key: cord-0044890-ndsjwbx4
authors: Kalina, Martin
title: Aggregation Functions Transformed by 0 - 1 Valued Monotone Systems of Functions
date: 2020-05-15
journal: Information Processing and Management of Uncertainty in Knowledge-Based Systems
DOI: 10.1007/978-3-030-50143-3_42
sha: f851847380dc601bd98874d709cd04f30dffe403
doc_id: 44890
cord_uid: ndsjwbx4

In the paper Jin et al. [8] the authors introduced a generalized phi-transformation of aggregation functions. This is a kind of two-step aggregation. This transformation was further developed in Jin et al. [9] into a Generalized-Convex-Sum-Transformation. A special case of the proposed Generalized-Convex-Sum-Transformation is the well-known *-product, also known as the Darsow product of copulas. This approach covers also the discrete Choquet integral. In this paper we study the monotone systems of functions, particularly the case when functions in these systems are just two-valued.

Jin et al. in [8] introduced a generalized ϕ-transformation of aggregation functions. This method is based on a so-called parametrized chain and an aggregation function F . The original aggregation function A is transformed into A F,c , where c is a vector-function and F (c(t)) = t. This method was modified by the same authors in [9] into a generalized-convex-sum-transformation. A special case of this generalized-convex-sum-transformation is the well-known * -product, also known as Darsow product, see [4] . As it is shown in [9] , this method generalizes the discrete Choquet integral. The transformation is based on systems of monotone functions as follows (we illustrate here the transformation of binary aggregation functions) 1] , {g y } y∈ [0, 1] influences the resulting transform A F of the (in this case) binary aggregation function A.

In this contribution we will study the case when the systems of functions {f x } x∈ [0, 1] , {g y } y∈ [0, 1] are two-valued, i.e., f x (t) ∈ {0, 1} and g y (t) ∈ {0, 1} for all x, y, t ∈ [0, 1].

After recalling some preliminary notions and results in Sect. 2, in Sect. 3 we provide the results of our study. Finally, conclusions are given in Sect. 4.

In this section we recall some basic definitions and known facts on aggregation functions. In the second part we provide basic idea of the generalized-convexsum-transformation that was introduced in [9].

In this contribution we will deal with (n-ary) aggregation function on [0, 1]. For more details including definitions and discussion concerning examples and properties of aggregation functions we recommend [1, 2, 7, 10, 12] .

Some distinguished families of n-ary aggregation functions are given in the following definition.

). An n-ary aggregation function A is said to be (1) an n-ary semi-copula if e = 1 is its neutral element, (2) a t-norm if it is an associative and symmetric semi-copula, (3) dual to a semi-copula if e = 0 is a neutral element, (4) a t-conorm if it is associative and symmetric and dual to a semi-copula, (5) an n-ary quasi-copula if it is a 1-Lipschitz semi-copula, i.e.,

Definition 2 ( [12] ). An n-ary aggregation function C n : [0, 1] n → [0, 1] is said to be an n-ary copula if it is an n-ary semi-copula which is n-increasing, i.e., if for all x (0) ∈ [0, 1] n and x (1) 

Lemma 1 ( [12] ). Every copula C is 1-Lipschitz.

Proposition 1 ( [12] ). Let C be a binary copula (or a quasi-copula). Then for every (x, y) ∈ [0, 1] 2

Let us remark that the functions related to the lower and upper bound occurring in inequality (2) are denoted by

and are called the lower and upper Fréchet-Hoeffding bounds, respectively. In the theory of t-norms the function W is usually denoted by T L and M is denoted by T M and are called the Łukasiewicz and minimum t-norm, respectively.

In this section we briefly recall some definitions and results from [9] explaining the construction method using transformation of a given aggregation function A by an n-tuple of monotone systems of functions F n . The notion of a monotone system of functions is crucial in the construction method in question. 1] be a family of functions such that

Then F is called a Monotone System of Functions, MSF for brevity. 1] are the least and the greatest Monotone Systems of Functions, respectively. 1] , i = 1, 2, . . . , n, be MSF (Definition 3).

is called an n-tuple of the Monotone Systems of Functions, n-MSF for brevity.

where f

xn ∈ F (n) . Then A Fn is an aggregation function.

Definition 5. An n-ary aggregation function A Fn defined by formula (4), where F n = (F (1) , . . . , F (n) ) is an n-tuple of monotone systems of functions, is said to be a Generalized-Convex-Sum-Transform of A by F n , or a GCS-transform in short. 1] , i = 1, 2, . . . , n, be an SMSF.

is called an n-tuple of the Standard Monotone Systems of Functions, or n-SMSF for brevity. Remark 1. It follows directly from Definition 3 and 6 that the set containing all the MSF and SMSF is convex.

Denote

Then for an arbitrary n ≥ 2, an arbitrary n-ary aggregation function A and the n-tuple F n = (F, . . . , F) the following holds

and Ch mA is the Choquet integral (see [3, 6] ) with respect to the capacity m A . This fact shows that our construction method can be seen as a significant extension of the Choquet integrals.

In the rest of the paper we will consider only standard monotone systems of

and t ∈ [0, 1]. We will call them 0-1-valued monotone systems of functions,

Let A be a binary aggregation function. Then, obviously

In other words, knowing C F we know the result of the GCS-transform A F of A.

We will focus our attention only to binary aggregation functions and their GCS-transforms. 1] be arbitrary 0-1-SMSF and F = (F, G) . Then, C F defined by formula (6) is a copula.

Proof. Directly by Definitions 3 and 6 we have that C F is a semi-copula. Let us prove the two-increasingness. Assume x 1 ≥ x 2 and y 1 ≥ y 2 be arbitrary elements of [0, 1]. Then for all t ∈ [0, 1] the following holds

These imply

Formulae (8) and (9) imply

and by inequalities (7), 1] be arbitrary 0-1-SMSF and F = (F, G). Then we denote C F the copula given by Eq. (6) and we say that C F is generated by F.

The following example illustrates some 0-1-SMSF and the copulas they generate. 1] where f x are given by formula (5) . Further, 1] , 1] , where i g x are given by formulae (10), (11), (12), (13) and (14), respectively: The copula sketched in the right part of Fig. 1 is the so-called tent copula (see, e.g., [4] ).

The following lemma is straightforward. 

Proposition 3. LetF be the 0-1-SMSF defined by formula (5) . There exists a pair of 0-1-SMSF G = (G (1) , G (2) ) and the copula C G generated by G such that

Then

There are three possibilities for the result of C F (x, y). 

3. There exists a countable (finite or infinite) system of pairwise disjoint intervals

and f y (t) = 0 almost everywhere for t ∈ [0, x] \ i∈I . There are two possibilities:

-there exists j ∈ I such that b j > x, then

This gives the following -setting G (1) = G 1 and G (2) = G 5 , where G 1 and G 5 are defined by formulae (10) and (14), respectively, and G = (G (1) , G (2) ) (see also Fig. 5) , then the copula C G cannot be generated by any pair of 0-1-SMSF F = (F, F).

Analysing the proof of Proposition 3 we see that if a copula has a component that is non-linear in the first variable, then it cannot be generated by a pair of 0-1-SMSF (F, F) whereF is the 0-1-SMSF defined by formula (5) .

Another consequence that can be derived is that the copula C G sketched on Fig. 5 cannot be generated by a pair of 0-1-SMSF (F, F) nor by (F,F).

A characterization by the second mixed partial derivatives of copulas generated by pairs of 0-1-SMSF is contained in the following proposition.

Proposition 4. Let F be a pair of 0-1-SMSF and C F the copula generated by F. Then

for all (x, y) ∈ [0, 1] 2 where the second mixed partial derivative exists.

Durante et al. [5] have shown that if formula (15) holds for a copula C F then C F still may have a density.

Finally, we show how we can construct an arbitrary shuffle of M by a pair of 0-1-SMSF. The family of all shuffles of min is very important, since, as it is proven in [11] , this family is dense in the system of all bivariate copulas.

A geometrical visualisation of a shuffle of min is quite straightforward. We choose a natural number n > 1, a system of nods 0 = a 0 < a 1 < · · · < a n = 1 and cut the minimum copula parallel to the y-axis into n strips using those nods. Then we shuffle the strips (this means we choose a permutation Π : 1] is given by formula (16) and the shuffle copula is then generated by the pair of 0-1-SMSF F 6 = (F, G 6 ) and displayed in Fig. 6 . Fig. 6 . Left the layout of G6, right the copula C F (6) where

The explicit formula for 0-1-SMSF G 6 = { 6 g x } x∈[0,1] is given by 

Shuffles of min are in [11] described in a more general way than we have illustrated by Example 4. Namely, the shuffles can be combined with flips (a flip of the minimum copula M is W ). This combination is sketched in Fig. 3 . The following proposition gives a characterization of all shuffles (possibly combined with flips) of min as special cases of copulas generated by a pair of 0-1-SMSF. In this case a shuffle of min is given by n, 

where, for i ∈ {1, 2, . . . , n},

This paper contributes to a study of Generalized-Convex-Sum-Transformation of (binary) aggregation functions. Particularly, we have studied copulas that can be generated by pairs of 0-1-SMSF (see formula (6)). Though, the 0-1-SMSFF given by formula (5) is, in a sense, a basic 0-1-SMSF, when we like to generate all possible copulas by pairs of 0-1-SMSF, it is not sufficient to consider only those pairs where one of the 0-1-SMSF isF. We have also shown that every shuffle of the minimum copula (possibly combined with flips) can be generated by a pair of 0-1-SMSF and in Proposition 5 we have written down an explicit formula for such 0-1-SMSF.

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Acknowledgements. The work on this paper has been supported from the Science and Technology Assistance Agency under the contract no. APVV-18-0052, and by the Slovak Scientific Grant Agency VEGA no. 1/0006/19 and 2/0142/20.