On fuzzy-qualitative descriptions and entropy

This paper models the assessments of a group of experts when evaluating different magnitudes, features or objects by using linguistic descriptions. A new general representation of linguistic descriptions is provided by unifying ordinal and fuzzy perspectives. Fuzzy qualitative labels are proposed as a generalization of the concept of qualitative labels over a well-ordered set. A lattice structure is established in the set of fuzzy-qualitative labels to enable the introduction of fuzzy-qualitative descriptions as L-fuzzy sets. A theorem is given that characterizes finite fuzzy partitions using fuzzy-qualitative labels, the cores and supports of which are qualitative labels. This theorem leads to a mathematical justification for commonly-used fuzzy partitions of real intervals via trapezoidal fuzzy sets. The information of a fuzzy-qualitative label is defined using a measure of specificity, in order to introduce the entropy of fuzzy-qualitative descriptions. (C) 2016 Elsevier Inc. All rights reserved.Peer ReviewedPostprint (author's final draft

Self-organization and evolution on large computer data structures

We study the long time evolution of a large data structure while inserting new items. It is implemented using a well known computer science approach based on 2-3 trees. We have seen self-organized critical behavior on this data structure. To tackle this problem we have introduced and studied experimentally three statistical magnitudes: the stress of a tree, the sequence of jump points and the distribution of subtrees inside a tree. The stress measures the amount of free space inside the 2-3 tree. When the stress increases some part of the tree is restructured in a way close to an avalanche. Experimentally we obtain a potential law for stress distribution. When the tree does not have more free space in any internal node, needs to grow up. When this happens, the height of the whole tree increases by one and we have a jump point. Experimentally these points have good expected behavior.A 2-3 tree is composed from a great number of other 2-3 trees called their subtrees. We have studied experimentally the distribution of the different subtrees inside the tree. Finally we analyze these results using simple theoretical models based on fringe analysis, Markov and branching processes. These models give us a quite good description of the long term process.Postprint (published version

Using qualitative reasoning in modelling consensus in group decision-making

Ordinal scales are commonly used in rating and evaluation processes. These processes usually involve group decision making by means of an experts’ committee. In this paper a mathematical framework based on the qualitative model of the absolute orders of magnitude is considered. The entropy of a qualitatively described system is defined in this framework. On the one hand, this enables us to measure the amount of information provided by each evaluator and, on the other hand, the coherence of the evaluation committee. The new approach is capable of managing situations where the assessment given by experts involves different levels of precision. The use of the proposed measures within an automatic system for group decision making will contribute towards avoiding the potential subjectivity caused by conflicts of interests of the evaluators in the group.Postprint (published version

A recommender system based on group consensus

This paper presents the foundation for a new methodology for a collaborative recommender system (RS). This methodology is based on the degree of consensus of a group of users stating their preferences via qualitative orders-of-magnitude. The structure of distributive lattice is considered in defining the distance between users and the RSs new users. This proposed methodology incorporates incomplete or partial knowledge into the recommendation process using qualitative reasoning techniques to obtain consensus of its users for recommendations.Postprint (published version

Self-organization and evolution on large computer data structures

We study the long time evolution of a large data structure while inserting new items. It is implemented using a well known computer science approach based on 2-3 trees. We have seen self-organized critical behavior on this data structure. To tackle this problem we have introduced and studied experimentally three statistical magnitudes: the stress of a tree, the sequence of jump points and the distribution of subtrees inside a tree. The stress measures the amount of free space inside the 2-3 tree. When the stress increases some part of the tree is restructured in a way close to an avalanche. Experimentally we obtain a potential law for stress distribution. When the tree does not have more free space in any internal node, needs to grow up. When this happens, the height of the whole tree increases by one and we have a jump point. Experimentally these points have good expected behavior.A 2-3 tree is composed from a great number of other 2-3 trees called their subtrees. We have studied experimentally the distribution of the different subtrees inside the tree. Finally we analyze these results using simple theoretical models based on fringe analysis, Markov and branching processes. These models give us a quite good description of the long term process

Self-organization and evolution on large computer data structures

We study the long time evolution of a large data structure while inserting new items. It is implemented using a well known computer science approach based on 2-3 trees. We have seen self-organized critical behavior on this data structure. To tackle this problem we have introduced and studied experimentally three statistical magnitudes: the stress of a tree, the sequence of jump points and the distribution of subtrees inside a tree. The stress measures the amount of free space inside the 2-3 tree. When the stress increases some part of the tree is restructured in a way close to an avalanche. Experimentally we obtain a potential law for stress distribution. When the tree does not have more free space in any internal node, needs to grow up. When this happens, the height of the whole tree increases by one and we have a jump point. Experimentally these points have good expected behavior.A 2-3 tree is composed from a great number of other 2-3 trees called their subtrees. We have studied experimentally the distribution of the different subtrees inside the tree. Finally we analyze these results using simple theoretical models based on fringe analysis, Markov and branching processes. These models give us a quite good description of the long term process

Measuring the Consensus in Group Decision by means of Qualitative Reasoning

This paper presents a mathematical framework to assess the consensus found among different evaluators who use ordinal scales in group decision-making and evaluation processes. This framework is developed on the basis of the absolute order-of-magnitude qualitative model through the use of qualitative entropy. As such, we study the algebraic structure induced in the set of qualitative descriptions given by evaluators. Our results demonstrate a weak, partial and semi lattice structure that in some conditions takes the form of a distributive lattice. We then define the entropy of a qualitatively-described system. This enables us, on the one hand, to measure the amount of information provided by each evaluator and, on the other hand, to consider a degree of consensus among the evaluation committee. This new approach is capable of managing situations where the assessment given by experts involves different levels of precision. In addition, when there is no consensus regarding the group decision, an automatic process assesses the effort required to achieve said consensus

Measuring the Consensus in Group Decision by means of Qualitative Reasoning

This paper presents a mathematical framework to assess the consensus found among different evaluators who use ordinal scales in group decision-making and evaluation processes. This framework is developed on the basis of the absolute order-of-magnitude qualitative model through the use of qualitative entropy. As such, we study the algebraic structure induced in the set of qualitative descriptions given by evaluators. Our results demonstrate a weak, partial and semi lattice structure that in some conditions takes the form of a distributive lattice. We then define the entropy of a qualitatively-described system. This enables us, on the one hand, to measure the amount of information provided by each evaluator and, on the other hand, to consider a degree of consensus among the evaluation committee. This new approach is capable of managing situations where the assessment given by experts involves different levels of precision. In addition, when there is no consensus regarding the group decision, an automatic process assesses the effort required to achieve said consensus.Postprint (published version

Using qualitative reasoning in modelling consensus in group decision-making

Ordinal scales are commonly used in rating and evaluation processes. These processes usually involve group decision making by means of an experts’ committee. In this paper a mathematical framework based on the qualitative model of the absolute orders of magnitude is considered. The entropy of a qualitatively described system is defined in this framework. On the one hand, this enables us to measure the amount of information provided by each evaluator and, on the other hand, the coherence of the evaluation committee. The new approach is capable of managing situations where the assessment given by experts involves different levels of precision. The use of the proposed measures within an automatic system for group decision making will contribute towards avoiding the potential subjectivity caused by conflicts of interests of the evaluators in the group

Measuring the Consensus in Group Decision by means of Qualitative Reasoning

This paper presents a mathematical framework to assess the consensus found among different evaluators who use ordinal scales in group decision-making and evaluation processes. This framework is developed on the basis of the absolute order-of-magnitude qualitative model through the use of qualitative entropy. As such, we study the algebraic structure induced in the set of qualitative descriptions given by evaluators. Our results demonstrate a weak, partial and semi lattice structure that in some conditions takes the form of a distributive lattice. We then define the entropy of a qualitatively-described system. This enables us, on the one hand, to measure the amount of information provided by each evaluator and, on the other hand, to consider a degree of consensus among the evaluation committee. This new approach is capable of managing situations where the assessment given by experts involves different levels of precision. In addition, when there is no consensus regarding the group decision, an automatic process assesses the effort required to achieve said consensus