Aggregation on bipolar scales

The paper addresses the problem of extending aggregation operators typically defined on $[0,1]$ to the symmetric interval $[-1,1]$, where the ``0'' value plays a particular role (neutral value). We distinguish the cases where aggregation operators are associative or not. In the former case, the ``0'' value may play the role of neutral or absorbant element, leading to pseudo-addition and pseudo-multiplication. We address also in this category the special case of minimum and maximum defined on some finite ordinal scale. In the latter case, we find that a general class of extended operators can be defined using an interpolation approach, supposing the value of the aggregation to be known for ternary vectors.bipolar scale; bi-capacity; aggregation

Contribution on some construction methods for aggregation functions

In this paper, based on [14], we present some well established construction methods for aggregation functions as well as some new ones

Bipolar rating scales: A survey and novel correlation measures based on nonlinear bipolar scoring functions

© 2017, Budapest Tech Polytechnical Institution. All rights reserved. A bipolar rating scale is a linearly ordered set with symmetry between elements considered as negative and positive categories. First, we present a survey of bipolar rating scales used in psychology, sociology, medicine, recommender systems, opinion mining, and sentiment analysis. We discuss different particular cases of bipolar scales and, in particular, typical structures of bipolar scales with verbal labels that can be used for construction of bipolar rating scales. Next, we introduce the concept of bipolar scoring function preserving linear ordering and the symmetry of bipolar scales, study its properties, and propose methods for construction of bipolar scoring functions. We show that Pearson’s correlation coefficient often used for analysis of relationship between profiles of ratings in recommender systems can be misleading if the rating scales are bipolar. Basing on the general methods of construction of association measures, we propose new correlation measures on bipolar scales free from the drawbacks of Pearson’s correlation coefficient. Our correlation measures can be used in recommender systems, sentiment analysis and opinion mining for analysis of possible relationship between opinions of users and their ratings of items

On the poset of computation rules for nonassociative calculus

The symmetric maximum, denoted by v, is an extension of the usual max operation so that 0 is the neutral element, and -x is the symmetric (or inverse) of x, i.e., x v(-x)=0. However, such an extension does not preserve the associativity of max. This fact asks for systematic ways of parenthesing (or bracketing) terms of a sequence (with more than two arguments) when using such an extended maximum. We refer to such systematic (predefined) ways of parenthesing as computation rules. As it turns out there are infinitely many computation rules each of which corresponding to a systematic way of bracketing arguments of sequences. Essentially, computation rules reduce to deleting terms of sequences based on the condition x v(-x)=0. This observation gives raise to a quasi-order on the set of such computation rules: say that rule 1 is below rule 2 if for all sequences of numbers, rule 1 deletes more terms in the sequence than rule 2. In this paper we present a study of this quasi-ordering of computation rules. In particular, we show that the induced poset of all equivalence classes of computation rules is uncountably infinite, has infinitely many maximal elements, has infinitely many atoms, and it embeds the powerset of natural numbers ordered by inclusion

Aggregation on bipolar scales

The paper addresses the problem of extending aggregation operators typically defined on $[0,1]$ to the symmetric interval $[-1,1]$, where the ``0'' value plays a particular role (neutral value). We distinguish the cases where aggregation operators are associative or not. In the former case, the ``0'' value may play the role of neutral or absorbant element, leading to pseudo-addition and pseudo-multiplication. We address also in this category the special case of minimum and maximum defined on some finite ordinal scale. In the latter case, we find that a general class of extended operators can be defined using an interpolation approach, supposing the value of the aggregation to be known for ternary vectors