Directions of directional, ordered directional and strengthened ordered directional increasingness of linear and ordered linear fusion operators

In this work we discuss the forms of monotonicity that have been recently introduced to relax the monotonicity condition in the definition of aggregation functions. We focus on directional, ordered directional and strengthened ordered directional monotonicity, study their main properties and provide some results about their links and relations among them. We also present two families of functions, the so-called linear fusion functions and ordered linear fusion functions and we study the set of directions for which these types of functions are directionally, ordered directionally and strengthened ordered directionally increasing. In particular, OWA operators are an example of ordered linear fusion functions.This work is partially supported by the Research Service of Universidad Pública de Navarra and the grants APVV-14-0013 and TIN2016-77356-P (AEI/ FEDER, UE)

Bounded Generalized Mixture Functions

In literature, it is common to find problems which require a way to encode a finite set of information into a single data; usually means are used for that. An important generalization of means are the so called Aggregation Functions, with a noteworthy subclass called OWA functions. There are, however, further functions which are able to provide such codification which do not satisfy the definition of aggregation functions; this is the case of pre-aggregation and mixture functions. In this paper we investigate two special types of functions: Generalized Mixture and Bounded Generalized Mixture functions. They generalize both: OWA and Mixture functions. Both Generalized and Bounded Generalized Mixture functions are developed in such way that the weight vectors are variables depending on the input vector. A special generalized mixture operator, H, is provided and applied in a simple toy example

Generalized forms of monotonicity in the data aggregation framework

El proceso de agregación trata el problema de combinar una colección de valores numéricos en un único valor que los represente y las funciones encargadas de esta operación se denominan funciones de agregación. A las funciones de agregación se les exige que cumplan dos condiciones de contorno y, además, han de ser monótonas con respecto a todos sus argumentos. Una de las tendencias en el área de investigación de las funciones de agregación es la relajación de la condición de monotonía. En este respecto, se han introducido varias formas de monotonía relajada. Tal es el caso de la monotonía débil, la monotonía direccional y la monotonía respecto a un cono. Sin embargo, todas estas relajaciones de monotonía están basadas en la idea de crecer, o decrecer, a lo largo de un rayo definido por un vector real. No existe noción de monotonía que permita que la dirección de crecimiento dependa de los valores a fusionar, ni tampoco existe noción de monotonía que considere el crecimiento a lo largo de caminos más generales, como son las curvas. Además, otra de las tendencias en la teoría de la agregación es la extensión a escalas más generales que la de los números reales y no existe relajación de monotonía disponible para este contexto general. En esta tesis, proponemos una colección de nuevas formas de monotonía relajada para las cuales las direcciones de monotonía pueden variar dependiendo del punto del dominio. En concreto, introducimos los conceptos de monotonía direccional ordenada, monotonía direccional ordenada reforzada y monotonía direccional punto a punto. Basándonos en funciones que cumplan las propiedades de monotonía direccional ordenada, proponemos un algoritmo de detección de bordes que justifica la aplicabilidad de estos conceptos. Por otro lado, generalizamos el concepto de monotonía direccional tomando, en lugar de direcciones en Rn, caminos más generales: definimos el concepto de monotonía basado en curvas. Por último, combinando ambas tendencias en la teoría de la agregación, generalizamos el concepto de monotonía direccional a funciones definidas en escalas más generales que la de los números reales.The process of aggregation addresses the problem of combining a collection of numerical values into a single representative number and the functions that perform this operation are called aggregation functions. Aggregation functions are required to satisfy two boundary conditions and to be monotone with respect to all their arguments. One of the trends in the research area of aggregation functions is the relaxation of the condition of monotonicity. In that attempt, various relaxed forms of monotonicity have been introduced. This is the case of weak, directional and cone monotonicity. However, all these relaxed forms of monotonicity are based on the idea of increasing, or decreasing, along a fixed ray defined by a real vector. There is no notion of monotonicity allowing the direction of increasingness to depend on the specific values to aggregate, nor there exists any other notion that considers increasingness along more general paths, such as curves. Additionally, another trend in the theory of aggregation is the extension to handle more general scales than real numbers and there is no relaxation of monotonicity available in that general context. In this dissertation, we propose a collection of new relaxed forms of monotonicity for which the directions of monotonicity may vary from one point of the domain to another. Specifically, we introduce the concepts of ordered directional monotonicity, strengthened ordered directional monotonicity and pointwise directional monotonicity. Based on the concept of ordered directionally monotone functions, we propose an edge detection algorithm that justifies the applicability of these concepts. Furthermore, we generalize the concept of directional monotonicity so that, instead of directions in Rn, more general paths are considered: we define curve-based monotonicity. Finally, combining both trends in the theory of aggregation, we generalize the concept of directional monotonicity to functions that are defined on more general scales than real numbers.Programa de Doctorado en Ciencias y Tecnologías Industriales (RD 99/2011)Industria Zientzietako eta Teknologietako Doktoretza Programa (ED 99/2011

Strengthened ordered directionally monotone functions. Links between the different notions of monotonicity

In this work, we propose a new notion of monotonicity: strengthened ordered directional monotonicity. This generalization of monotonicity is based on directional monotonicity and ordered directional monotonicity, two recent weaker forms of monotonicity. We discuss the relation between those different notions of monotonicity from a theoretical point of view. Additionally, along with the introduction of two families of functions and a study of their connection to the considered monotonicity notions, we define an operation between functions that generalizes the Choquet integral and the Lukasiewicz implication.This work is partially supported by the grants APVV-14-0013 and TIN2016-77356-P (AEI/FEDER, UE)

Curve-based monotonicity: a generalization of directional monotonicity

In this work we propose a generalization of the notion of directional monotonicity. Instead of considering increasingness or decreasingness along rays, we allow more general paths defined by curves in the n-dimensional space. These considerations lead us to the notion of α-monotonicity, where α is the corresponding curve. We study several theoretical properties of α-monotonicity and relate it to other notions of monotonicity, such as weak monotonicity and directional monotonicity.This work is supported by the research group FQM268 of Junta de Andalucía, by the projects TIN2017-89517-P and TIN2016-77356-P (AEI/FEDER, UE), by the the Slovak Scientic Grant Agency VEGA no. 1/0093/17 Identication of risk factors and their impact on products of the insurance and savings schemes, by Slovak grant APVV- 14-0013, and by Czech Project LQ1602 \IT4Innovations excellence in science

On some classes of directionally monotone functions

In this work we consider some classes of functions with relaxed monotonicity conditions generalizing some other given classes of fusion functions. In particular, directionally increasing aggregation functions (called also pre-aggregation functions), directionally increasing conjunctors, or directionally increasing implications, etc., generalize the standard classes of aggregation functions, conjunctors, or implication functions, respectively. We analyze different properties of these classes of functions and we discuss a construction method in terms of linear combinations of t-norms.H. Bustince and J. Fernandez were supported by research project TIN2016-77356-P of the Spanish Government. G. Dimuro was supported by CNPq/Brazil Proc.305882/2016-3. R. Mesiar was supported by the project APVV14-0013 and by the project of Grant Agency of the Czech Republic (GACR) no. 18-06915S. A. Kolesárová was supported by the project VEGA 1/0614/18. I. Díaz was supported by research project TIN2017-87600-P of the Spanish Government. S. Montes was supported by research project TIN2014-59543-P of the Spanish Government

Strengthened ordered directional and other generalizations of monotonicity for aggregation functions

Trabajo presentado a la 17th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2018. Cádiz, junio de 2018A tendency in the theory of aggregation functions is the generalization of the monotonicity condition. In this work, we examine the latest developments in terms of different generalizations. In particular, we discuss strengthened ordered directional monotonicity, its relation to other types of monotonicity, such as directional and ordered directional monotonicity and the main properties of the class of functions that are strengthened ordered directionally monotone. We also study some construction methods for such functions and provide a characterization of usual monotonicity in terms of these notions of monotonicity.This work is partially supported by the Research Service of Universidad Pública de Navarra and the grants APVV-14-0013 and TIN2016-77356-P (AEI/FEDER, UE)

Local properties of strengthened ordered directional and other forms of monotonicity

In this study we discuss some of the recent generalized forms of monotonicity, introduced in the attempt of relaxing the monotonicity condition of aggregation functions. Specifically, we deal with weak, directional, ordered directional and strengthened ordered directional monotonicity. We present some of the most relevant properties of the functions that satisfy each of these monotonicity conditions and, using the concept of pointwise directional monotonicity, we carry out a local study of the discussed relaxations of monotonicity. This local study enables to highlight the differences between each notion of monotonicity. We illustrate such differences with an example of a restricted equivalence function.This work is supported by the project TIN2016-77356-P (AEI/FEDER, UE), by the Public University of Navarra under the project PJUPNA13 and by Slovak grant APVV-14-0013

Description and properties of curve-based monotone functions

Curve-based monotonicity is one of the lately introduced relaxations of monotonicity. As directional monotonicity regards monotonicity along fixed rays, which are given by real vectors, curve-based monotonicity studies the increase of functions with respect to a general curve. In this work we study some theoretical properties of this type of monotonicity and we relate this concept with previous relaxations of monotonicity.This work is supported by the research group FQM268 of Junta de Andalucía, by the project TIN2016-77356-P (AEI/FEDER, UE), by the Slovak Scientific Grant Agency VEGA no. 1/0093/17 Identification of risk factors and their impact on products of the insurance and savings schemes, by Slovak grant APVV-14-0013, and by Czech Project LQ1602 'IT4Innovations excellence in science'

Ordered directional monotonicity in the construction of edge detectors

In this paper we provide a specific construction method of ordered directionally monotone functions. We show that the functions obtained with this construction method can be used to build edge detectors for grayscale images. We compare the results of these detectors to those obtained with some other ones that are widely used in the literature. Finally, we show how a consensus edge detector can be built improving the results obtained both by our proposal and by those in the literature when applied individually.This work was supported by the Slovak Research and Development Agency through grant APVV-18-0052 and the Grant Agency of the Czech Republic, through grant GACR 1806915S, and by the Spanish Government through project PID2019-108392GB-I00 (AEI/FEDER, UE)