Computing a T-transitive lower approximation or opening of a proximity relation

Fuzzy Sets and Systems. IMPACT FACTOR: 1,181. Fuzzy Sets and Systems. IMPACT FACTOR: 1,181. Since transitivity is quite often violated even by decision makers that accept transitivity in their preferences as a condition for consistency, a standard approach to deal with intransitive preference elicitations is the search for a close enough transitive preference relation, assuming that such a violation is mainly due to decision maker estimation errors. In some way, the more number of elicitations, the more probable inconsistency is. This is mostly the case within a fuzzy framework, even when the number of alternatives or object to be classified is relatively small. In this paper we propose a fast method to compute a T-indistinguishability from a reflexive and symmetric fuzzy relation, being T any left-continuous t-norm. The computed approximation we propose will take O(n3) time complexity, where n is the number of elements under consideration, and is expected to produce a T-transitive opening. To the authors¿ knowledge, there are no other proposed algorithm that computes T-transitive lower approximations or openings while preserving the reflexivity and symmetry properties

An algorithm to compute the transitive closure, a transitive approximation and a transitive opening of a fuzzy proximity

A method to compute the transitive closure, a transitive opening and a transitive approximation of a reflexive and symmetric fuzzy relation is given. Other previous methods in literature compute just the transitive closure, some transitive approximations or some transitive openings. The proposed algorithm computes the three different similarities that approximate a proximity for the computational cost of computing just one. The shape of the binary partition tree for the three output similarities are the same.Peer ReviewedPostprint (published version

An algorithm to compute the transitive closure, a transitive approximation and a transitive opening of a proximity

A method to get the transitive closure, a transitive opening and a transitive approximation of a reflexive and symmetric fuzzy relation is presented. The method builds at the same time a binary partition tree for the output similarities.Peer ReviewedPreprin

Approximating proximities by similarities

In this paper an algorithm to find a similarity close to a proximity or tolerance relation R is given. The obtained similarity is closer to R than its transitive closure or any transitive opening of R.Peer ReviewedPostprint (published version

Dynamics in the European Air Transport Network, 2003-9 : an explanatory framework drawing on stochastic actor-based modeling

In this paper, we outline and test an explanatory framework drawing on stochastic actor-based modeling to understand changes in the outline of European air transport networks between 2003 and 2009. Stochastic actor-based models show their capabilities to estimate and test the effect of exogenous and endogenous drivers on network changes in this application to the air transport network. Our results reveal that endogenous structural effects, such as transitivity triads, indirect relations and betweenness effects impact the development of the European air transport network in the period under investigation. In addition, exogenous nodal and dyadic covariates also play a role, with above all the enlargement of the European Common Aviation Area having benefitted its new members to open more air routes between them. The emergence of major low-cost airline-focused airports also significantly contributed to these changes. We conclude by outlining some avenues for further research

Neuronal modulation in the prefrontal cortex in a transitive inference task: evidence of neuronal correlates of mental schema management

When informed that A > B and B > C, humans and other animals can easily conclude that A > C. This remarkable trait of advanced animals, which allows them to manipulate knowledge flexibly to infer logical relations, has only recently garnered interest in mainstream neuroscience. How the brain controls these logical processes remains an unanswered question that has been merely superficially addressed in neuroimaging and lesion studies, which are unable to identify the underlying neuronal computations. We observed that the activation pattern of neurons in the prefrontal cortex (PFC) during pair comparisons in a highly demanding transitive inference task fully supports the behavioral performance of the two monkeys that we tested. Our results indicate that the PFC contributes to the construction and use of a mental schema to represent premises. This evidence provides a novel framework for understanding the function of various areas of brain in logic processes and impairments to them in degenerative, traumatic, and psychiatric pathologies. SIGNIFICANCE STATEMENT: In cognitive neuroscience, it is unknown how information that leads to inferential deductions are encoded and manipulated at the neuronal level. We addressed this question by recording single-unit activity from the dorsolateral prefrontal cortex of monkeys that were performing a transitive inference (TI) task. The TI required one to choose the higher ranked of two items, based on previous, indirect experience. Our results demonstrated that single-neuron activity supports the construction of an abstract, mental schema of ordered items in solving the task and that this representation is independent of the reward value that is experienced for the single items. These findings identify the neural substrates of abstract mental representations that support inferential thinking

Approximation of proximities by aggregating T-indistinguishability operators

For a continuous Archimedean t-norm T a method to approximate a proximity relation R (i.e. a reflexive and symmetric fuzzy relation) by a T-transitive one is provided. It consists of aggregating the transitive closure R of R with a (maximal) T-transitive relation B contained in R using a suitable weighted quasi-arithmetic mean to maximize the similarity or minimize the distance to R.Peer Reviewe

Fifty years of similarity relations: a survey of foundations and applications

On the occasion of the 50th anniversary of the publication of Zadeh's significant paper Similarity Relations and Fuzzy Orderings, an account of the development of similarity relations during this time will be given. Moreover, the main topics related to these fuzzy relations will be reviewed.Peer ReviewedPostprint (author's final draft

Finding close T-indistinguishability operators to a given proximity

Two ways to approximate a proximity relation R (i.e. a reflexive and symmetric fuzzy relation) by a T-transitive one where T is a continuous archimedean t-norm are given. The first one aggregates the transitive closure R of R with a (maximal) T-transitive relation B contained in R. The second one modifies the values of R or B to better fit them with the ones of R.Peer ReviewedPostprint (published version