arities). In some cases, properties of general real functions will be introduced.
2.2 Monotonicity properties
Definition 4. The function F : In^ → R is strictly increasing (in each argument) if, for any x, x′^ ∈ In, x < x′^ ⇒ F(x) < F(x′).
Thus, a function is strictly increasing if it is nondecreasing and if it presents a positive reaction to any increase of at least one input value. An intermediate kind of monotonicity (between nondecreasingness and strict increasigness) is the unanimeonus increasigness, also called jointly strict increas- ingness.
Definition 5. The function F : In^ → R is unanimously increasing if it is nondecreasing and if, for any x, x′^ ∈ In,
xi < x′ i, ∀i ∈ { 1 ,... , n} ⇒ F(x) < F(x′).
Clearly, strict increasing monotonicity ensures unanimous increasing mono- tonicity. For example, the arithmetic mean AM is strictly increasing, hence unanimously increasing. Functions Min and Max are unanimously increasing but not strictly increasing. The product Π is unanimously increasing on [0, 1]n. However, if 0 occurs among inputs, the strict monotonicity of Π is violated. The bounded sum SL(x) = Min(
∑n i=1 xi,^ 1) on [0,^ 1] n (^) is nondecreasing but not
unanimously increasing.
2.3 Continuity properties
As already mentioned, the continuity of an extended aggregation function A means the classical continuity of all n-ary functions A(n). The same holds for the other kinds of continuity which are therefore introduced for n-ary functions only. We recall only few of them, more details can be found in [?], section 2.2.2. The continuity property can be strengthened into the well-known Lipschitz condition [?]; see Zygmund [?].
Definition 6. Let ‖ · ‖ : Rn^ → [0, ∞[ be a norm. If a function F : In^ → R satisfies the inequality
|F(x) − F(y)| 6 c ‖x − y‖ (x, y ∈ In), (4)
for some constant c ∈ ]0, ∞[ , then we say that F satisfies the Lipschitz condition or is Lipschitzian (with respect to ‖ · ‖). More precisely, any function F : In^ → R satisfying (4) is said to be c-Lipschitzian. The greatest lower bound d of constants c > 0 in (4) is called the best Lipschitz constant (which means that F is d-Lipschitzian but, for any u ∈ ]0, d[, F is not u-Lipschitzian).
Important examples of norms are given by the Minkowski norm of order p ∈ [1, ∞[, namely
‖x‖p :=
( (^) n ∑
i=
|xi|p
) 1 /p ,
also called the Lp-norm, and its limiting case ‖x‖∞ := maxi |xi|, which is the Chebyshev norm. The c-Lipschitz condition has an interesting interpretation when applied in aggregation. It allows us to estimate the relative output error in comparison with input errors |F(x) − F(y)| 6 c ε
whenever ‖x − y‖ 6 ε for some ε > 0. We also have the following result.
Proposition 1. For arbitrary reals p, q ∈ [1, ∞], a function F : In^ → R is Lipschitzian with respect to the norm ‖ · ‖p if and only if it is Lipschitzian with respect to the norm ‖ · ‖q. Moreover, if F is c-Lipschitzian with respect to the norm ‖ · ‖p then it is also c-Lipschitzian with respect to the norm ‖ · ‖q for q 6 p, and for q > p it is nc-Lipschitzian.
Example 2. (i) An important example of a left-continuous (lower semi-continuous) but noncontinuous aggregation function is the nilpotent minimum TnM^ : [0, 1]^2 → [0, 1],
TnM(x 1 , x 2 ) :=
Min(x 1 , x 2 ) if x 1 + x 2 > 1 0 otherwise.
(ii) The drastic product TD : [0, 1]n^ → [0, 1], given by
TD(x) :=
Min(x) if |{i ∈ { 1 ,... , n} | xi < 1 }| < 2 0 otherwise,
is a noncontinuous but upper semi-continuous aggregation function.
2.4 Symmetry
The next property we consider is symmetry, also called commutativity, neutral- ity, or anonymity. The standard commutativity of binary operations x∗y = y∗x, well known in algebra, can be easily generalized to n-ary functions, with n > 2, as follows.
Definition 9. F : In^ → R is a symmetric function if
F(x) = F(σ(x))
for any x ∈ In^ and for any permutation σ of (1,... , n), where σ(x) = (xσ(1),... , xσ(n)).
The symmetry property essentially means that the aggregated value does not depend on the order of the arguments. This is required when combining criteria of equal importance or anonymous experts’ opinions.^1 Many aggregation functions introduced thus far are symmetric. For exam- ple, AM, GM, SL, TL,
, Min, Max are symmetric functions. A prominent ex- ample of non-symmetric aggregation functions is the weighted arithmetic mean WAMw, WAMw(x 1 ,... , xn) =
∑n i=1 wixi,^ where the nonnegative weights^ wi^ are constrained by
∑n i=1 wi^ = 1 (and at least one weight^ wi^6 =^
1 n ,^ else^ WAMw^ =^ AM is symmetric). The following result, well-known in group theory, shows that the symmetry property can be checked with only two equalities; see for instance Rotman [?, Exercise 2.9 p. 24].
Proposition 5. F : In^ → R is a symmetric function if and only if, for all x ∈ In, we have
(i) F(x 2 , x 1 , x 3 ,... , xn) = F(x 1 , x 2 , x 3 ,... , xn),
(ii) F(x 2 , x 3 ,... , xn, x 1 ) = F(x 1 , x 2 , x 3 ,... , xn). This simple test is very efficient, especially when symmetry does not appear immediately, like in the 4-variable expression
(x 1 ∧ x 2 ∧ x 3 ) ∨ (x 1 ∧ x 2 ∧ x 4 ) ∨ (x 1 ∧ x 3 ∧ x 4 ) ∨ (x 2 ∧ x 3 ∧ x 4 ),
which is nothing other than the 4-ary order statistic x(2).
(^1) Of course, symmetry is more natural in voting procedures than in multicriteria decision making, where criteria usually have different importances.
2.5 Idempotency
Definition 10. F : In^ → R is an idempotent function if δF = id, that is,
F(n · x) = x (x ∈ I).
Idempotency is in some areas supposed to be a natural property of aggrega- tion functions, e.g., in multicriteria decision making (see for instance Fodor and Roubens [?]), where it is commonly accepted that if all criteria are satisfied at the same degree x, implicitly assuming the commensurateness of criteria, then also the overall score should be x. It is evident that AM, WAMw, Min, Max, and Med are idempotent functions, while Σ and Π are not. Recall also that any nondecreasing and idempotent function F : In^ → R is an aggregation function.
2.5.1 Idempotent elements
Definition 11. An element x ∈ I is idempotent for F : In^ → R if δF(x) = x.
In [0, 1]n^ the product Π has no idempotent elements other than the extreme elements 0 and 1. As an example of a function in [0, 1]n^ which is not idempotent but has a non-extreme idempotent element, take an arbitrarily chosen element c ∈ ]0, 1[ and define the aggregation function A{c} : [0, 1]n^ → [0, 1] as follows:
A{c}(x) := Med
0 , c +
∑n i=
(xi − c), 1
where Med is the standard median function (i.e., in ternary case the ”midle” input, between the smallest one and the greatest one). It is easy to see that the only idempotent elements for A{c} are 0, 1, and c.
2.5.2 Strong idempotency
The idempotency property has been generalized to extended functions as follows; see Calvo et al. [?].
Definition 12. F : ∪n∈NIn^ → R is strongly idempotent if, for any n ∈ N,
F(n · x) = F(x) (x ∈ ∪m∈NIm).
For instance, if F : ∪n∈NIn^ → R is strongly idempotent then we have
F(x 1 , x 2 , x 1 , x 2 ) = F(x 1 , x 2 ).
Proposition 6. Suppose F : ∪n∈NIn^ → R is strongly idempotent. Then F is idempotent if and only if F(x) = x for all x ∈ I.
According to our convention on unary aggregation functions, namely A(x) = x for all x ∈ I, it follows immediately from the previous proposition that any strongly idempotent extended aggregation function is idempotent.
Example 3. (i) Let
∆ = (wi,n | n ∈ N, i ∈ { 1 ,... , n})
Definition 14. F : ∪n∈NIn^ → I is associative if F(x) = x for all x ∈ I and if
F(x, x′) = F(F(x), F(x′))
for all x, x′^ ∈ ∪n∈N 0 In.
As the next proposition shows, associativity means that each subset of consecutive arguments can be replaced with their partial aggregation without changing the overall aggregation.
Proposition 7. F : ∪n∈NIn^ → I is associative if and only if F(x) = x for all x ∈ I and F(x, F(x′), x′′) = F(x, x′, x′′)
for all x, x′, x′′^ ∈ ∪n∈N 0 In.
Associativity is also a well-known algebraic property which allows one to omit “parentheses” in an aggregation of at least three elements. Implicit in the assumption of associativity is a consistent way of going unambiguously from the aggregation of n elements to n + 1 elements, which implies that any associa- tive extended function F is completely determined by its binary function F(2). Indeed, by associativity, we clearly have
F(x 1 ,... , xn+1) = F
F(x 1 ,... , xn), xn+
For practical purpose we can start with the aggregation procedure before knowing all inputs to be aggregated. Additional input data are then simply aggregated with the current aggregated output. Each associative idempotent extended function is necessary strongly idem- potent. For a fixed arity n > 2, we can introduce the associativity as follows.
Definition 15. Let F : In^ → R be an n-ary function. Then it is associative if, for all x 1 ,... , x 2 n− 1 ∈ I, we have
F(F(x 1 ,... , xn), xn+1,... , x 2 n− 1 ) = F(x 1 , F(x 2 ,... , xn+1), xn+2,... , x 2 n− 1 ) = F(x 1 ,... , xn− 1 , F(xn,... , x 2 n− 1 )).
2.7 Decomposability
Introduced first in Bemporad [?, p. 87] in a characterization of the arith- metic mean, associativity of means has been used by Kolmogoroff [?] and Nagumo [?] to characterize the so-called mean values. More recently, Marichal and Roubens [?] proposed to call this property “decomposability” in order not to confuse it with classical associativity. Alternative names, such as associativ- ity with repetitions or weighted associativity, could be naturally considered as well. When symmetry is not assumed, it is necessary to rewrite this property in such a way that the first variables are not privileged. To abbreviate notations, for nonnegative integers m, n, we write F(m·x, n·y) what means the repetition of arguments, i.e., F(x,... , x ︸ ︷︷ ︸ m
, y,... , y ︸ ︷︷ ︸ n
). We then consider the following definition.
Definition 16. F : ∪n∈NIn^ → I is decomposable if F(x) = x for all x ∈ I and if
F(x, x′) = F
k · F(x), k′^ · F(x′)
for all k, k′^ ∈ N 0 , all x ∈ Ik, and all x′^ ∈ Ik
′ .
By considering k = 0 or k′^ = 0 in (6), we see that any decomposable function is range-idempotent. Moreover, as the following proposition shows, decompos- ability means that each element of any subset of consecutive arguments can be replaced with their partial aggregation without changing the overall aggregation.
Proposition 8. F : ∪n∈NIn^ → I is decomposable if and only if F(x) = x for all x ∈ I and F
x, k′^ · F(x′), x′′
= F(x, x′, x′′)
for all k′^ ∈ N 0 , all x′^ ∈ Ik
′ , and all x, x′′^ ∈ ∪n∈N 0 In.
2.8 Bisymmetry
Another grouping property is the bisymmetry.
Definition 17. F : I^2 → I is bisymmetric if for all x ∈ I^4 , we have
F
F(x 1 , x 2 ), F(x 3 , x 4 )
= F
F(x 1 , x 3 ), F(x 2 , x 4 )
The bisymmetry property is very easy to handle and has been investigated from the algebraic point of view by using it mostly in structures without the property of associativity. For a list of references see Acz´el [?, Section 6.4] (see also Acz´el and Dhombres [?, Chapter 17], and Soublin [?]). For n arguments, bisymmetry takes the following form (see Acz´el [?]).
Definition 18. F : In^ → I is bisymmetric if
F
F(x 11 ,... , x 1 n),... , F(xn 1 ,... , xnn)
= F
F(x 11 ,... , xn 1 ),... , F(x 1 n,... , xnn)
for all square matrices
x 11 · · · x 1 n .. .
xn 1 · · · xnn
∈ In×n.
Bisymmetry expresses the condition that aggregation of all the elements of any square matrix can be performed first on the rows, then on the columns, or conversely. However, since only square matrices are involved, this property seems not to have a good interpretation in terms of aggregation. Its usefulness remains theoretical. We then consider it for extended functions as follows; see Marichal et al. [?].
Definition 19. F : ∪n∈NIn^ → I is strongly bisymmetric if F(x) = x for all x ∈ I, and if, for any n, p ∈ N, we have
F
F(x 11 ,... , x 1 n),... , F(xp 1 ,... , xpn)
= F
F(x 11 ,... , xp 1 ),... , F(x 1 n,... , xpn)
Definition 21. An element e ∈ I is called a neutral element of a function F : In^ → R if, for any i ∈ { 1 ,... , n} and any x ∈ I, we have F(x{i}e) = x.
Clearly, if e ∈ I is an extended neutral element of an extended function F : ∪n∈NIn^ → I, with F(1)(x) = x, then e is a neutral element of all F(n), n ∈ N. For instance, e = 0 is an extended neutral element for the extended sum function Σ.
Definition 22. An element a ∈ I is called an annihilator element of a function F : In^ → R if, for any x ∈ In^ such that a ∈ {x 1 ,... , xn}, we have F(x) = a.
Proposition 9. Consider an aggregation function A : In^ → I. If A is con- junctive and a := inf I ∈ I then a is an annihilator element. Dually, if A is disjunctive and b := sup I ∈ I then b is an annihilator element.
The converse of Proposition 9 is false. For instance, in [0, 1]n, 0 is an anni- hilator of the geometric mean GM, which is not conjunctive. For more specific properties of aggregation functions we recommend to con- sider [?], chapter 2.
3 Means and averages
It would be very unnatural to propose a monograph on aggregation functions without dealing somehow with means and averaging functions. Already dis- covered and studied by the ancient Greeks,^2 the concept of mean has given rise today to a very wide field of investigation with a huge variety of applica- tions. Actually, a tremendous amount of literature on the properties of several means (such as the arithmetic mean, the geometric mean, etc.) has already been produced, especially since the 19th century, and is still developing today. For a good overview, see the expository paper by Frosini [?] and the remarkable monograph by Bullen [?]. The first modern definition of mean was probably due to Cauchy [?] who considered in 1821 a mean as an internal function. We adopt this approach and assume further that a mean should be a nondecreasing function. As usual, I represents a nonempty real interval, bounded or not. The more general cases where I includes −∞ and/or ∞ will always be mentioned explicitly.
Definition 23. An n-ary mean in In^ is an internal aggregation function M : In^ → I. An extended mean in ∪n∈NIn^ is an extended function M : ∪n∈NIn^ → I whose restriction to each In^ is a mean.
It follows that a mean is nothing other than an idempotent aggregation function. Moreover, if M : In^ → I is a mean in In, then it is also a mean in Jn, for any subinterval J ⊆ I. The concept of mean as an average or numerical equalizer is usually ascribed to Chisini [?, p. 108], who gave in 1929 the following definition:
Let y = F(x 1 ,... , xn) be a function of n independent variables x 1 ,... , xn. A mean of x 1 ,... , xn with respect to the function F (^2) See Antoine [?, Chapter 3] for a historical discussion of the various Greek notions of “mean”.
is a number M such that, if each of x 1 ,... , xn is replaced by M , the function value is unchanged, that is, F(M,... , M ) = F(x 1 ,... , xn).
When F is considered as the sum, the product, the sum of squares, the sum of inverses, or the sum of exponentials, the solution of Chisini’s equation corre- sponds respectively to the arithmetic mean, the geometric mean, the quadratic mean, the harmonic mean, and the exponential mean. Unfortunately, as noted by de Finetti [?, p. 378] in 1931, Chisini’s definition is so general that it does not even imply that the “mean” (provided there exists a real and unique solution to Chisini’s equation) fulfills Cauchy’s internality property. To ensure existence, uniqueness, and internality of the solution of Chisini’s equation, we assume that F is nondecreasing and idempotizable. Therefore we propose the following definition:
Definition 24. A function M : In^ → I is an average in In^ if there exists a nondecreasing and idempotizable function F : In^ → R such that F = δF ◦ M. In this case, we say that M is an average associated with F in In.
Averages are also known as Chisini means or level surface means. The average associated with F is also called the F-level mean (see Bullen [?, VI.4.1]). The following result shows that, thus defined, the concepts of mean and average coincide.
Proposition 10. The following assertions hold:
(i) Any average is a mean. (ii) Any mean is the average associated with itself.
(iii) Let M be the average associated with a function F : In^ → R. Then M is the average associated with a function G : In^ → R if and only if there exists an increasing bijection ϕ : ran(F) → ran(G) such that G = ϕ ◦ F. Proposition 10 shows that, thus defined, the concepts of mean and average are identical and, in a sense, rather general. Note that some authors (see for instance Bullen [?, p. xxvi], Sahoo and Riedel [?, Section 7.2], and Bhatia [?, Chapter 4]) define the concept of mean by adding conditions such as continuity, symmetry, and homogeneity, which is M(r x) = r M(x) for all admissible r ∈ R.
3.1 Quasi-arithmetic means
A well-studied class of means is the class of quasi-arithmetic means (see for instance Bullen [?, Chapter IV]), introduced as extended aggregation functions as early as 1930 by Kolmogoroff [?], Nagumo [?], and then as n-ary functions in 1948 by Acz´el [?]. In this section we introduce the quasi-arithmetic means and describe some of their properties and axiomatizations.
Definition 25. Let f : I → R be a continuous and strictly monotonic function. The n-ary quasi-arithmetic mean generated by f is the function Mf : In^ → I defined as
Mf (x) := f −^1
n
∑^ n
i=
f (xi)
We now present an axiomatization of the class of quasi-arithmetic means as extended aggregation functions, originally called mean values. The next theorem brings an axiomatization of quasi-arithmetic means as extended aggregation functions. This axiomatization was obtained independently by Kolmogoroff [?] and Nagumo [?] in 1930.
Theorem 1. F : ∪n∈NIn^ → R is symmetric, continuous, strictly increasing, idempotent, and decomposable if and only if there is a continuous and strictly monotonic function f : I → R such that F = Mf is the extended quasi-arithmetic mean generated by f.
Another axiomatization of n-ary quasi-arithmetic means is due to Acz´el [?].
Theorem 2. The function F : In^ → R is symmetric, continuous, strictly in- creasing, idempotent, and bisymmetric if and only if there is a continuous and strictly monotonic function f : I → R such that F = Mf is the quasi-arithmetic mean generated by f.
Remark 3. Note that the results given in Theorem 1, and Theorem 2 can be extended to subintervals I of the extended real line containing ∞ or −∞ with a slight modification of the requirements. Namely, the codomain of F should be [−∞, ∞], and the strict monotonicity and continuity are required for bounded input vectors only. Observe also that if I = [−∞, ∞] then the corresponding quasi-arithmetic means are no more continuous due to the non-continuity of the standard summation on [−∞, ∞].
Adding some particular property, a special subfamily of quasi-arithmetic means is obtained. Due to Nagumo [?] we have the next result.
Theorem 3. (i) The quasi-arithmetic mean M : In^ → I is difference scale invariant, i.e., M(x + c) = M(x) + c for all x ∈ In^ and c = (c,... , c) ∈ Rn such that x + c ∈ In, if and only if either M is the arithmetic mean AM or M is the exponential mean, i.e., there exists α ∈ R \ { 0 } such that
M(x) =
α
ln
n
∑^ n
i=
eαxi
(ii) Assume I ⊆ ]0, ∞[. The quasi-arithmetic mean M : In^ → I is ratio scale invariant, i.e., M(cx) = cM(x) for all x ∈ In^ and c > 0 such that cx ∈ In, if and only if either M is the geometric mean GM or M is the root-mean- power, i.e., there exists α ∈ R \ { 0 } such that
M(x) =
n
∑^ n
i=
xαi
) (^1) α .
A modification of Theorem 2, where the symmetry requirements is omitted, yields an axiomatic characterization of weighted quasi-arithmetic means, see Acz´el [?] (these are called also quasi-linear means in some sources).
Theorem 4. The function F : In^ → R is continuous, strictly increasing, idempo- tent, and bisymmetric if and only if there is a continuous and strictly monotonic
f (x) M(x) name notation
x
∑n i=
wi xi weighted arithmetic mean WAMw
x^2
( (^) ∑n
i=
wi x^2 i
weighted quadratic mean WQMw
log x
∏n i=
xw i i weighted geometric mean WGMw
xα^ (α ∈ R \ { 0 })
( (^) ∑n
i=
wi xαi
) 1 /α weighted root-mean-power WMidα,w
Table 2: Examples of quasi-linear means
function f : I → R and real numbers w 1 ,... , wn > 0 satisfying
i wi^ = 1 such that
F(x) = f −^1
( (^) ∑n
i=
wi f (xi)
(x ∈ In). (8)
Weighted quasi-arithmetic means can be seen as transformed weighted arith- metic means. The later means are trivially characterized by the additivity prop- erty, i.e., F(x + y) = F(x) + F(y) for all x, y, x + y ∈ Dom(F).
Proposition 12. F : Rn^ → R is additive, nondecreasing, and idempotent if and only if there exists a weight vector w ∈ [0, 1]n^ satisfying
i wi^ = 1 such that F = WAMw.
Corollary 1. F : Rn^ → R is additive, nondecreasing, symmetric, and idempo- tent if and only if F = AM is the arithmetic mean.
Remark 4. As will be discussed in Section 4, the weighted arithmetic means WAMw are exactly the Choquet integrals with respect to additive normalized capacities; see Proposition 14 (v). If we further assume the symmetry property, we obtain the arithmetic mean AM.
A natural way to generalize the quasi-arithmetic mean consists in incorpo- rating weights as in the quasi-linear mean (8). To generalize a step further, we could assume that the weights are not constant. On this issue, Losonczi [?, ?] considered and investigated in 1971 nonsymmetric functions of the form
M(x) = f −^1
n i ∑=1n pi(xi)f^ (xi) i=1 pi(xi)
where f : I → R is a continuous and strictly monotonic function and p 1 ,... , pn : I → ]0, ∞[ are positive valued functions. The special case where p 1 = · · · = pn
3.2 Constructions of means
There are several methods how to construct means (binary, n-ary, extended). Integral-based methods are discussed in Section 4. Means can also be constructed by minimization of functions. This construc- tion method will be thoroughly discussed in Part II, Section 4. To give here a simple example, consider weights w 1 , w 2 ∈ ]0, ∞[ and minimize (in r) the expression
f (r) = w 1 |x 1 − r| + w 2 (x 2 − r)^2.
This minimization problem leads to the unique solution
r = M(x 1 , x 2 ) = Med
x 1 , x 2 − w 1 2 w 2
, x 2 + w 1 2 w 2
which defines a mean M : R^2 → R. Any nonsymmetric function F can be symmetrized by replacing it variables x 1 ,... , xn with corresponding order statistics functions x(1) (minimal input), x(2),... , x(n) (maximal input). One of the simplest examples is given by the ordered weighted averaging function
OWAw(x) =
∑^ n
i=
wi x(i), (9)
which merely results from the symmetrization of the corresponding weighted arithmetic mean WAMw.
Remark 7. The concept of ordered weighted averaging function was intro- duced by Yager^4 in 1988; see Yager [?], and also the book [?] edited by Yager and Kacprzyk. Since then, the family of these functions has been axiomatized in various ways; see for instance Fodor et al. [?] and Marichal and Mathonet [?]. Also, these functions are exactly the Choquet integrals with respect to symmet- ric normalized capacities; see Proposition 14 (vi).
The symmetrization process can naturally be applied to the quasi-linear mean (i.e., to the weighted quasi-arithmetic mean) (8) to produce the quasi- ordered weighted averaging function OWAw,f : In^ → R, which is defined as
OWAw,f (x) := f −^1
( (^) ∑n
i=
wif (x(i))
where the generator f : I → R is a continuous and strictly monotonic function; see Fodor et al. [?]. The classical mean value formulas (Lagrange, Cauchy) lead to the concept of lagrange (Cauchy) means, see [?] and [?, ?].
Definition 27. Let f : I → R be a continuous and strictly monotonic function. The Lagrangian mean M[f ] : I^2 → I associated with f is a mean defined as
M[f ](x, y) :=
f −^1
y − x
∫ (^) y
x
f (t) dt
if x 6 = y
x if x = y.
(^4) Note however that linear (not necessarily convex) combinations of ordered statistics were already studied previously in statistics; see for instance Weisberg [?] (and David and Na- garaja [?, Section 6.5] for a more recent overview).
The uniqueness of the generator is the same as for quasi-arithmetic means: Let f and g be two generators of the same Lagrangian mean. Then, there exist r, s ∈ R, r 6 = 0 such that g(x) = rf (x) + s; see [?, Corollary 7], [?, p. 344], and [?, Theorem 1]. Many classical means are Lagrangian. The arithmetic mean, the geometric mean, and the so-called Stolarsky means [?], defined by
MS (x, y) :=
xr^ − yr r(x − y)
) (^) r−^11 if x 6 = y
x if x = y,
correspond to taking f (x) = x, f (x) = 1/x^2 , and f (x) = xr−^1 , respectively, in (10). The harmonic mean, however, is not Lagrangian. In general, some of the most common means are both quasi-arithmetic and Lagrangian, but there are quasi-arithmetic means, like the harmonic one, which are not Lagrangian. Conversely, the logarithmic mean
M(x, y) :=
x − y log x − log y
for x, y > 0, x 6 = y
x for x = y > 0 ,
is an example of a Lagrangian mean (actually a Stolarsky mean, f (x) = 1/x), that is not quasi-arithmetic. A characterization of the class of Lagrangian means and a study of its connections with the class of quasi-arithmetic means can be found in Berrone and Moro [?]. Further properties of Lagrangian means and other extensions are investigated for instance in Acz´el and Kuczma [?], Berrone [?], Glazowska [?], Horwitz [?, ?], Kuczma [?], S´andor [?], and Wimp [?].
Definition 28. Let f, g : I → R be continuous and strictly monotonic functions. The Cauchy mean M[f,g] : I^2 → I associated with the pair (f, g) is a mean defined as
M[f,g](x, y) :=
f −^1
g(y) − g(x)
∫ (^) y
x
f (t) dg(t)
if x 6 = y
x if x = y.
We note that any Cauchy mean is continuous, idempotent, symmetric, and strictly increasing. When g = f (respectively, g is the identity function), we retrieve the quasi- arithmetic (respectively, the Lagrangian) mean generated by f. The anti- Lagrangian mean [?] is obtained when f is the identity function. For exam- ple, the harmonic mean is an anti-Lagrangian mean generated by the function g = 1/x^2. We also note that the generator of an anti-Lagrangian mean is defined up to a non-zero affine transformation. Further studies on Cauchy means can be found for instance in Berrone [?], Lorenzen [?], and Losonczi [?, ?]. Extensions of Lagrangian and Cauchy means, called generalized weighted mean values, including discussions on their mono- tonicity properties, can be found in Chen and Qi [?, ?], Qi et al. [?, ?, ?], and Witkowski [?].
